diff --git a/-NFRT4oBgHgl3EQfrTfI/content/tmp_files/2301.13620v1.pdf.txt b/-NFRT4oBgHgl3EQfrTfI/content/tmp_files/2301.13620v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fcbdba7f7782be8ad5c06fbc33e8ebf90e5f6a0f --- /dev/null +++ b/-NFRT4oBgHgl3EQfrTfI/content/tmp_files/2301.13620v1.pdf.txt @@ -0,0 +1,1801 @@ +A Maximum Principle for Optimal Control Problems involving +Sweeping Processes with a Nonsmooth Set +M. d. R. de Pinho, M. Margarida A. Ferreira ∗and +Georgi Smirnov † +February 1, 2023 +Abstract +We generalize a Maximum Principle for optimal control problems involving sweeping systems +previously derived in [14] to cover the case where the moving set may be nonsmooth. Noteworthy, +we consider problems with constrained end point. A remarkable feature of our work is that we rely +upon an ingenious smooth approximating family of standard differential equations in the vein of that +used in [10]. +Keywords: Sweeping Process Optimal Control, Maximum Principle, Approximations +1 +Introduction +In recent years, there has been a surge of interest in optimal control problems involving the controlled +sweeping process of the form +˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), u(t) ∈ U, +x(0) ∈ C0. +(1.1) +In this respect, we refer to, for example, [3], [4], [5], [8], [9], [16], [23], [10] (see also accompanying correction +[11]), [6], [15] and [14]. Sweeping processes first appeared in the seminal paper [18] by J.J. Moreau as a +mathematical framework for problems in plasticity and friction theory. They have proved of interest to +tackle problems in mechanics, engineering, economics and crowd motion problems; to name but a few, +see [1], [5], [16], [17] and [21]. In the last decades, systems in the form (1.1) have caught the attention and +interest of the optimal control community. Such interest resides not only in the range of applications but +also in the remarkable challenge they rise concerning the derivation of necessary conditions. This is due +to the presence of the normal cone NC(t)(x(t)) in the dynamics. Indeed, the presence of the normal cone +renders the discontinuity of the right hand of the differential inclusion in (1.1) destroying a regularity +property central to many known optimal control results. +Lately, there has been several successful attempts to derive necessary conditions for optimal control +problems involving (1.1). Assuming that the set C is time independent, necessary conditions for optimal +control problems with free end point have been derived under different assumptions and using different +techniques. In [10], the set C has the form C = {x : +ψ(x) ≤ 0} and an approximating sequence of +optimal control problems, where (1.1) is approximated by the differential equation +˙xγk(t) = f(t, xγk(t), u(t)) − γkeγkψ(xγk (t))∇ψ(xγk(t)), +(1.2) +for some positive sequence γk → +∞, is used. Similar techniques are also applied to somehow more +general problems in [23]. A useful feature of those approximations is explored in [12] to define numerial +schemes to solve such problems. +∗MdR de Pinho and MMA Ferreira are at Faculdade de Engenharia da Universidade do Porto, DEEC, SYSTEC. Portugal, +mrpinho, mmf@fe.up.pt +†G. Smirnov is at Universidade do Minho, Dep. Matem´atica, Physics Center of Minho and Porto Universities (CF-UM- +UP), Campus de Gualtar, Braga, Portugal, smirnov@math.uminho.pt +1 +arXiv:2301.13620v1 [math.OC] 31 Jan 2023 + +More recently, an adaptation of the family of approximating systems (1.2) is used in [14] to generalize +the results in [10] to cover problems with additional end point constraints and with a moving set of the +form C(t) = {x : ψ(t, x) ≤ 0}. +In this paper we generalize the Maximum Principle proved in [14] to cover problems with possibly +nonsmooth sets. Our problem of interest is +(P) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Minimize φ(x(T)) +over processes (x, u) such that +˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), a.e. t ∈ [0, T], +u(t) ∈ U, +a.e. t ∈ [0, T], +(x(0), x(T)) ∈ C0 × CT ⊂ C(0) × C(T), +where T > 0 is fixed, φ : Rn → R, f : [0, T] × Rn × Rm → Rn, U ⊂ Rm and +C(t) := +� +x ∈ Rn : ψi(t, x) ≤ 0, i = 1, . . . , I +� +(1.3) +for some functions ψi : [0, T] × Rn → R, i = 1, . . . , I. +The case where I = 1 in (1.3) and ψ1 is C2 is covered in [14]. Here, we assume I > 1 and that +the functions ψi are also C2. Although going from I = 1 in (1.3) to I > 1 may be seen as a small +generalization, it demands a significant revision of the technical approach and, plus, the introduction +of a constraint qualification. This is because the set (1.3) may be nonsmooth. We focus on sets (1.3), +satisfying a certain constraint qualification, introduced in assumption (A1) in section 2 below. This is, +indeed, a restriction on the nonsmoothness of (1.3). A similar problem with nonsmooth moving set is +considered in [15]. Our results cannot be obtained from the results of [15] and do not generalize them. +This paper is organized in the following way. In section 2, we introduce the main notation and we state +and discuss the assumptions under which we work. In this same section, we also introduce the family of +approximating systems to ˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)) and establish a crucial convergence result, +Theorem 2.2. In section 3, we dwell on the approximating family of optimal control problems to (P) +and we state the associated necessary conditions. The Maximum Principle for (P) is then deduced and +stated in Theorem 4.1, covering additionally, problems in the form of (P) where the end point constraint +x(T) ∈ CT is absent. Before finishing, we present an illustrative example of our main result, Theorem +4.1. +2 +Preliminaries +In this section, we introduce a summary of the notation and state the assumptions on the data of (P) +enforced throughout. Furthermore, we extract information from the assumptions establishing relations +crucial for the forthcoming analysis. +Notation +For a set S ⊂ Rn, ∂S, cl S and int S denote the boundary, closure and interior of S. +If g : Rp → Rq, ∇g represents the derivative and ∇2g the second derivative. If g : R × Rp → Rq, then +∇xg represents the derivative w.r.t. x ∈ Rp and ∇2 +xg the second derivative, while ∂tg(t, x) represents the +derivative w.r.t. t ∈ R. +The Euclidean norm or the induced matrix norm on Rp×q is denoted by |·|. We denote by Bn the +closed unit ball in Rn centered at the origin. The inner product of x and y is denoted by ⟨x, y⟩. For +some A ⊂ Rn, d(x, A) denotes the distance between x and A. We denote the support function of A at z +by S(z, A) = sup{⟨z, a⟩ | a ∈ A} +The space L∞([a, b]; Rp) (or simply L∞ when the domains are clearly understood) is the Lebesgue +space of essentially bounded functions h : [a, b] → Rp. We say that h ∈ BV ([a, b]; Rp) if h is a function +of bounded variation. The space of continuous functions is denoted by C([a, b]; Rp). +2 + +Standard concepts from nonsmooth analysis will also be used. Those can be found in [7], [19] or [22], +to name but a few. The Mordukhovich normal cone to a set S at s ∈ S is denoted by NS(s) and ∂f(s) +is the Mordukhovich subdifferential of f at s (also known as limiting subdifferential). +For any set A ⊂ Rn, cone A is the cone generated by the set A. +We now turn to problem (P). We first state the definition of admissible processes for (P) and then +we describe the assumptions under which we will derive our main results. +Definition 2.1 A pair (x, u) is called an admissible process for (P) when x is an absolutely continuous +function and u is a measurable function satisfying the constraints of (P). +Assumptions on the data of (P) +A1: The function ψi, i = 1, . . . , I, are C2. The graph of C(·) is compact and it is contained in the +interior of a ball rBn+1, for some r > 0. There exist constants β > 0, η > 0 and ρ ∈]0, 1[ such that +ψi(t, x) ∈ [−β, β] =⇒ |∇xψi(t, x)|> η forall (t, x) ∈ [0, T] × Rn, +(2.1) +and, for I(t, x) = {i = 1, . . . , I | ψi(t, x) ∈] − 2β, β]}, +⟨∇xψi(t, x), ∇xψj(t, x)⟩ ≥ 0, i, j ∈ I(t, x). +(2.2) +Moreover, if i ∈ I(t, x), then +� +j∈I(t,x)\{i} +|⟨∇xψi(t, x), ∇xψj(t, x)⟩| ≤ ρ|∇xψi(t, x)|2 +(2.3) +and +ψi(t, x) ≤ −2β =⇒ ∇ψi(t, x) = 0 for i = 1, . . . I. +(2.4) +A2: The function f is continuous, x → f(t, x, u) is continuously differentiable for all (t, u) ∈ [0, T]×Rm. +The constant M > 0 is such that |f(t, x, u)|≤ M and |∇xf(t, x, u)|≤ M for all (t, x, u) ∈ rBn+1×U. +A3: For each (t, x), the set f(t, x, U) is convex. +A4: The set U is compact. +A5: The sets C0 and CT are compact. +A6: There exists a constant Lφ such that |φ(x) − φ(x′)|≤ Lφ|x − x′| for all x, x′ ∈ Rn. +Assumption (A1) concerns the functions ψi defining the set C and it plays a crucial role in the analysis. +All ψi are assumed to be smooth with gradients bounded away from the origin when ψi takes values in +a neighorhood of zero. Moreover, the boundary of C may be nonsmooth at the intersection points of the +level sets +� +x : ψi(t, x) = 0 +� +. However, nonsmoothness at those corner points is restricted to (2.2) which +excludes the cases where the angle between the two gradients of the functions defining the boundary of +C is obtuse; see figure 1. +On the other hand, (2.3) guarantees that the Gramian matrix of the gradients of the functions +taking values near the boundary of C(t) is diagonally dominant and, hence, the gradients are linearly +independent. +In many situations, as in the example we present in the last section, we can guarantee the fulfillment +of (A1), in particular (2.4), replacing the function ψi by +˜ψi(t, x) = h ◦ ψi(t, x), +(2.5) +3 + +c +Y1=0 +Y2=0 +▽ Y2 +Not allowed +Allowed +▽ Y1 +C +Y2=0 +▽ Y2 +▽ Y1 +Y1=0 +Figure 1: Examples of two diferent sets C. On the left size, a set that does not satisfies (2.2). On the +right side, the set C is nonsmooth and it fulfils (2.2). +where +h(z) = +� +� +� +z +if +z > −β, +hs(z) +if +−2β ≤ z ≤ −β, +−2β +if +z < −2β, +Here, h is an C2 function, with hs an increasing function defined on [−2β, −β]. For example, h may be +a cubic polinomial with positive derivative on the interval ] − 2β, −β[. For all t ∈ [0, T], set +˜C(t) := +� +x ∈ R : +˜ψi(t, x) ≤ 0, i = 1, . . . , I +� +. +It is then a simple matter to see that +C(t) = ˜C(t) for all t ∈ [0, T]. +and that the functions ˜ψi(·) satisfy the assumption (A1). +The assumption that the graph of C(·) is compact and contained in the interior of a ball is introduced +to avoid technicalities in our forthcoming analysis. In applied problems, this may be easily side tracked +by considering the intersection of the graph of C(·) with a tube around the optimal trajectory. +We now proceed introducing an approximation family of controlled systems to (1.1). Let x(·) be a +solution to the differential inclusion +˙x(t) ∈ f(t, x(t), U) − NC(t)(x(t)). +Under our assumptions, measurable selection theorems assert the existence of measurable functions u +and ξi such that u(t) ∈ U, ξi(t) ≥ 0 a.e. t ∈ [0, T], ξi(t) = 0 if ψi(t, x(t)) < 0, and +˙x(t) = f(t, x(t), u(t)) − +I +� +i=1 +ξi(t)∇xψi(t, x(t)) a.e. t ∈ [0, T]. +Considering the trajectory x, some observations are called for. Let µ be such that +max +� +(|∇xψi(t, x)||f(t, x, u)|+|∂tψi(t, x)|) + 1 : +t ∈ [0, T], u ∈ U, x ∈ C(t) + Bn, i = 1, . . . , I} ≤ µ. +The properties of the graph of C(·) in (A1) guarantee the existence of such maximum. +4 + +Consider now some t such that, for some j ∈ {1, . . . I}, ψj(t, x(t)) = 0 and ˙x(t) exists. Since the +trajectory x is always in C, we have (see (2.2)) +0 = d +dtψj(t, x(t)) = ⟨∇xψj(t, x(t)), ˙x(t)⟩ + ∂tψj(t, x(t)) += ⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ − ξj(t)|∇xψj(t, x(t))|2 +− +� +i∈I(t,x(t))\{j} +ξi(t)⟨∇xψi(t, x(t)), ∇xψj(t, x(t))⟩ + ∂tψj(t, x(t)) +≤ ⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ − ξj(t)|∇xψj(t, x(t))|2+∂tψj(t, x(t)), +and, hence (see (2.1)), +ξj(t) ≤ +1 +|∇xψj(t, x(t))|2 (⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ + ∂tψj(t, x(t))) ≤ µ +η2 . +Define the function +µ(γ) = 1 +γ log +� µ +η2γ +� +, +γ > 0, +consider a sequence {σk} such that σk ↓ 0 and choose another sequence {γk} with γk ↑ +∞ and +C(t) ⊂ int Ck(t) = int +� +x : ψi(t, x) − σk ≤ µk, i = 1, . . . , I +� +, +where +µk = µ(γk). +Let xk be a solution to the differential equation +˙xk(t) = f(t, xk(t), uk(t)) − +I +� +i=1 +γkeγk(ψi(t,xk(t))−σk)∇xψi(t, xk(t)) +(2.6) +for some uk(t) ∈ U a.e. t ∈ [0, T]. Take any t ∈ [0, T] such that ˙xk(t) exists and ψj(t, xk(t)) − σk = µk. +5 + +Assume k is such that j ∈ I(t, xk(t)). Then, whenever γk is sufficiently large, we have +d +dtψj(t, xk(t)) = ⟨∇xψj(t, xk(t)), f(t, xk(t), uk(t))⟩ +− γkeγk(ψj(t,xk(t))−σk)|∇xψj(t, xk(t))|2 +− +� +i∈I(t,xk(t))\{j} +γkeγk(ψi(t,xk(t))−σk)⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩ +− +� +i̸∈I(t,xk(t)) +γkeγk(ψi(t,xk(t))−σk)⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩ ++ ∂tψj(t, xk(t)) +≤ ⟨∇xψj(t, xk(t)), f(t, xk(t), uk(t))⟩ +− γkeγk(ψj(t,xk(t))−σk)|∇xψj(t, xk(t))|2 +− +� +i̸∈I(t,xk(t)) +γkeγk(ψi(t,xk(t))−σk)⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩ ++ ∂tψj(t, xk(t)) +≤ ⟨∇xψj(t, xk(t)), f(t, xk(t), uk(t))⟩ +− γkeγk(ψj(t,xk(t))−σk)|∇xψj(t, xk(t))|2 ++ +� +i̸∈I(t,xk(t)) +γkeγk(−2β−σk)|⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩| ++ ∂tψj(t, xk(t)) +≤ µ − 1 +2 − η2γkeγkµk += −1 +2. +Above, we have used the definition of µ and the inequality +� +i̸∈I(t,xk(t)) +γkeγk(−2β−σk)|⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩|≤ 1 +2, +which holds for γk sufficiently large. +Now, if xk(0) ∈ Ck(0), we assure that xk(t) ∈ Ck(t), for all t ∈ [0, T], and +γkeγk(ψj(t,xk(t))−σk) ≤ γkeγkµk = µ +η2 . +(2.7) +It follows that, for k sufficienttly large, we have +| ˙xk(t)|≤ (const). +We are now a in position to state and prove our first result, Theorem 2.2 below. This is in the vein of +Theorem 4.1 in [23] (see also Lemma 1 in [10] when ψ is independent of t and convex) deviating from it +in so far as the approximating sequence of control systems (2.6) differs from the one introduced in [10]1. +The proof of Theorem 2.2 relies on (2.7). +Theorem 2.2 Let {(xk, uk)}, with uk(t) ∈ U a.e., be a sequence of solutions of Cauchy problems +� +� +� +� +� +˙xk(t) += +f(t, xk(t), uk(t)) − +I +� +i=1 +γkeγk(ψi(t,xk(t))−σk)∇xψi(t, xk(t)), +xk(0) += +bk ∈ Ck(0). +(2.8) +1See also Theorem 2.2 in [14] +6 + +If bk → x0, then there exists a subsequence {xk} (we do not relabel) converging uniformly to x, a unique +solution to the Cauchy problem +˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), +x(0) = x0, +(2.9) +where u is a measurable function such that u(t) ∈ U a.e. t ∈ [0, T]. +If, moreover, all the controls uk are equal, i.e., uk = u, then the subsequence converges to a unique +solution of (2.9), i.e., any solution of +˙x(t) ∈ f(t, x(t), U) − NC(t)(x(t)), +x(0) = x0 ∈ C(0) +(2.10) +can be approximated by solutions of (2.8). +Proof Consider the sequence {xk}, where (xk, uk) solves (2.8). Recall that xk(t) ∈ Ck(t) for all +t ∈ [0, T], and +| ˙xk(t)|≤ (const) +and +ξi +k(t) = γkeγk(ψi(t,xk(t))−σk) ≤ (const). +(2.11) +Then there exist subsequences (we do not relabel) weakly-∗ converging in L∞ to some v and ξi. Hence +xk(t) = x0 + +� t +0 +˙xk(s)ds −→ x(t) = x0 + +� t +0 +v(s)ds, ∀ t ∈ [0, T], +for an absolutely continuous function x. Obviously, x(t) ∈ C(t) for all t ∈ [0, T]. Considering the sequence +{xk}, recall that +˙xk(t) ∈ f(t, xk(t), U) − +I +� +i=1 +ξi +k(t)∇xψi(t, xk(t)). +(2.12) +Inclusion (2.12) is equivalent to +⟨z, ˙xk(t)⟩ ≤ S(z, f(t, xk(t), U)) − +I +� +i=1 +ξi +k(t)⟨z, ∇xψi(t, xk(t))⟩, +∀ z ∈ Rn. +Integrating this inequality, we get +� +z, xk(t + τ) − xk(t) +τ +� +≤ 1 +τ +� t+τ +t +� +S(z, f(s, xk(s), U)) − +I +� +i=1 +ξi +k(s)⟨z, ∇xψi(s, xk(s))⟩ +� +ds += 1 +τ +� t+τ +t +� +S(z, f(s, xk(s), U)) − +I +� +i=1 +ξi +k(s)⟨z, ∇xψi(s, x(s))⟩ ++ +I +� +i=1 +ξi +k(s)⟨z, ∇xψi(s, x(s)) − ∇xψi(s, xk(s))⟩ +� +ds. +(2.13) +Passing to the limit as k → ∞, we obtain +� +z, x(t + τ) − x(t) +τ +� +≤ 1 +τ +� t+τ +t +� +S(z, f(s, x(s), U)) − +I +� +i=1 +ξi(s)⟨z, ∇xψi(s, x(s))⟩ +� +ds. +(2.14) +7 + +Let t ∈ [0, T] be a Lebesgue point of x and ξ. Passing in the last inequality to the limit as τ ↓ 0, it leads +to +⟨z, ˙x(t)⟩ ≤ S(z, f(t, x(t), U)) − +I +� +i=1 +ξi(t)⟨z, ∇xψi(t, x(t))⟩. +Since z ∈ Rn is an arbitrary vector and the set f(t, x(t), U) is convex, we conclude that +˙x(t) ∈ f(t, x(t), U) − +I +� +i=1 +ξi(t)∇xψi(t, x(t)). +By the Filippov lemma there exists a measurable control u(t) ∈ U such that +˙x(t) = f(t, x(t), u(t)) − +I +� +i=1 +ξi(t)∇xψi(t, x(t)). +Furthermore, observe that ξi is zero if ψi(t, x(t)) < 0. If for some u such that u(t) ∈ U a.e., uk = u for +all k, then the sequence xk converges to the solution of +˙x(t) = f(t, x(t), u(t)) − +I +� +i=1 +ξi(t)∇xψi(t, x(t)). +Indeed, to see this, it suffices to pass to the limit as k → ∞ and then as τ ↓ 0, in the equality +xk(t + τ) − xk(t) +τ += 1 +τ +� t+τ +t +� +f(s, xk(s), u(s)) − +I +� +i=1 +ξi +k(s)∇xψi(s, xk(s)) +� +ds. +We now prove the uniqueness of the solution. We follow the proof of Theorem 4.1 in [23]. Notice, +however, that we now consider a special case and not the general case treated in [23]. Suppose that there +exist two different solutions of (2.9): x1 and x2. We have +1 +2 +d +dt|x1(t) − x2(t)|2= ⟨x1(t) − x2(t), ˙x1(t) − ˙x2(t)⟩ += ⟨x1(t) − x2(t), f(t, x1(t), u(t)) − f(t, x2(t), u(t))⟩ +− +� +x1(t) − x2(t), +I +� +i=1 +ξi +1(t)∇ψi(t, x1(t)) − +I +� +i=1 +ξi +2(t)∇ψi(t, x2(t)) +� +. +(2.15) +If, for all i, ψi(t, x1(t)) < 0 and ψi(t, x2(t)) < 0, then ξi +1(t) = ξi +2(t) = 0 and we obtain +1 +2 +d +dt|x1(t) − x2(t)|2≤ Lf|x1(t) − x2(t)|2. +Suppose that ψj(t, x1(t)) = 0. Then by the Taylor formula we get +ψj(t, x2(t)) = ψj(t, x1(t)) + ⟨∇xψj(t, x1(t)), x2(t) − x1(t)⟩ ++ 1 +2⟨x2(t) − x1(t), ∇2 +xψj(t, θx2(t) + (1 − θ)x1(t))(x2(t) − x1(t))⟩, +(2.16) +where θ ∈ [0, 1]. Since ψj(t, x2(t)) ≤ 0, we have +⟨∇xψj(t, x1(t)), x2(t) − x1(t)⟩ +≤ −1 +2⟨x2(t) − x1(t), ∇2 +xψj(t, θx2(t) + (1 − θ)x1(t))(x2(t) − x1(t))⟩ +≤ (const)|x1(t) − x2(t)|2. +(2.17) +8 + +Now, if ψj(t, x2(t)) = 0, we deduce in the same way that +⟨∇xψj(t, x2(t)), x1(t) − x2(t)⟩ ≤ (const)|x1(t) − x2(t)|2. +Thus we have +1 +2 +d +dt|x1(t) − x2(t)|2≤ (const)|x1(t) − x2(t)|2. +Hence |x1(t) − x2(t)|= 0. +2 +3 +Approximating Family of Optimal Control Problems +In this section we define an approximating family of optimal control problems to (P) and we state the +corresponding necessary conditions. +Let (ˆx, ˆu) be a global solution to (P) and consider sequences {γk} and {σk} as defined above. Let +ˆxk(·) be the solution to +� +� +� +� +� +˙x(t) = f(t, x(t), ˆu(t)) − +I +� +i=1 +γkeγk(ψi(t,x(t))−σk)∇xψi(t, x(t)), +x(0) = ˆx(0). +(3.1) +Set ϵk = |ˆxk(T) − ˆx(T)|. It follows from Theorem 2.2 that ϵk ↓ 0. Take α > 0 and define the problem +(P α +k ) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Minimize φ(x(T)) + |x(0) − ˆx(0)|2+α +� T +0 +|u(t) − ˆu(t)|dt +over processes (x, u) such that +˙x(t) = f(t, x(t), u(t)) − +I +� +i=1 +∇xeγk(ψi(t,x(t))−σk) a.e. t ∈ [0, T], +u(t) ∈ U +a.e. t ∈ [0, T], +x(0) ∈ C0, +x(T) ∈ CT + ϵkBn, +Clearly, the problem (P α +k ) has admissible solutions. Consider the space +W = {(c, u) | c ∈ C0, u ∈ L∞ with u(t) ∈ U} +and the distance +dW ((c1, u1), (c2, u2)) = |c1 − c2|+ +� T +0 +|u1(t) − u2(t)|dt. +Endowed with dW , W is a complete metric space. Take any (c, u) ∈ W and a solution y to the Cauchy +problem +� +� +� +� +� +˙y(t) += +f(t, y(t), u(t)) − +I +� +i=1 +∇xeγk(ψi(t,y(t))−σk) a.e. t ∈ [0, T], +y(0) += +c. +Under our assumptions, the function +(c, u) → φ(y(T)) + |c − ˆx(0)|2+α +� T +0 +|u − ˆu| dt +9 + +is continuous on (W, dW ) and bounded below. Appealing to Ekeland’s Theorem we deduce the existence +of a pair (xk, uk) solving the following problem +(APk) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Minimize Φ(x, u) = φ(x(T)) + |x(0) − ˆx(0)|2+α +� T +0 +|u(t) − ˆu(t)|dt ++ϵk +� +|x(0) − xk(0)|+ +� T +0 +|u(t) − uk(t)|dt +� +, +over processes (x, u) such that +˙x(t) = f(t, x(t), u(t)) − +I +� +i=1 +∇xeγk(ψi(t,x(t))−σk) a.e. t ∈ [0, T], +u(t) ∈ U +a.e. t ∈ [0, T], +x(0) ∈ C0, +x(T) ∈ CT + ϵkBn, +Lemma 3.1 Take γk → ∞, σk → 0 and ϵk → 0 as defined above. For each k, let (xk, uk) be the solution +to (APk). Then there exists a subsequence (we do not relabel) such that +uk(t) → ˆu(t) a.e., +xk → ˆx uniformly in [0, T]. +Proof We deduce from Theorem 2.2 that {xk} uniformly converges to an admissible solution ˜x to (P). +Since U and C0 are compact, we have U ⊂ KBm and C0 ⊂ KBn. Without loss of generality, uk weakly-∗ +converges to a function ˜u ∈ L∞([0, T], U). Hence it weakly converges to ˜u in L1. From optimality of the +processes (xk, uk) we have +φ(xk(T)) + |xk(0) − ˆx(0)|2+α +� T +0 +|uk(t) − ˆu(t)|dt +≤ φ(ˆxk(T)) + ϵk +� +|ˆxk(0) − xk(0)|+ +� T +0 +|uk(t) − ˆu(t)|dt +� +≤ φ(ˆxk(T)) + 2K(1 + T)ϵk. +Since (ˆx, ˆu) is a global solution of the problem, passing to the limit, we get +φ(˜x(T)) + |˜x(0) − ˆx(0)|2+α +� T +0 +|˜u(t) − ˆu(t)|dt +≤ lim +k→∞(φ(xk(T)) + |xk(0) − ˆx(0)|2) + α lim inf +k→∞ +� T +0 +|uk(t) − ˆu(t)|dt +≤ lim +k→∞ φ(ˆxk(T)) = φ(ˆx(T)) ≤ φ(˜x(T)). +Hence ˜x(0) = ˆx(0), ˜u = ˆu a.e., and uk converges to ˆu in L1, and some subsequence converges to ˆu almost +everywhere (we do not relabel). +2 +We now finish this section with the statement of the optimality necessary conditions for the family of +problems (APk). These can be seen as a direct consequence of Theorem 6.2.1 in [22]. +Proposition 3.2 For each k, let (xk, uk) be a solution to (APk). Then there exist absolutely continous +functions pk and scalars λk ≥ 0 such that +(a) (nontriviality condition) +λk + |pk(T)|= 1, +(3.2) +10 + +(b) (adjoint equation) +˙pk = −(∇xfk)∗pk + �I +i=1 γkeγk(ψi +k−σk)∇2 +xψi +kpk ++ �I +i=1 γ2 +keγk(ψi +k−σk)∇xψi +k⟨∇xψi +k, pk⟩, +(3.3) +where the superscript ∗ stands for transpose, +(c) (maximization condition) +max +u∈U {⟨f(t, xk, u), pk⟩ − αλk|u − ˆu|−ϵkλk|u − uk|} +(3.4) +is attained at uk(t), for almost every t ∈ [0, T], +(d) (transversality condition) +(pk(0), −pk(T)) ∈ λk (2(xk(0) − ˆx(0)) + ϵkBn, ∂φ(xk(T))) ++NC0(xk(0)) × NCT +ϵkBn(xk(T)). +(3.5) +To simplify the notation above, we drop the t dependance in pk, ˙pk, xk, uk, ˆx and ˆu. Moreover, in +(b), we write ψk instead of ψ(t, xk(t)), fk instead of f(t, xk(t), uk(t)). The same holds for the derivatives +of ψ and f. +4 +Maximum Principle for (P) +In this section, we establish our main result, a Maximum Principle for (P). This is done by taking limits +of the conclusions of Proposition 3.2, following closely the analysis done in the proof of [10, Theorem 2]. +Observe that +1 +2 +d +dt|pk(t)|2 = −⟨∇xfkpk, pk⟩ + +I +� +i=1 +γkeγk(ψi +k−σk)⟨∇2 +xψi +kpk, pk⟩ ++ +I +� +i=1 +γ2 +keγk(ψi +k−σk)⟨∇xψi +k, pk⟩2 +≥ −⟨∇xfkpk, pk⟩ + +I +� +i=1 +γkeγk(ψi +k−σk)⟨∇2 +xψi +kpk, pk⟩ +≥ −M|pk|2+ +I +� +i=1 +γkeγk(ψi +k−σk)⟨∇2 +xψi +kpk, pk⟩, +where M is the constant of (A2). Taking into account hypothesis (A1) and (2.7) we deduce the existence +of a constant K0 > 0 such that +1 +2 +d +dt|pk(t)|2≥ −K0|pk(t)|2. +This last inequality leads to +|pk(t)|2 ≤ e2K0(T −t)|pk(T)|2≤ e2K0T |pk(T)|2. +Since, by (a) of Proposition 3.2, |pk(T)|≤ 1, we deduce from the above that there exists M0 > 0 such +that +|pk(t)| ≤ M0. +(4.1) +Now, we claim that the sequence { ˙pk} is uniformly bounded in L1. To prove our claim, we need to establish +bounds for the three terms in (3.3). Following [10] and [14], we start by deducing some inequalities that +will be of help. +11 + +Denote Ik = I(t, xk(t)) and Sj +k = sign +� +⟨∇xψj +k, pk⟩ +� +. We have +I +� +j=1 +d +dt +���⟨∇xψj +k, pk⟩ +��� += +I +� +j=1 +� +⟨∇2 +xψj +k ˙xk, pk⟩ + ⟨∂t∇xψj +k, pk⟩ + ⟨∇xψj +k, ˙pk⟩ +� +Sj +k += +I +� +j=1 +� +⟨pk, ∇2 +xψj +kfk⟩ − +I +� +i=1 +γkeγk(ψi +k−σk)⟨pk, ∇2ψj +k∇xψi +k⟩ +� +Sj +k ++ +I +� +j=1 +� +⟨∂t∇xψj +k, pk⟩ − ⟨∇xψj +k, (∇xfk)∗pk⟩ +� +Sj +k ++ +I +� +j=1 +� I +� +i=1 +γkeγk(ψi +k−σk)⟨∇xψj +k, ∇2 +xψi +kpk⟩ +� +Sj +k ++ +I +� +i=1 +I +� +j=1 +γ2 +keγk(ψi +k−σk)⟨∇xψj +k, ∇xψi +k⟩⟨∇xψi +k, pk⟩Sj +k +Observe that (see (2.3) and (2.4)) +I +� +i=1 +I +� +j=1 +γ2 +keγk(ψi +k−σk)⟨∇xψj +k, ∇xψi +k⟩⟨∇xψi +k, pk⟩Sj +k += +I +� +i=1 +� +j∈Ik +γ2 +keγk(ψi +k−σk)⟨∇xψj +k, ∇xψi +k⟩⟨∇xψi +k, pk⟩Sj +k += +� +i̸∈Ik +γ2 +keγk(ψi +k−σk) � +j∈Ik +⟨∇xψj +k, ∇xψi +k⟩⟨∇xψi +k, pk⟩Sj +k ++ +� +i∈Ik +γ2 +keγk(ψi +k−σk) +� +�|∇xψi +k|2+ +� +j∈Ik\{i} +⟨∇xψj +k, ∇xψi +k⟩Sj +k Si +k +� +� |⟨∇xψi +k, pk⟩| += +� +i∈Ik +γ2 +keγk(ψi +k−σk) +� +�|∇xψi +k|2+ +� +j∈Ik\{i} +⟨∇xψj +k, ∇xψi +k⟩Sj +k Si +k +� +� |⟨∇xψi +k, pk⟩| +≥ (1 − ρ) +� +i∈Ik +γ2 +keγk(ψi +k−σk)|∇xψi +k|2|⟨∇xψi +k, pk⟩| += (1 − ρ) +I +� +i=1 +γ2 +keγk(ψi +k−σk)|∇xψi +k|2|⟨∇xψi +k, pk⟩|. +Using this and integrating the previous equality, we deduce the existence of M1 > 0 such that: +� T +0 +I +� +i=1 +γ2 +keγk(ψi +k−σk)|∇xψi +k|2|⟨∇xψi +k, pk⟩|dt ≤ M1. +(4.2) +We are now in a position to show that +� T +0 +I +� +i=1 +γ2 +keγk(ψi +k−σk)|∇xψi +k| +��⟨∇xψi +k, pk⟩ +�� dt +12 + +is bounded. For simplicity, set Li +k(t) = γ2 +keγk(ψi +k−σk)|∇xψi +k| +��⟨∇xψi +k, pk⟩ +��. Notice that +I +� +i=1 +� T +0 +Li +k(t)dt = +I +� +i=1 +�� +{t:|∇xψi +k|<η} +Li +k(t) dt + +� +{t:|∇xψi +k|≥η} +Li +k(t)dt +� +. +Using (A1) and (4.2), we deduce that +I +� +i=1 +� T +0 +Li +k(t) dt ≤ +I +� +i=1 +� +γ2 +ke−γk(β+σk)η2 max +t |pk(t)| +� ++ +I +� +i=1 +� +γ2 +k +� +{t:|∇xψi +k|≥η} +eγk(ψi +k−σk) |∇xψi +k|2 +|∇xψi +k| +��⟨∇xψi +k, pk⟩ +�� dt +� +≤ γ2 +kI e−γk(β+σk)η2M0 ++ 1 +η +I +� +i=1 +�� T +0 +γ2 +keγk(ψi +k−σk)|∇xψi +k|2��⟨∇xψi +k, pk⟩ +�� dt +� +≤ η2M0I + M1 +η , +for k large enough. Summarizing, there exists a M2 > 0 such that +I +� +i=1 +γ2 +k +� T +0 +eγk(ψi +k−σk)|∇ψi +k| +��⟨∇ψi +k, pk⟩ +�� dt +≤ M2. +(4.3) +Mimicking the analysis conducted in Step 1, b) and c) of the proof of Theorem 2 in [10] and taking into +account (b) of Proposition 3.2 we conclude that there exist constants N1 > 0 such that +� T +0 +| ˙pγk(t)| dt ≤ N1, +(4.4) +for k sufficiently large, proving our claim. +Before proceeding, observe that it is a simple matter to assert the existence of a constant N2 such +that +I +� +i=1 +� T +0 +γ2 +keγk(ψi +k−σk)|⟨∇ψi +k, pγk⟩|dt ≤ N2. +(4.5) +This inequality will be of help in what follows. +Let us now recall that +ξi +k(t) = γkeγk(ψi(t,xk(t))−σk) +and that the second inequality in (2.11) holds. We turn to the analysis of Step 2 in the proof of Theorem +2 in [10] (see also [14]). Adapting those arguments, we can conclude the existence of some function p ∈ +BV ([0, T], Rn) and, for i = 1, . . . , I, functions ξi ∈ L∞([0, T], R) with ξi(t) ≥ 0 a. e. t, ξi(t) = 0, t ∈ Ii +b, +where +Ii +b = +� +t ∈ [0, T] : ψi(t, ˆx(t)) < 0 +� +, +and finite signed Radon measures ηi, null in Ii +b, such that, for any z ∈ C([0, T], Rn) +� T +0 +⟨z, dp⟩ = − +� T +0 +⟨z, (∇ ˆf)∗p⟩dt + +I +� +i=1 +�� T +0 +ξi⟨z, ∇2 ˆψip⟩dt + +� T +0 +⟨z, ∇ ˆψi(t)⟩dηi +� +, +where ∇ ˆψi(t) = ∇ψi(t, ˆx(t)). The finite signed Radon measures ηi are weak-∗ limits of +γ2 +keγk(ψi +k−σk)⟨∇ψi +k(xk(t), pk(t)⟩dt. +13 + +Observe that the measures +⟨∇ψi(ˆx(t), p(t)⟩dηi(t) +(4.6) +are nonnegative. +For each i = 1, . . . , I, the sequence ξi +k is weakly-∗ convergent in L∞ to ξi ≥ 0. Following [14], we +deduce from (4.5) that, for each i = 1, . . . , I, +� T +0 +|ξi⟨∇x ˆψi, p⟩|dt = lim +k→∞ +� T +0 +|ξi +k⟨∇x ˆψi, p⟩|dt +≤ lim +k→∞ +�� T +0 +ξi +k|⟨∇x ˆψi, p⟩ − ⟨∇xψi +k, pk⟩|dt + +� T +0 +ξi +k|⟨∇xψi +k, pk⟩|dt +� +≤ lim +k→∞ +����ξi +k +��� +L∞ +���⟨∇x ˆψi, p⟩ − ⟨∇xψi +k, pk⟩ +��� +L1 + N2 +γk +� += 0. +It turns out that +ξi⟨∇x ˆψi, p⟩ = 0 a.e.. +(4.7) +Consider now the sequence of scalars {λk}. It is an easy matter to show that there exists a subsequence +of {λk} converging to some λ ≥ 0. This, together with the convergence of pk to p, allows us to take limits +in (a) and (c) of Proposition 3.2 to deduce that +λ + |p(T)|= 1 +and +⟨p(t), f(t, ˆx(t), u)⟩ − αλ|u − ˆu(t)|≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ ∀u ∈ U, a.e. t ∈ [0, T]. +It remains to take limits of the transversality conditions (d) in Proposition 3.2. First, observe that +CT + ϵkBn = {x : d(x, CT ) ≤ ϵk} . +From the basic properties of the Mordukhovich normal cone and subdifferential (see [19], section 1.3.3) +we have +NCT +ϵkBn(xk(T)) ⊂ cl cone ∂d(xk(T), CT ) +and +NCT (ˆx(T)) = cl cone ∂d(ˆx(T), CT ). +Passing to the limit as k → ∞ we get +(p(0), −p(T)) ∈ NC0(ˆx(0)) × NCT (ˆx(T)) + {0} × λ ∂φ(ˆx(T)). +Finally, and mimicking Step 3 in the proof of Theorem 2 in [10], we remove the dependence of the +conditions on the parameter α. This is done by taking further limits, this time considering a sequence of +αj ↓ 0. +We then summarize our conclusions in the following Theorem. +Theorem 4.1 Let (ˆx, ˆu) be the optimal solution to (P). Suppose that assumption A1–A6 are satisfied. +For i = 1, · · · , I, set +Ii +b = {t ∈ [0, T] : ψi(t, ˆx(t)) < 0}. +There exist λ ≥ 0, p ∈ BV ([0, T], Rn), finite signed Randon measures ηi, null in Ii +b, for i = 1, · · · , I, +ξi ∈ L∞([0, T], R), with i = 1, · · · , I, where ξi(t) ≥ 0 a. e. t and ξi(t) = 0, t ∈ Ii +b, such that +a) λ + |p(T)|̸= 0, +14 + +b) ˙ˆx(t) = f(t, ˆx(t), ˆu(t)) − +I +� +i=1 +ξi(t)∇x ˆψi(t), +c) for any z ∈ C([0, T]; Rn) +� T +0 +⟨z(t), dp(t)⟩ = − +� T +0 +⟨z(t), (∇x ˆf(t))∗p(t)⟩dt ++ +I +� +i=1 +�� T +0 +ξi(t)⟨z(t), ∇2 +x ˆψi(t)p(t)⟩dt + +� T +0 +⟨z(t), ∇x ˆψi(t)⟩dηi +� +, +where ∇ ˆf(t) = ∇xf(t, ˆx(t), ˆu(t)), +∇ ˆψi(t) = ∇ψi(t, ˆx(t)) and ∇2 ˆψi(t) = ∇2ψi(t, x(t)), +d) ξi(t)⟨∇xψi(t, ˆx(t)), p(t)⟩ = 0, a.e. t for all i = 1, . . . , I, +e) for all i = 1, . . . , I, the meaures ⟨∇ψi(ˆx(t), p(t)⟩dηi(t) are nonnegative, +f) ⟨p(t), f(t, ˆx(t), u)⟩ ≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ for all u ∈ U, a.e. t, +g) +(p(0), −p(T)) ∈ NC0(ˆx(0)) × NCT (ˆx(T)) + {0} × λ∂φ(ˆx(T)). +Noteworthy, condition e) is not considered in any of our previous works. +We now turn to the free end point case, i. e., to the problem +(Pf) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Minimize φ(x(T)) +over processes (x, u) such that +˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), a.e. t ∈ [0, T], +u(t) ∈ U, +a.e. t ∈ [0, T], +x(0) ∈ C0 ⊂ C(0). +Problem (Pf) differs from (P) because x(T) is not constrained to take values in CT . We apply Theorem +4.1 to (Pf). Since x(T) is free, we deduce from (f) in the above Theorem that −p(T) = λ∂φ(ˆx(T)). +Suppose that λ = 0. +Then p(T) = 0 contradicting the nontriviality condition (a) of Theorem 4.1. +Without loss of generality, we then conclude that the conditions of Theorem 4.1 hold with λ = 1. We +summarize our findings in the following Corollary. +Corollary 4.2 Let (ˆx, ˆu) be the optimal solution to (Pf). Suppose that assumption A1–A6 are satisfied. +For i = 1, · · · , I, set +Ii +b = {t ∈ [0, T] : ψi(t, ˆx(t)) < 0}. +There exist p ∈ BV ([0, T], Rn), finite signed Randon measures ηi, null in Ii +b, for i = 1, · · · , I, ξi ∈ +L∞([0, T], R), with i = 1, · · · , I, where ξi(t) ≥ 0 a.e. t and ξi(t) = 0 for t ∈ Ii +b, such that +a) ˙ˆx(t) = f(t, ˆx(t), ˆu(t)) − +I +� +i=1 +ξi(t)∇x ˆψi(t), +15 + +b) for any z ∈ C([0, T]; Rn) +� T +0 +⟨z(t), dp(t)⟩ = − +� T +0 +⟨z(t), (∇x ˆf(t))∗p(t)⟩dt ++ +I +� +i=1 +�� T +0 +ξi(t)⟨z(t), ∇2 +x ˆψi(t)p(t)⟩dt + +� T +0 +⟨z(t), ∇x ˆψi(t)⟩dηi +� +, +where ∇ ˆf(t) = ∇xf(t, ˆx(t), ˆu(t)), +∇ ˆψi(t) = ∇ψi(t, ˆx(t)) and ∇2 ˆψi(t) = ∇2ψi(t, x(t)), +c) ξi(t)⟨∇xψi(t, ˆx(t)), p(t)⟩ = 0 for a.e. t and for all i = 1, . . . , I, +d) for all i = 1, . . . , I, the meaures ⟨∇ψi(ˆx(t), p(t)⟩dηi(t) are nonnegative, +e) ⟨p(t), f(t, ˆx(t), u)⟩ ≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ for all u ∈ U, a.e. t, +f) +(p(0), −p(T)) ∈ NC0(ˆx(0)) × {0} + {0} × ∂φ(ˆx(T)). +5 +Example +Let us consider the following problem +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Minimize +− x(T) +over processes ((x, y, z), u) such that +� +� +˙x(t) +˙y(t) +˙z(t) +� +� ∈ +� +� +0 +σ +0 +0 +0 +0 +0 +0 +0 +� +� +� +� +x +y +z +� +� + +� +� +0 +u +0 +� +� − NC(x, y, z), +u ∈ [−1, 1], +(x, y, z)(0) = (x0, y0, z0), +(x, y, z)(T) ∈ CT , +where +• 0 < σ ≪ 1, +• C = {(x, y, z) | x2 + y2 + (z + h)2 ≤ 1, x2 + y2 + (z − h)2 ≤ 1}, 2h2 < 1, +• (x0, y0, z0) ∈ intC, with x0 < −δ, y0 = 0 and z0 > 0, +• CT = {(x, y, z) | x ≤ 0, y ≥ 0, δy − y2x ≤ δy2} ∩ C, where +δ < y2|x0| +y1 +, with y1 = +� +1 − x2 +0 − (z0 + h)2 and y2 = +� +1 − h2. +We choose T > 0 small and, nonetheless, sufficiently large to guarantee that, when σ = 0, the system +can reach the interior of CT but not the segment {(x, 0, 0) | x ∈ [−δ, 0]}. Since σ and T are small, it +follows that the optimal trajectory should reach CT at the face δy − y2x = δy2 of CT . +16 + +To significantly increase the value of the x(T), the optimal trajectory needs to live on the boundary +of C for some interval of time. Then, before reaching and after leaving the boundary of C, the optimal +trajectory lives in the interior of C. Since δ is small, the trajectory cannot reach CT from any point of +the sphere x2 +y2 +(z +h)2 = 1 with z > 0. This means that, while on the boundary of C the trajectory +should move on the sphere x2 + y2 + (z + h)2 = 1 untill reaching the plane z = 0 and then it moves on +the intersection of the two spheres. +While in the interior of C, the control can change sign from −1 to 1 or from 1 to −1. Certainly, the +control should be 1 right before reaching the boundary and −1 right before arriving at CT . Changes of +the control from 1 to −1 or −1 to 1 before reaching the boundary translate into time waste and leads to +smaller values of x(T). It then follows that the optimal control should be of the form +u(t) = +� +1, +t ∈ [0, ˜t], +−1, +t ∈ ]˜t, T], +(5.1) +for some value ˜t ∈]0, T[. +After the modification (2.5), the data of the problem satisfy the conditions under which Theorem 4.1 +holds. We now show that the conclusions of Theorem 4.1 completly identify the structure (5.1) of the +optimal control. +From Theorem 4.1 we deduce the existence of λ ≥ 0, p, q, r ∈ BV ([0, T], R), finite signed Randon +measures η1 and η2, null respectively in +I1 +b = +� +(x, y, z) | x2 + y2 + (z + h)2 − 1 < 0 +� +and +I2 +b = +� +(x, y, z) | x2 + y2 + (z − h)2 − 1 < 0 +� +, +ξi ∈ L∞([0, T], R), with i = 1, 2, where ξi(t) ≥ 0 a. e. t and ξi(t) = 0, t ∈ Ii +b, such that +(i) +� +� +˙x(t) +˙y(t) +˙z(t) +� +� = +� +� +0 +σ +0 +0 +0 +0 +0 +0 +0 +� +� +� +� +x +y +z +� +� + +� +� +0 +u +0 +� +� − 2ξ1 +� +� +x +y +z + h +� +� − 2ξ2 +� +� +x +y +z − h +� +� +(ii) +d +� +� +p +q +r +� +� = +� +� +0 +0 +0 +−σ +0 +0 +0 +0 +0 +� +� +� +� +p +q +r +� +� dt ++2(ξ1 + ξ2) +� +� +p +q +r +� +� dt + 2 +� +� +x +y +z + h +� +� dη1 + 2 +� +� +x +y +z − h +� +� dη2, +(iii) +� +� +p +q +r +� +� (T) = +� +� +λ +0 +0 +� +� + µ +� +� +y2 +−δ +0 +� +� , where µ ≥ 0, +(iv) +ξ1(xp + yq + (z + h)r) = 0, ξ2(xp + yq + (z − h)r) = 0, +(v) +the meaures (xp + yq + (z + h)r)dη1 and (xp + yq + (z − h)r)dη2 +are nonnegative, +(vi) +maxu∈[−1,1] uq = ˆuq. +where ˆu is the optimal control. +Let t1 be the instant of time when the trajectory reaches the shere x2 + y2 + (z + h)2 = 1, t2 the +instant of time when the trajectory reaches the intersection of the two spheres and t3 be the instant of +time the trajectory leaves the boundary of C. We have 0 < t1 < t2 < t3 < T. +Next we show that the multiplier q changes sign only once and so identifing the structure (5.1) of the +optimal control in a unique way. We start by looking at the case when t = T. We have +� +p +q +� +(T) = +� +λ +0 +� ++ µ +� +y2 +−δ +� +. +17 + +Starting from t = T, let us go backwards in time until the instant t3 when the trajectory leaves the +boundary of C. If q(T) = 0, then p(T) = λ > 0 and we would have q(t) > 0 for t ∈]t3, T[ (see (ii) above), +which is impossible. We then have p(T) > 0 and q(T) < 0 and, in ]t3, T[, since σ is small, the vector +(p(t), q(t)) does not change much. At t = t3, the vector (p, q) has a jump and such jump can only occur +along the vector (x(t3), y(t3)). Therefore, we have p(t3 − 0) > 0 and q(t3 − 0) < 0. +Let us now consider t ∈]t2, t3[. We have the following +1. when t ∈ [t2, t3], we have z = 0; +2. condition (i) above implies that ξ1 = ξ2 = ξ, ξ > 0 since, otherwise the motion along x2+y2 = 1−h2 +would not be possible; +3. from 0 = d +dt(x2 + y2) = σ2xy − 8ξx2 + 2uy − 8ξy2 we get ξ = σxy+uy +4(1−h2); +4. condition (iv) implies that r = 0 leading to xp + yq = 0. Since x < 0, y > 0, then q = 0 implies +p = 0; +5. condition (ii) implies dη1 = dη2 = dη; +6. 0 = d(xp + yq) = uqdt + 4(1 − h2)dη ⇒ dη +dt = − +uq +4(1−h2); +7. from the above analysis we deduce that +˙p = σxy + uy +(1 − h2) p − +xuq +(1 − h2), +˙q = −σp + +σxy +(1 − h2) q. +Thus, (p, q) is a solution to a linear system and it can never be equal to zero. It follows that q +cannot be zero because q = 0 implies p = 0. Since q ̸= 0, we have q > 0. +Let us consider the case when t = t2. We claim that +(p(t2 − 0), q(t2 − 0)) ̸= (0, 0). +Seeking a contradiction, assume that it is (p(t2 − 0), q(t2 − 0)) = (0, 0). Then we have +(p(t2 + 0), q(t2 + 0)) = (0, 0) + (2x2(t2), 2y2(t2))(dη1 + dη2) +and such jump has to be normal to (x(t2), y(t2)) since r(t2 + 0) = 0 (see (iv)). It follows that (x2(t2) + +y2(t2))(dη1 + dη2) = 0 and, since x2(t2) + y2(t2) > 0, we get dη1 + dη2 = 0, proving our claim. +We now consider t ∈]t1, t2[. It is easy to see that ξ2 = 0 and dη2 = 0. We also deduce that +1. 0 = d +dt(x2+y2+(z+h)2) = 2σxy+2uy−4ξ1y2−4ξ1x2−4ξ1(z+h)2 which implies that ξ1 = σxy+uy +2 +; +2. also 0 = d(xp + yq + (z + h)r) = uqdt + 2dη1 implies that dη1 +dt = − uq +2 ; +3. from the above we deduce that +˙p = (σxy + uy)p − xuq, +˙q = −σp + σxyq. +Thus (p, q) is a solution to a linear system and never is equal to zero. Second equation implies that +if q = 0 then ˙q ̸= 0. Hence q > 0. +18 + +Now we need to consider t = t1. We claim that +(p(t1 − 0), q(t1 − 0), r(t1 − 0)) ̸= (0, 0, 0). +Let us then assume that it is (p(t1−0), q(t1−0), r(t1−0)) = (0, 0, 0). It then follows that (p(t1+0), q(t1+ +0), r(t1 +0)) = (0, 0, 0)+(2x(t1)dη1, 2y(t1)dη1, 2(z(t1)+h)dη1). We now show that there is no such jump. +Set r(t1 − 0) = r0. Then it follows from (iv) that (x(t1) · 0 + y(t1) · 0 + (z(t1) + h))r0 = 0 which implies +that r0 = 0. We also have (x2(t1) + y2(t1) + (z(t1) + h)2)dη1 = 0 from (v). But this implies that dη1 = 0. +Consequently, the multipliers do not exhibit a jump at t1. +From the previous analysis we deduce that q should be positive almost everywhere on the boundary. It +then follows that to find the optimal solution we have to analyze admissible trajectories with the controls +with the structure (5.1) and choose the optimal value of ˜t. +Acknowledgements +The authors gratefully thank the support of Portuguese Foundation for Science and Technology (FCT) +in the framework of the Strategic Funding UIDB/04650/2020. +Also we thank the support by the ERDF - European Regional Development Fund through the Oper- +ational Programme for Competitiveness and Internationalisation - COMPETE 2020, INCO.2030, under +the Portugal 2020 Partnership Agreement and by National Funds, Norte 2020, through CCDRN and +FCT, within projects To Chair (POCI-01-0145-FEDER-028247), Upwind (PTDC/EEI-AUT/31447/2017 +- POCI-01-0145-FEDER-031447) and Systec R&D unit (UIDB/00147/2020). +References +[1] Addy K, Adly S, Brogliato B, Goeleven D, A method using the approach of Moreau and Pana- +giotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear +Anal. Hybrid Syst., vol. 1, 30–43, (2013), https://doi.org/10.1016/j.nahs.2006.04.00. +[2] Arroud +C +and +Colombo +G, +Necessary +conditions +for +a +nonclassical +control +problem +with state constraints, +20th IFAC World Congress, +Toulouse, +France, +July 9-14, +2017, +https://doi.org/10.1016/j.ifacol.2017.08.110. +[3] Arroud C and Colombo G, A maximum principle for the controlled sweeping process, Set-Valued +Var. Anal 26, 607–629 (2018) DOI: 10.1007/s11228-017-0400-4. +[4] Brokate M, Krejˇc´ı P Optimal control of ODE systems Involving a rate independent variational in- +equality, Disc. Cont. Dyn. Syst. Ser. B, vol. 18 (2) 331–348 (2013), doi: 10.3934/dcdsb.2013.18.331. +[5] Cao TH, Mordukhovich B, Optimality conditions for a controlled sweeping process with applica- +tions to the crowd motion model, Disc. Cont. Dyn. Syst. Ser. B, vol. 22, 267–306 (2017). +[6] Cao +TH, +Colombo +G, +Mordukhovich +B, +Nguyen +D., +Optimization +of +fully +con- +trolled +sweeping +processes, +Journal +of +Differential +Equations, +295, +138–186 +(2021) +https://doi.org/10.1016/j.jde.2021.05.042 +[7] Clarke F, Optimization and nonsmooth analysis, John Wiley, New York (1983). +[8] Colombo G, Palladino M, The minimum time function for the controlled Moreau’s sweeping +process, SIAM, vol. 54, no. 4,2036– 2062 (2016), https://doi.org/10.1137/15M1043364. +[9] Colombo G,Henrion R, Hoang ND, Mordukhovich BS, Optimal control of the sweeping process +over polyhedral controlled sets, Journal of Differential Equations, vol. 260, 4, 3397–3447, (2016), +https://doi.org/10.1016/j.jde.2015.10.039. +19 + +[10] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control involving Sweeping Processes, Set- +Valued Var. Anal 27, 523–548, (2019), https://doi.org/10.1007/s11228-018-0501-8. +[11] de Pinho MdR, Ferreira MMA, Smirnov G, Correction to: Optimal Control Involving Sweeping +Processes, Set-Valued Var. Anal 27, 1025–1027 (2019) https://doi.org/10.1007/s11228-019-00520- +5. +[12] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control with Sweeping Processes: Numerical +Method, J Optim Theory Appl 185, 845– 858 (2020) https://doi.org/10.1007/s10957-020-01670-5 +[13] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control Involving Sweeping Processes with +End Point Constraints, 2021 60th IEEE Conference on Decision and Control (CDC), 2021, 96– +101(2019) doi: 10.1109/CDC45484.2021.9683291 +[14] de Pinho MdR, Ferreira MMA, Smirnov G, Necessary conditions for optimal control problems +with sweeping systems and end point constraints, Optimization, to appear (2022). +[15] Hermosilla C, Palladino M, Optimal Control of the Sweeping Process with a Non-Smooth Moving +Set, SIAM j. Cont. Optim., to appear (2022). +[16] Kunze M, Monteiro Marques MDP, An Introduction to Moreau’s sweeping process. Impacts in +Mechanical Systems, Lecture Notes in Physics, vol. 551, 1–60, (2000). +[17] Maury B, Venel J (2011), A discrete contact model for crowd motion, ESAIM: M2AN 45 1, +145–168. +[18] Moreau JJ, On unilateral constraints, friction and plasticity, In: Capriz G., Stampacchia G. (Eds.) +New Variational Techniques in Mathematical Physics, CIME ciclo Bressanone 1973. Edizioni +Cremonese, Rome, 171–322 (1974). +[19] Mordukhovich B, Variational analysis and generalized differentiation. Basic Theory. Fundamental +Principles of Mathematical Sciences 330, Springer-Verlag, Berlin (2006). +[20] Mordukhovich B, Variational analysis and generalized differentiation II. Applications, Fundamen- +tal Principles of Mathematical Sciences 330, Springer-Verlag, Berlin (2006). +[21] Thibault L, Moreau sweeping process with bounded truncated retraction, J. Convex Anal, vol. +23, pp. 1051–1098 (2016). +[22] Vinter RB, Optimal Control, Birkh¨auser, Systems and Control: Foundations and Applications, +Boston MA (2000). +[23] Zeidan V, Nour C, Saoud H, A nonsmooth maximum principle for a controlled noncon- +vex sweeping process, Journal of Differential Equations, vol. 269 (11), 9531–9582 (2020), +https://doi.org/10.1016/j.jde.2020.06.053 +20 + diff --git a/-NFRT4oBgHgl3EQfrTfI/content/tmp_files/load_file.txt b/-NFRT4oBgHgl3EQfrTfI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..710a69132eb43a02dcc4f18a72fa6eb7cb5e32a4 --- /dev/null +++ b/-NFRT4oBgHgl3EQfrTfI/content/tmp_files/load_file.txt @@ -0,0 +1,722 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf,len=721 +page_content='A Maximum Principle for Optimal Control Problems involving Sweeping Processes with a Nonsmooth Set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' de Pinho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Margarida A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Ferreira ∗and Georgi Smirnov † February 1, 2023 Abstract We generalize a Maximum Principle for optimal control problems involving sweeping systems previously derived in [14] to cover the case where the moving set may be nonsmooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Noteworthy, we consider problems with constrained end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' A remarkable feature of our work is that we rely upon an ingenious smooth approximating family of standard differential equations in the vein of that used in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Keywords: Sweeping Process Optimal Control, Maximum Principle, Approximations 1 Introduction In recent years, there has been a surge of interest in optimal control problems involving the controlled sweeping process of the form ˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), u(t) ∈ U, x(0) ∈ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) In this respect, we refer to, for example, [3], [4], [5], [8], [9], [16], [23], [10] (see also accompanying correction [11]), [6], [15] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Sweeping processes first appeared in the seminal paper [18] by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Moreau as a mathematical framework for problems in plasticity and friction theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' They have proved of interest to tackle problems in mechanics, engineering, economics and crowd motion problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' to name but a few, see [1], [5], [16], [17] and [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In the last decades, systems in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) have caught the attention and interest of the optimal control community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Such interest resides not only in the range of applications but also in the remarkable challenge they rise concerning the derivation of necessary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This is due to the presence of the normal cone NC(t)(x(t)) in the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Indeed, the presence of the normal cone renders the discontinuity of the right hand of the differential inclusion in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) destroying a regularity property central to many known optimal control results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Lately, there has been several successful attempts to derive necessary conditions for optimal control problems involving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Assuming that the set C is time independent, necessary conditions for optimal control problems with free end point have been derived under different assumptions and using different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In [10], the set C has the form C = {x : ψ(x) ≤ 0} and an approximating sequence of optimal control problems, where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) is approximated by the differential equation ˙xγk(t) = f(t, xγk(t), u(t)) − γkeγkψ(xγk (t))∇ψ(xγk(t)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2) for some positive sequence γk → +∞, is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Similar techniques are also applied to somehow more general problems in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' A useful feature of those approximations is explored in [12] to define numerial schemes to solve such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∗MdR de Pinho and MMA Ferreira are at Faculdade de Engenharia da Universidade do Porto, DEEC, SYSTEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Portugal, mrpinho, mmf@fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='pt †G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Smirnov is at Universidade do Minho, Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Matem´atica, Physics Center of Minho and Porto Universities (CF-UM- UP), Campus de Gualtar, Braga, Portugal, smirnov@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='uminho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='pt 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='13620v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='OC] 31 Jan 2023 More recently, an adaptation of the family of approximating systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2) is used in [14] to generalize the results in [10] to cover problems with additional end point constraints and with a moving set of the form C(t) = {x : ψ(t, x) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In this paper we generalize the Maximum Principle proved in [14] to cover problems with possibly nonsmooth sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Our problem of interest is (P) � � � � � � � � � � � � � � � � � Minimize φ(x(T)) over processes (x, u) such that ˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], u(t) ∈ U, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], (x(0), x(T)) ∈ C0 × CT ⊂ C(0) × C(T), where T > 0 is fixed, φ : Rn → R, f : [0, T] × Rn × Rm → Rn, U ⊂ Rm and C(t) := � x ∈ Rn : ψi(t, x) ≤ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) for some functions ψi : [0, T] × Rn → R, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The case where I = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) and ψ1 is C2 is covered in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Here, we assume I > 1 and that the functions ψi are also C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Although going from I = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) to I > 1 may be seen as a small generalization, it demands a significant revision of the technical approach and, plus, the introduction of a constraint qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This is because the set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) may be nonsmooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We focus on sets (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3), satisfying a certain constraint qualification, introduced in assumption (A1) in section 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This is, indeed, a restriction on the nonsmoothness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' A similar problem with nonsmooth moving set is considered in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Our results cannot be obtained from the results of [15] and do not generalize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This paper is organized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In section 2, we introduce the main notation and we state and discuss the assumptions under which we work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In this same section, we also introduce the family of approximating systems to ˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)) and establish a crucial convergence result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In section 3, we dwell on the approximating family of optimal control problems to (P) and we state the associated necessary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The Maximum Principle for (P) is then deduced and stated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1, covering additionally, problems in the form of (P) where the end point constraint x(T) ∈ CT is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Before finishing, we present an illustrative example of our main result, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 2 Preliminaries In this section, we introduce a summary of the notation and state the assumptions on the data of (P) enforced throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Furthermore, we extract information from the assumptions establishing relations crucial for the forthcoming analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Notation For a set S ⊂ Rn, ∂S, cl S and int S denote the boundary, closure and interior of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' If g : Rp → Rq, ∇g represents the derivative and ∇2g the second derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' If g : R × Rp → Rq, then ∇xg represents the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' x ∈ Rp and ∇2 xg the second derivative, while ∂tg(t, x) represents the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The Euclidean norm or the induced matrix norm on Rp×q is denoted by |·|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We denote by Bn the closed unit ball in Rn centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The inner product of x and y is denoted by ⟨x, y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For some A ⊂ Rn, d(x, A) denotes the distance between x and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We denote the support function of A at z by S(z, A) = sup{⟨z, a⟩ | a ∈ A} The space L∞([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Rp) (or simply L∞ when the domains are clearly understood) is the Lebesgue space of essentially bounded functions h : [a, b] → Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We say that h ∈ BV ([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Rp) if h is a function of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The space of continuous functions is denoted by C([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 2 Standard concepts from nonsmooth analysis will also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Those can be found in [7], [19] or [22], to name but a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The Mordukhovich normal cone to a set S at s ∈ S is denoted by NS(s) and ∂f(s) is the Mordukhovich subdifferential of f at s (also known as limiting subdifferential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For any set A ⊂ Rn, cone A is the cone generated by the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We now turn to problem (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We first state the definition of admissible processes for (P) and then we describe the assumptions under which we will derive our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 A pair (x, u) is called an admissible process for (P) when x is an absolutely continuous function and u is a measurable function satisfying the constraints of (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Assumptions on the data of (P) A1: The function ψi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, are C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The graph of C(·) is compact and it is contained in the interior of a ball rBn+1, for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' There exist constants β > 0, η > 0 and ρ ∈]0, 1[ such that ψi(t, x) ∈ [−β, β] =⇒ |∇xψi(t, x)|> η forall (t, x) ∈ [0, T] × Rn, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) and, for I(t, x) = {i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I | ψi(t, x) ∈] − 2β, β]}, ⟨∇xψi(t, x), ∇xψj(t, x)⟩ ≥ 0, i, j ∈ I(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2) Moreover, if i ∈ I(t, x), then � j∈I(t,x)\\{i} |⟨∇xψi(t, x), ∇xψj(t, x)⟩| ≤ ρ|∇xψi(t, x)|2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) and ψi(t, x) ≤ −2β =⇒ ∇ψi(t, x) = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='4) A2: The function f is continuous, x → f(t, x, u) is continuously differentiable for all (t, u) ∈ [0, T]×Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The constant M > 0 is such that |f(t, x, u)|≤ M and |∇xf(t, x, u)|≤ M for all (t, x, u) ∈ rBn+1×U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' A3: For each (t, x), the set f(t, x, U) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' A4: The set U is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' A5: The sets C0 and CT are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' A6: There exists a constant Lφ such that |φ(x) − φ(x′)|≤ Lφ|x − x′| for all x, x′ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Assumption (A1) concerns the functions ψi defining the set C and it plays a crucial role in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' All ψi are assumed to be smooth with gradients bounded away from the origin when ψi takes values in a neighorhood of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Moreover, the boundary of C may be nonsmooth at the intersection points of the level sets � x : ψi(t, x) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' However, nonsmoothness at those corner points is restricted to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2) which excludes the cases where the angle between the two gradients of the functions defining the boundary of C is obtuse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' On the other hand, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) guarantees that the Gramian matrix of the gradients of the functions taking values near the boundary of C(t) is diagonally dominant and, hence, the gradients are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In many situations, as in the example we present in the last section, we can guarantee the fulfillment of (A1), in particular (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='4), replacing the function ψi by ˜ψi(t, x) = h ◦ ψi(t, x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='5) 3 c Y1=0 Y2=0 ▽ Y2 Not allowed Allowed ▽ Y1 C Y2=0 ▽ Y2 ▽ Y1 Y1=0 Figure 1: Examples of two diferent sets C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' On the left size, a set that does not satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' On the right side, the set C is nonsmooth and it fulfils (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' where h(z) = � � � z if z > −β, hs(z) if −2β ≤ z ≤ −β, −2β if z < −2β, Here, h is an C2 function, with hs an increasing function defined on [−2β, −β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For example, h may be a cubic polinomial with positive derivative on the interval ] − 2β, −β[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For all t ∈ [0, T], set ˜C(t) := � x ∈ R : ˜ψi(t, x) ≤ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It is then a simple matter to see that C(t) = ˜C(t) for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' and that the functions ˜ψi(·) satisfy the assumption (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The assumption that the graph of C(·) is compact and contained in the interior of a ball is introduced to avoid technicalities in our forthcoming analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' In applied problems, this may be easily side tracked by considering the intersection of the graph of C(·) with a tube around the optimal trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We now proceed introducing an approximation family of controlled systems to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let x(·) be a solution to the differential inclusion ˙x(t) ∈ f(t, x(t), U) − NC(t)(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Under our assumptions, measurable selection theorems assert the existence of measurable functions u and ξi such that u(t) ∈ U, ξi(t) ≥ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], ξi(t) = 0 if ψi(t, x(t)) < 0, and ˙x(t) = f(t, x(t), u(t)) − I � i=1 ξi(t)∇xψi(t, x(t)) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Considering the trajectory x, some observations are called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let µ be such that max � (|∇xψi(t, x)||f(t, x, u)|+|∂tψi(t, x)|) + 1 : t ∈ [0, T], u ∈ U, x ∈ C(t) + Bn, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I} ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The properties of the graph of C(·) in (A1) guarantee the existence of such maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 4 Consider now some t such that, for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' I}, ψj(t, x(t)) = 0 and ˙x(t) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since the trajectory x is always in C, we have (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2)) 0 = d dtψj(t, x(t)) = ⟨∇xψj(t, x(t)), ˙x(t)⟩ + ∂tψj(t, x(t)) = ⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ − ξj(t)|∇xψj(t, x(t))|2 − � i∈I(t,x(t))\\{j} ξi(t)⟨∇xψi(t, x(t)), ∇xψj(t, x(t))⟩ + ∂tψj(t, x(t)) ≤ ⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ − ξj(t)|∇xψj(t, x(t))|2+∂tψj(t, x(t)), and, hence (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1)), ξj(t) ≤ 1 |∇xψj(t, x(t))|2 (⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ + ∂tψj(t, x(t))) ≤ µ η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Define the function µ(γ) = 1 γ log � µ η2γ � , γ > 0, consider a sequence {σk} such that σk ↓ 0 and choose another sequence {γk} with γk ↑ +∞ and C(t) ⊂ int Ck(t) = int � x : ψi(t, x) − σk ≤ µk, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I � , where µk = µ(γk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let xk be a solution to the differential equation ˙xk(t) = f(t, xk(t), uk(t)) − I � i=1 γkeγk(ψi(t,xk(t))−σk)∇xψi(t, xk(t)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='6) for some uk(t) ∈ U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Take any t ∈ [0, T] such that ˙xk(t) exists and ψj(t, xk(t)) − σk = µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 5 Assume k is such that j ∈ I(t, xk(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' whenever γk is sufficiently large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' we have d dtψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)) = ⟨∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' uk(t))⟩ − γkeγk(ψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t))−σk)|∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t))|2 − � i∈I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t))\\{j} γkeγk(ψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t))−σk)⟨∇xψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t))⟩ − � i̸∈I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t)) γkeγk(ψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t))−σk)⟨∇xψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t))⟩ + ∂tψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)) ≤ ⟨∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' uk(t))⟩ − γkeγk(ψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t))−σk)|∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t))|2 − � i̸∈I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t)) γkeγk(ψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t))−σk)⟨∇xψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t))⟩ + ∂tψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)) ≤ ⟨∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' uk(t))⟩ − γkeγk(ψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t))−σk)|∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t))|2 + � i̸∈I(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='xk(t)) γkeγk(−2β−σk)|⟨∇xψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t))⟩| + ∂tψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' xk(t)) ≤ µ − 1 2 − η2γkeγkµk = −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Above, we have used the definition of µ and the inequality � i̸∈I(t,xk(t)) γkeγk(−2β−σk)|⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩|≤ 1 2, which holds for γk sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Now, if xk(0) ∈ Ck(0), we assure that xk(t) ∈ Ck(t), for all t ∈ [0, T], and γkeγk(ψj(t,xk(t))−σk) ≤ γkeγkµk = µ η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='7) It follows that, for k sufficienttly large, we have | ˙xk(t)|≤ (const).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We are now a in position to state and prove our first result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This is in the vein of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 in [23] (see also Lemma 1 in [10] when ψ is independent of t and convex) deviating from it in so far as the approximating sequence of control systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='6) differs from the one introduced in [10]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 relies on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 Let {(xk, uk)}, with uk(t) ∈ U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', be a sequence of solutions of Cauchy problems � � � � � ˙xk(t) = f(t, xk(t), uk(t)) − I � i=1 γkeγk(ψi(t,xk(t))−σk)∇xψi(t, xk(t)), xk(0) = bk ∈ Ck(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='8) 1See also Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 in [14] 6 If bk → x0, then there exists a subsequence {xk} (we do not relabel) converging uniformly to x, a unique solution to the Cauchy problem ˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), x(0) = x0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='9) where u is a measurable function such that u(t) ∈ U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' If, moreover, all the controls uk are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', uk = u, then the subsequence converges to a unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='9), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', any solution of ˙x(t) ∈ f(t, x(t), U) − NC(t)(x(t)), x(0) = x0 ∈ C(0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='10) can be approximated by solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Proof Consider the sequence {xk}, where (xk, uk) solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Recall that xk(t) ∈ Ck(t) for all t ∈ [0, T], and | ˙xk(t)|≤ (const) and ξi k(t) = γkeγk(ψi(t,xk(t))−σk) ≤ (const).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='11) Then there exist subsequences (we do not relabel) weakly-∗ converging in L∞ to some v and ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Hence xk(t) = x0 + � t 0 ˙xk(s)ds −→ x(t) = x0 + � t 0 v(s)ds, ∀ t ∈ [0, T], for an absolutely continuous function x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Obviously, x(t) ∈ C(t) for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Considering the sequence {xk}, recall that ˙xk(t) ∈ f(t, xk(t), U) − I � i=1 ξi k(t)∇xψi(t, xk(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='12) Inclusion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='12) is equivalent to ⟨z, ˙xk(t)⟩ ≤ S(z, f(t, xk(t), U)) − I � i=1 ξi k(t)⟨z, ∇xψi(t, xk(t))⟩, ∀ z ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Integrating this inequality, we get � z, xk(t + τ) − xk(t) τ � ≤ 1 τ � t+τ t � S(z, f(s, xk(s), U)) − I � i=1 ξi k(s)⟨z, ∇xψi(s, xk(s))⟩ � ds = 1 τ � t+τ t � S(z, f(s, xk(s), U)) − I � i=1 ξi k(s)⟨z, ∇xψi(s, x(s))⟩ + I � i=1 ξi k(s)⟨z, ∇xψi(s, x(s)) − ∇xψi(s, xk(s))⟩ � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='13) Passing to the limit as k → ∞, we obtain � z, x(t + τ) − x(t) τ � ≤ 1 τ � t+τ t � S(z, f(s, x(s), U)) − I � i=1 ξi(s)⟨z, ∇xψi(s, x(s))⟩ � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='14) 7 Let t ∈ [0, T] be a Lebesgue point of x and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Passing in the last inequality to the limit as τ ↓ 0, it leads to ⟨z, ˙x(t)⟩ ≤ S(z, f(t, x(t), U)) − I � i=1 ξi(t)⟨z, ∇xψi(t, x(t))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since z ∈ Rn is an arbitrary vector and the set f(t, x(t), U) is convex, we conclude that ˙x(t) ∈ f(t, x(t), U) − I � i=1 ξi(t)∇xψi(t, x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' By the Filippov lemma there exists a measurable control u(t) ∈ U such that ˙x(t) = f(t, x(t), u(t)) − I � i=1 ξi(t)∇xψi(t, x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Furthermore, observe that ξi is zero if ψi(t, x(t)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' If for some u such that u(t) ∈ U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', uk = u for all k, then the sequence xk converges to the solution of ˙x(t) = f(t, x(t), u(t)) − I � i=1 ξi(t)∇xψi(t, x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Indeed, to see this, it suffices to pass to the limit as k → ∞ and then as τ ↓ 0, in the equality xk(t + τ) − xk(t) τ = 1 τ � t+τ t � f(s, xk(s), u(s)) − I � i=1 ξi k(s)∇xψi(s, xk(s)) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We now prove the uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We follow the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Notice, however, that we now consider a special case and not the general case treated in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Suppose that there exist two different solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='9): x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We have 1 2 d dt|x1(t) − x2(t)|2= ⟨x1(t) − x2(t), ˙x1(t) − ˙x2(t)⟩ = ⟨x1(t) − x2(t), f(t, x1(t), u(t)) − f(t, x2(t), u(t))⟩ − � x1(t) − x2(t), I � i=1 ξi 1(t)∇ψi(t, x1(t)) − I � i=1 ξi 2(t)∇ψi(t, x2(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='15) If, for all i, ψi(t, x1(t)) < 0 and ψi(t, x2(t)) < 0, then ξi 1(t) = ξi 2(t) = 0 and we obtain 1 2 d dt|x1(t) − x2(t)|2≤ Lf|x1(t) − x2(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Suppose that ψj(t, x1(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then by the Taylor formula we get ψj(t, x2(t)) = ψj(t, x1(t)) + ⟨∇xψj(t, x1(t)), x2(t) − x1(t)⟩ + 1 2⟨x2(t) − x1(t), ∇2 xψj(t, θx2(t) + (1 − θ)x1(t))(x2(t) − x1(t))⟩, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='16) where θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since ψj(t, x2(t)) ≤ 0, we have ⟨∇xψj(t, x1(t)), x2(t) − x1(t)⟩ ≤ −1 2⟨x2(t) − x1(t), ∇2 xψj(t, θx2(t) + (1 − θ)x1(t))(x2(t) − x1(t))⟩ ≤ (const)|x1(t) − x2(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='17) 8 Now, if ψj(t, x2(t)) = 0, we deduce in the same way that ⟨∇xψj(t, x2(t)), x1(t) − x2(t)⟩ ≤ (const)|x1(t) − x2(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Thus we have 1 2 d dt|x1(t) − x2(t)|2≤ (const)|x1(t) − x2(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Hence |x1(t) − x2(t)|= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 2 3 Approximating Family of Optimal Control Problems In this section we define an approximating family of optimal control problems to (P) and we state the corresponding necessary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let (ˆx, ˆu) be a global solution to (P) and consider sequences {γk} and {σk} as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let ˆxk(·) be the solution to � � � � � ˙x(t) = f(t, x(t), ˆu(t)) − I � i=1 γkeγk(ψi(t,x(t))−σk)∇xψi(t, x(t)), x(0) = ˆx(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) Set ϵk = |ˆxk(T) − ˆx(T)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 that ϵk ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Take α > 0 and define the problem (P α k ) � � � � � � � � � � � � � � � � � � � � � � � � � Minimize φ(x(T)) + |x(0) − ˆx(0)|2+α � T 0 |u(t) − ˆu(t)|dt over processes (x, u) such that ˙x(t) = f(t, x(t), u(t)) − I � i=1 ∇xeγk(ψi(t,x(t))−σk) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], u(t) ∈ U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], x(0) ∈ C0, x(T) ∈ CT + ϵkBn, Clearly, the problem (P α k ) has admissible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Consider the space W = {(c, u) | c ∈ C0, u ∈ L∞ with u(t) ∈ U} and the distance dW ((c1, u1), (c2, u2)) = |c1 − c2|+ � T 0 |u1(t) − u2(t)|dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Endowed with dW , W is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Take any (c, u) ∈ W and a solution y to the Cauchy problem � � � � � ˙y(t) = f(t, y(t), u(t)) − I � i=1 ∇xeγk(ψi(t,y(t))−σk) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], y(0) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Under our assumptions, the function (c, u) → φ(y(T)) + |c − ˆx(0)|2+α � T 0 |u − ˆu| dt 9 is continuous on (W, dW ) and bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Appealing to Ekeland’s Theorem we deduce the existence of a pair (xk, uk) solving the following problem (APk) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Minimize Φ(x, u) = φ(x(T)) + |x(0) − ˆx(0)|2+α � T 0 |u(t) − ˆu(t)|dt +ϵk � |x(0) − xk(0)|+ � T 0 |u(t) − uk(t)|dt � , over processes (x, u) such that ˙x(t) = f(t, x(t), u(t)) − I � i=1 ∇xeγk(ψi(t,x(t))−σk) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], u(t) ∈ U a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], x(0) ∈ C0, x(T) ∈ CT + ϵkBn, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 Take γk → ∞, σk → 0 and ϵk → 0 as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For each k, let (xk, uk) be the solution to (APk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then there exists a subsequence (we do not relabel) such that uk(t) → ˆu(t) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', xk → ˆx uniformly in [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Proof We deduce from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 that {xk} uniformly converges to an admissible solution ˜x to (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since U and C0 are compact, we have U ⊂ KBm and C0 ⊂ KBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Without loss of generality, uk weakly-∗ converges to a function ˜u ∈ L∞([0, T], U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Hence it weakly converges to ˜u in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' From optimality of the processes (xk, uk) we have φ(xk(T)) + |xk(0) − ˆx(0)|2+α � T 0 |uk(t) − ˆu(t)|dt ≤ φ(ˆxk(T)) + ϵk � |ˆxk(0) − xk(0)|+ � T 0 |uk(t) − ˆu(t)|dt � ≤ φ(ˆxk(T)) + 2K(1 + T)ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since (ˆx, ˆu) is a global solution of the problem, passing to the limit, we get φ(˜x(T)) + |˜x(0) − ˆx(0)|2+α � T 0 |˜u(t) − ˆu(t)|dt ≤ lim k→∞(φ(xk(T)) + |xk(0) − ˆx(0)|2) + α lim inf k→∞ � T 0 |uk(t) − ˆu(t)|dt ≤ lim k→∞ φ(ˆxk(T)) = φ(ˆx(T)) ≤ φ(˜x(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Hence ˜x(0) = ˆx(0), ˜u = ˆu a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', and uk converges to ˆu in L1, and some subsequence converges to ˆu almost everywhere (we do not relabel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 2 We now finish this section with the statement of the optimality necessary conditions for the family of problems (APk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' These can be seen as a direct consequence of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 For each k, let (xk, uk) be a solution to (APk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then there exist absolutely continous functions pk and scalars λk ≥ 0 such that (a) (nontriviality condition) λk + |pk(T)|= 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2) 10 (b) (adjoint equation) ˙pk = −(∇xfk)∗pk + �I i=1 γkeγk(ψi k−σk)∇2 xψi kpk + �I i=1 γ2 keγk(ψi k−σk)∇xψi k⟨∇xψi k, pk⟩, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) where the superscript ∗ stands for transpose, (c) (maximization condition) max u∈U {⟨f(t, xk, u), pk⟩ − αλk|u − ˆu|−ϵkλk|u − uk|} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='4) is attained at uk(t), for almost every t ∈ [0, T], (d) (transversality condition) (pk(0), −pk(T)) ∈ λk (2(xk(0) − ˆx(0)) + ϵkBn, ∂φ(xk(T))) +NC0(xk(0)) × NCT +ϵkBn(xk(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='5) To simplify the notation above, we drop the t dependance in pk, ˙pk, xk, uk, ˆx and ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Moreover, in (b), we write ψk instead of ψ(t, xk(t)), fk instead of f(t, xk(t), uk(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The same holds for the derivatives of ψ and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 4 Maximum Principle for (P) In this section, we establish our main result, a Maximum Principle for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This is done by taking limits of the conclusions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2, following closely the analysis done in the proof of [10, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Observe that 1 2 d dt|pk(t)|2 = −⟨∇xfkpk, pk⟩ + I � i=1 γkeγk(ψi k−σk)⟨∇2 xψi kpk, pk⟩ + I � i=1 γ2 keγk(ψi k−σk)⟨∇xψi k, pk⟩2 ≥ −⟨∇xfkpk, pk⟩ + I � i=1 γkeγk(ψi k−σk)⟨∇2 xψi kpk, pk⟩ ≥ −M|pk|2+ I � i=1 γkeγk(ψi k−σk)⟨∇2 xψi kpk, pk⟩, where M is the constant of (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Taking into account hypothesis (A1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='7) we deduce the existence of a constant K0 > 0 such that 1 2 d dt|pk(t)|2≥ −K0|pk(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This last inequality leads to |pk(t)|2 ≤ e2K0(T −t)|pk(T)|2≤ e2K0T |pk(T)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since, by (a) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2, |pk(T)|≤ 1, we deduce from the above that there exists M0 > 0 such that |pk(t)| ≤ M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) Now, we claim that the sequence { ˙pk} is uniformly bounded in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' To prove our claim, we need to establish bounds for the three terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Following [10] and [14], we start by deducing some inequalities that will be of help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 11 Denote Ik = I(t, xk(t)) and Sj k = sign � ⟨∇xψj k, pk⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We have I � j=1 d dt ���⟨∇xψj k, pk⟩ ��� = I � j=1 � ⟨∇2 xψj k ˙xk, pk⟩ + ⟨∂t∇xψj k, pk⟩ + ⟨∇xψj k, ˙pk⟩ � Sj k = I � j=1 � ⟨pk, ∇2 xψj kfk⟩ − I � i=1 γkeγk(ψi k−σk)⟨pk, ∇2ψj k∇xψi k⟩ � Sj k + I � j=1 � ⟨∂t∇xψj k, pk⟩ − ⟨∇xψj k, (∇xfk)∗pk⟩ � Sj k + I � j=1 � I � i=1 γkeγk(ψi k−σk)⟨∇xψj k, ∇2 xψi kpk⟩ � Sj k + I � i=1 I � j=1 γ2 keγk(ψi k−σk)⟨∇xψj k, ∇xψi k⟩⟨∇xψi k, pk⟩Sj k Observe that (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='4)) I � i=1 I � j=1 γ2 keγk(ψi k−σk)⟨∇xψj k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψi k⟩⟨∇xψi k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' pk⟩Sj k = I � i=1 � j∈Ik γ2 keγk(ψi k−σk)⟨∇xψj k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψi k⟩⟨∇xψi k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' pk⟩Sj k = � i̸∈Ik γ2 keγk(ψi k−σk) � j∈Ik ⟨∇xψj k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψi k⟩⟨∇xψi k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' pk⟩Sj k + � i∈Ik γ2 keγk(ψi k−σk) � �|∇xψi k|2+ � j∈Ik\\{i} ⟨∇xψj k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψi k⟩Sj k Si k � � |⟨∇xψi k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' pk⟩| = � i∈Ik γ2 keγk(ψi k−σk) � �|∇xψi k|2+ � j∈Ik\\{i} ⟨∇xψj k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ∇xψi k⟩Sj k Si k � � |⟨∇xψi k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' pk⟩| ≥ (1 − ρ) � i∈Ik γ2 keγk(ψi k−σk)|∇xψi k|2|⟨∇xψi k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' pk⟩| = (1 − ρ) I � i=1 γ2 keγk(ψi k−σk)|∇xψi k|2|⟨∇xψi k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' pk⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Using this and integrating the previous equality, we deduce the existence of M1 > 0 such that: � T 0 I � i=1 γ2 keγk(ψi k−σk)|∇xψi k|2|⟨∇xψi k, pk⟩|dt ≤ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2) We are now in a position to show that � T 0 I � i=1 γ2 keγk(ψi k−σk)|∇xψi k| ��⟨∇xψi k, pk⟩ �� dt 12 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For simplicity, set Li k(t) = γ2 keγk(ψi k−σk)|∇xψi k| ��⟨∇xψi k, pk⟩ ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Notice that I � i=1 � T 0 Li k(t)dt = I � i=1 �� {t:|∇xψi k|<η} Li k(t) dt + � {t:|∇xψi k|≥η} Li k(t)dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Using (A1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2), we deduce that I � i=1 � T 0 Li k(t) dt ≤ I � i=1 � γ2 ke−γk(β+σk)η2 max t |pk(t)| � + I � i=1 � γ2 k � {t:|∇xψi k|≥η} eγk(ψi k−σk) |∇xψi k|2 |∇xψi k| ��⟨∇xψi k, pk⟩ �� dt � ≤ γ2 kI e−γk(β+σk)η2M0 + 1 η I � i=1 �� T 0 γ2 keγk(ψi k−σk)|∇xψi k|2��⟨∇xψi k, pk⟩ �� dt � ≤ η2M0I + M1 η , for k large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Summarizing, there exists a M2 > 0 such that I � i=1 γ2 k � T 0 eγk(ψi k−σk)|∇ψi k| ��⟨∇ψi k, pk⟩ �� dt ≤ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) Mimicking the analysis conducted in Step 1, b) and c) of the proof of Theorem 2 in [10] and taking into account (b) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 we conclude that there exist constants N1 > 0 such that � T 0 | ˙pγk(t)| dt ≤ N1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='4) for k sufficiently large, proving our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Before proceeding, observe that it is a simple matter to assert the existence of a constant N2 such that I � i=1 � T 0 γ2 keγk(ψi k−σk)|⟨∇ψi k, pγk⟩|dt ≤ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='5) This inequality will be of help in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let us now recall that ξi k(t) = γkeγk(ψi(t,xk(t))−σk) and that the second inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We turn to the analysis of Step 2 in the proof of Theorem 2 in [10] (see also [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Adapting those arguments, we can conclude the existence of some function p ∈ BV ([0, T], Rn) and, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, functions ξi ∈ L∞([0, T], R) with ξi(t) ≥ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t, ξi(t) = 0, t ∈ Ii b, where Ii b = � t ∈ [0, T] : ψi(t, ˆx(t)) < 0 � , and finite signed Radon measures ηi, null in Ii b, such that, for any z ∈ C([0, T], Rn) � T 0 ⟨z, dp⟩ = − � T 0 ⟨z, (∇ ˆf)∗p⟩dt + I � i=1 �� T 0 ξi⟨z, ∇2 ˆψip⟩dt + � T 0 ⟨z, ∇ ˆψi(t)⟩dηi � , where ∇ ˆψi(t) = ∇ψi(t, ˆx(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' The finite signed Radon measures ηi are weak-∗ limits of γ2 keγk(ψi k−σk)⟨∇ψi k(xk(t), pk(t)⟩dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 13 Observe that the measures ⟨∇ψi(ˆx(t), p(t)⟩dηi(t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='6) are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, the sequence ξi k is weakly-∗ convergent in L∞ to ξi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Following [14], we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='5) that, for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, � T 0 |ξi⟨∇x ˆψi, p⟩|dt = lim k→∞ � T 0 |ξi k⟨∇x ˆψi, p⟩|dt ≤ lim k→∞ �� T 0 ξi k|⟨∇x ˆψi, p⟩ − ⟨∇xψi k, pk⟩|dt + � T 0 ξi k|⟨∇xψi k, pk⟩|dt � ≤ lim k→∞ ����ξi k ��� L∞ ���⟨∇x ˆψi, p⟩ − ⟨∇xψi k, pk⟩ ��� L1 + N2 γk � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It turns out that ξi⟨∇x ˆψi, p⟩ = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='. (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='7) Consider now the sequence of scalars {λk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It is an easy matter to show that there exists a subsequence of {λk} converging to some λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This, together with the convergence of pk to p, allows us to take limits in (a) and (c) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 to deduce that λ + |p(T)|= 1 and ⟨p(t), f(t, ˆx(t), u)⟩ − αλ|u − ˆu(t)|≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ ∀u ∈ U, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It remains to take limits of the transversality conditions (d) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' First, observe that CT + ϵkBn = {x : d(x, CT ) ≤ ϵk} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' From the basic properties of the Mordukhovich normal cone and subdifferential (see [19], section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3) we have NCT +ϵkBn(xk(T)) ⊂ cl cone ∂d(xk(T), CT ) and NCT (ˆx(T)) = cl cone ∂d(ˆx(T), CT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Passing to the limit as k → ∞ we get (p(0), −p(T)) ∈ NC0(ˆx(0)) × NCT (ˆx(T)) + {0} × λ ∂φ(ˆx(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Finally, and mimicking Step 3 in the proof of Theorem 2 in [10], we remove the dependence of the conditions on the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This is done by taking further limits, this time considering a sequence of αj ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We then summarize our conclusions in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 Let (ˆx, ˆu) be the optimal solution to (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Suppose that assumption A1–A6 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For i = 1, · · · , I, set Ii b = {t ∈ [0, T] : ψi(t, ˆx(t)) < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' There exist λ ≥ 0, p ∈ BV ([0, T], Rn), finite signed Randon measures ηi, null in Ii b, for i = 1, · · · , I, ξi ∈ L∞([0, T], R), with i = 1, · · · , I, where ξi(t) ≥ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t and ξi(t) = 0, t ∈ Ii b, such that a) λ + |p(T)|̸= 0, 14 b) ˙ˆx(t) = f(t, ˆx(t), ˆu(t)) − I � i=1 ξi(t)∇x ˆψi(t), c) for any z ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Rn) � T 0 ⟨z(t), dp(t)⟩ = − � T 0 ⟨z(t), (∇x ˆf(t))∗p(t)⟩dt + I � i=1 �� T 0 ξi(t)⟨z(t), ∇2 x ˆψi(t)p(t)⟩dt + � T 0 ⟨z(t), ∇x ˆψi(t)⟩dηi � , where ∇ ˆf(t) = ∇xf(t, ˆx(t), ˆu(t)), ∇ ˆψi(t) = ∇ψi(t, ˆx(t)) and ∇2 ˆψi(t) = ∇2ψi(t, x(t)), d) ξi(t)⟨∇xψi(t, ˆx(t)), p(t)⟩ = 0, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, e) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, the meaures ⟨∇ψi(ˆx(t), p(t)⟩dηi(t) are nonnegative, f) ⟨p(t), f(t, ˆx(t), u)⟩ ≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ for all u ∈ U, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t, g) (p(0), −p(T)) ∈ NC0(ˆx(0)) × NCT (ˆx(T)) + {0} × λ∂φ(ˆx(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Noteworthy, condition e) is not considered in any of our previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We now turn to the free end point case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', to the problem (Pf) � � � � � � � � � � � � � � � � � Minimize φ(x(T)) over processes (x, u) such that ˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], u(t) ∈ U, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ [0, T], x(0) ∈ C0 ⊂ C(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Problem (Pf) differs from (P) because x(T) is not constrained to take values in CT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 to (Pf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since x(T) is free, we deduce from (f) in the above Theorem that −p(T) = λ∂φ(ˆx(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Suppose that λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then p(T) = 0 contradicting the nontriviality condition (a) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Without loss of generality, we then conclude that the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 hold with λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We summarize our findings in the following Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2 Let (ˆx, ˆu) be the optimal solution to (Pf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Suppose that assumption A1–A6 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' For i = 1, · · · , I, set Ii b = {t ∈ [0, T] : ψi(t, ˆx(t)) < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' There exist p ∈ BV ([0, T], Rn), finite signed Randon measures ηi, null in Ii b, for i = 1, · · · , I, ξi ∈ L∞([0, T], R), with i = 1, · · · , I, where ξi(t) ≥ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t and ξi(t) = 0 for t ∈ Ii b, such that a) ˙ˆx(t) = f(t, ˆx(t), ˆu(t)) − I � i=1 ξi(t)∇x ˆψi(t), 15 b) for any z ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Rn) � T 0 ⟨z(t), dp(t)⟩ = − � T 0 ⟨z(t), (∇x ˆf(t))∗p(t)⟩dt + I � i=1 �� T 0 ξi(t)⟨z(t), ∇2 x ˆψi(t)p(t)⟩dt + � T 0 ⟨z(t), ∇x ˆψi(t)⟩dηi � , where ∇ ˆf(t) = ∇xf(t, ˆx(t), ˆu(t)), ∇ ˆψi(t) = ∇ψi(t, ˆx(t)) and ∇2 ˆψi(t) = ∇2ψi(t, x(t)), c) ξi(t)⟨∇xψi(t, ˆx(t)), p(t)⟩ = 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t and for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, d) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' , I, the meaures ⟨∇ψi(ˆx(t), p(t)⟩dηi(t) are nonnegative, e) ⟨p(t), f(t, ˆx(t), u)⟩ ≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ for all u ∈ U, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t, f) (p(0), −p(T)) ∈ NC0(ˆx(0)) × {0} + {0} × ∂φ(ˆx(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 5 Example Let us consider the following problem � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Minimize − x(T) over processes ((x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' u) such that � � ˙x(t) ˙y(t) ˙z(t) � � ∈ � � 0 σ 0 0 0 0 0 0 0 � � � � x y z � � + � � 0 u 0 � � − NC(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' u ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z)(0) = (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z)(T) ∈ CT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' where 0 < σ ≪ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' C = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z) | x2 + y2 + (z + h)2 ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' x2 + y2 + (z − h)2 ≤ 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 2h2 < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z0) ∈ intC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' with x0 < −δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y0 = 0 and z0 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' CT = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' z) | x ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' y ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' δy − y2x ≤ δy2} ∩ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' where δ < y2|x0| y1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' with y1 = � 1 − x2 0 − (z0 + h)2 and y2 = � 1 − h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We choose T > 0 small and, nonetheless, sufficiently large to guarantee that, when σ = 0, the system can reach the interior of CT but not the segment {(x, 0, 0) | x ∈ [−δ, 0]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since σ and T are small, it follows that the optimal trajectory should reach CT at the face δy − y2x = δy2 of CT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 16 To significantly increase the value of the x(T), the optimal trajectory needs to live on the boundary of C for some interval of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then, before reaching and after leaving the boundary of C, the optimal trajectory lives in the interior of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since δ is small, the trajectory cannot reach CT from any point of the sphere x2 +y2 +(z +h)2 = 1 with z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' This means that, while on the boundary of C the trajectory should move on the sphere x2 + y2 + (z + h)2 = 1 untill reaching the plane z = 0 and then it moves on the intersection of the two spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' While in the interior of C, the control can change sign from −1 to 1 or from 1 to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Certainly, the control should be 1 right before reaching the boundary and −1 right before arriving at CT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Changes of the control from 1 to −1 or −1 to 1 before reaching the boundary translate into time waste and leads to smaller values of x(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It then follows that the optimal control should be of the form u(t) = � 1, t ∈ [0, ˜t], −1, t ∈ ]˜t, T], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) for some value ˜t ∈]0, T[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' After the modification (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='5), the data of the problem satisfy the conditions under which Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We now show that the conclusions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 completly identify the structure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) of the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1 we deduce the existence of λ ≥ 0, p, q, r ∈ BV ([0, T], R), finite signed Randon measures η1 and η2, null respectively in I1 b = � (x, y, z) | x2 + y2 + (z + h)2 − 1 < 0 � and I2 b = � (x, y, z) | x2 + y2 + (z − h)2 − 1 < 0 � , ξi ∈ L∞([0, T], R), with i = 1, 2, where ξi(t) ≥ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t and ξi(t) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' t ∈ Ii b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' such that (i) � � ˙x(t) ˙y(t) ˙z(t) � � = � � 0 σ 0 0 0 0 0 0 0 � � � � x y z � � + � � 0 u 0 � � − 2ξ1 � � x y z + h � � − 2ξ2 � � x y z − h � � (ii) d � � p q r � � = � � 0 0 0 −σ 0 0 0 0 0 � � � � p q r � � dt +2(ξ1 + ξ2) � � p q r � � dt + 2 � � x y z + h � � dη1 + 2 � � x y z − h � � dη2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (iii) � � p q r � � (T) = � � λ 0 0 � � + µ � � y2 −δ 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' where µ ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (iv) ξ1(xp + yq + (z + h)r) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' ξ2(xp + yq + (z − h)r) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (v) the meaures (xp + yq + (z + h)r)dη1 and (xp + yq + (z − h)r)dη2 are nonnegative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (vi) maxu∈[−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1] uq = ˆuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' where ˆu is the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let t1 be the instant of time when the trajectory reaches the shere x2 + y2 + (z + h)2 = 1, t2 the instant of time when the trajectory reaches the intersection of the two spheres and t3 be the instant of time the trajectory leaves the boundary of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We have 0 < t1 < t2 < t3 < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Next we show that the multiplier q changes sign only once and so identifing the structure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) of the optimal control in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We start by looking at the case when t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We have � p q � (T) = � λ 0 � + µ � y2 −δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 17 Starting from t = T, let us go backwards in time until the instant t3 when the trajectory leaves the boundary of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' If q(T) = 0, then p(T) = λ > 0 and we would have q(t) > 0 for t ∈]t3, T[ (see (ii) above), which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We then have p(T) > 0 and q(T) < 0 and, in ]t3, T[, since σ is small, the vector (p(t), q(t)) does not change much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' At t = t3, the vector (p, q) has a jump and such jump can only occur along the vector (x(t3), y(t3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Therefore, we have p(t3 − 0) > 0 and q(t3 − 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let us now consider t ∈]t2, t3[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We have the following 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' when t ∈ [t2, t3], we have z = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' condition (i) above implies that ξ1 = ξ2 = ξ, ξ > 0 since, otherwise the motion along x2+y2 = 1−h2 would not be possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' from 0 = d dt(x2 + y2) = σ2xy − 8ξx2 + 2uy − 8ξy2 we get ξ = σxy+uy 4(1−h2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' condition (iv) implies that r = 0 leading to xp + yq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since x < 0, y > 0, then q = 0 implies p = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' condition (ii) implies dη1 = dη2 = dη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 0 = d(xp + yq) = uqdt + 4(1 − h2)dη ⇒ dη dt = − uq 4(1−h2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' from the above analysis we deduce that ˙p = σxy + uy (1 − h2) p − xuq (1 − h2), ˙q = −σp + σxy (1 − h2) q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Thus, (p, q) is a solution to a linear system and it can never be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It follows that q cannot be zero because q = 0 implies p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Since q ̸= 0, we have q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let us consider the case when t = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We claim that (p(t2 − 0), q(t2 − 0)) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Seeking a contradiction, assume that it is (p(t2 − 0), q(t2 − 0)) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then we have (p(t2 + 0), q(t2 + 0)) = (0, 0) + (2x2(t2), 2y2(t2))(dη1 + dη2) and such jump has to be normal to (x(t2), y(t2)) since r(t2 + 0) = 0 (see (iv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It follows that (x2(t2) + y2(t2))(dη1 + dη2) = 0 and, since x2(t2) + y2(t2) > 0, we get dη1 + dη2 = 0, proving our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We now consider t ∈]t1, t2[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It is easy to see that ξ2 = 0 and dη2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We also deduce that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 0 = d dt(x2+y2+(z+h)2) = 2σxy+2uy−4ξ1y2−4ξ1x2−4ξ1(z+h)2 which implies that ξ1 = σxy+uy 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' also 0 = d(xp + yq + (z + h)r) = uqdt + 2dη1 implies that dη1 dt = − uq 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' from the above we deduce that ˙p = (σxy + uy)p − xuq, ˙q = −σp + σxyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Thus (p, q) is a solution to a linear system and never is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Second equation implies that if q = 0 then ˙q ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Hence q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 18 Now we need to consider t = t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We claim that (p(t1 − 0), q(t1 − 0), r(t1 − 0)) ̸= (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Let us then assume that it is (p(t1−0), q(t1−0), r(t1−0)) = (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It then follows that (p(t1+0), q(t1+ 0), r(t1 +0)) = (0, 0, 0)+(2x(t1)dη1, 2y(t1)dη1, 2(z(t1)+h)dη1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We now show that there is no such jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Set r(t1 − 0) = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Then it follows from (iv) that (x(t1) · 0 + y(t1) · 0 + (z(t1) + h))r0 = 0 which implies that r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' We also have (x2(t1) + y2(t1) + (z(t1) + h)2)dη1 = 0 from (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' But this implies that dη1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Consequently, the multipliers do not exhibit a jump at t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' From the previous analysis we deduce that q should be positive almost everywhere on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' It then follows that to find the optimal solution we have to analyze admissible trajectories with the controls with the structure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1) and choose the optimal value of ˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Acknowledgements The authors gratefully thank the support of Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Also we thank the support by the ERDF - European Regional Development Fund through the Oper- ational Programme for Competitiveness and Internationalisation - COMPETE 2020, INCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2030, under the Portugal 2020 Partnership Agreement and by National Funds, Norte 2020, through CCDRN and FCT, within projects To Chair (POCI-01-0145-FEDER-028247), Upwind (PTDC/EEI-AUT/31447/2017 POCI-01-0145-FEDER-031447) and Systec R&D unit (UIDB/00147/2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' References [1] Addy K, Adly S, Brogliato B, Goeleven D, A method using the approach of Moreau and Pana- giotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Hybrid Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 1, 30–43, (2013), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='nahs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [2] Arroud C and Colombo G, Necessary conditions for a nonclassical control problem with state constraints, 20th IFAC World Congress, Toulouse, France, July 9-14, 2017, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='ifacol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [3] Arroud C and Colombo G, A maximum principle for the controlled sweeping process, Set-Valued Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Anal 26, 607–629 (2018) DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1007/s11228-017-0400-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [4] Brokate M, Krejˇc´ı P Optimal control of ODE systems Involving a rate independent variational in- equality, Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' B, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 18 (2) 331–348 (2013), doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='3934/dcdsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [5] Cao TH, Mordukhovich B, Optimality conditions for a controlled sweeping process with applica- tions to the crowd motion model, Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' B, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 22, 267–306 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [6] Cao TH, Colombo G, Mordukhovich B, Nguyen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', Optimization of fully con- trolled sweeping processes, Journal of Differential Equations, 295, 138–186 (2021) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='jde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='042 [7] Clarke F, Optimization and nonsmooth analysis, John Wiley, New York (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [8] Colombo G, Palladino M, The minimum time function for the controlled Moreau’s sweeping process, SIAM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 4,2036– 2062 (2016), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1137/15M1043364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [9] Colombo G,Henrion R, Hoang ND, Mordukhovich BS, Optimal control of the sweeping process over polyhedral controlled sets, Journal of Differential Equations, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 260, 4, 3397–3447, (2016), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='jde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 19 [10] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control involving Sweeping Processes, Set- Valued Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Anal 27, 523–548, (2019), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1007/s11228-018-0501-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [11] de Pinho MdR, Ferreira MMA, Smirnov G, Correction to: Optimal Control Involving Sweeping Processes, Set-Valued Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Anal 27, 1025–1027 (2019) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1007/s11228-019-00520- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [12] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control with Sweeping Processes: Numerical Method, J Optim Theory Appl 185, 845– 858 (2020) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1007/s10957-020-01670-5 [13] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control Involving Sweeping Processes with End Point Constraints, 2021 60th IEEE Conference on Decision and Control (CDC), 2021, 96– 101(2019) doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1109/CDC45484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='9683291 [14] de Pinho MdR, Ferreira MMA, Smirnov G, Necessary conditions for optimal control problems with sweeping systems and end point constraints, Optimization, to appear (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [15] Hermosilla C, Palladino M, Optimal Control of the Sweeping Process with a Non-Smooth Moving Set, SIAM j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', to appear (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [16] Kunze M, Monteiro Marques MDP, An Introduction to Moreau’s sweeping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Impacts in Mechanical Systems, Lecture Notes in Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 551, 1–60, (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [17] Maury B, Venel J (2011), A discrete contact model for crowd motion, ESAIM: M2AN 45 1, 145–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [18] Moreau JJ, On unilateral constraints, friction and plasticity, In: Capriz G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=', Stampacchia G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=') New Variational Techniques in Mathematical Physics, CIME ciclo Bressanone 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Edizioni Cremonese, Rome, 171–322 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [19] Mordukhovich B, Variational analysis and generalized differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Basic Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Fundamental Principles of Mathematical Sciences 330, Springer-Verlag, Berlin (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [20] Mordukhovich B, Variational analysis and generalized differentiation II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Applications, Fundamen- tal Principles of Mathematical Sciences 330, Springer-Verlag, Berlin (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [21] Thibault L, Moreau sweeping process with bounded truncated retraction, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' Convex Anal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 23, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 1051–1098 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [22] Vinter RB, Optimal Control, Birkh¨auser, Systems and Control: Foundations and Applications, Boston MA (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' [23] Zeidan V, Nour C, Saoud H, A nonsmooth maximum principle for a controlled noncon- vex sweeping process, Journal of Differential Equations, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content=' 269 (11), 9531–9582 (2020), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='jde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} +page_content='053 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFRT4oBgHgl3EQfrTfI/content/2301.13620v1.pdf'} diff --git a/.gitattributes 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sha256:49964442a7d70d5e496ea4162197df4e7518a7d05209a38ec81ff4056a4463ff +size 265408 diff --git a/1NFPT4oBgHgl3EQfUTQQ/content/tmp_files/2301.13056v1.pdf.txt b/1NFPT4oBgHgl3EQfUTQQ/content/tmp_files/2301.13056v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b9bda01d75098959a2719b0ac9f908ddba85ad5d --- /dev/null +++ b/1NFPT4oBgHgl3EQfUTQQ/content/tmp_files/2301.13056v1.pdf.txt @@ -0,0 +1,1506 @@ +arXiv:2301.13056v1 [math.AG] 30 Jan 2023 +EQUIVARIANT ORIENTED HOMOLOGY OF THE AFFINE +GRASSMANNIAN +CHANGLONG ZHONG +Abstract. We generalize the property of small-torus equivariant K-homology of the affine Grass- +mannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool +we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we +prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the +GKM condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to +the centralizer of the equivariant oriented cohomology of a point in the the formal affine Demazure +algebra. +0. Introduction +Let h be an oriented cohomology theory in the sense of Levine and Morel. Let G be a semi-simple +linear algebraic group over C with maximal torus T and a Borel subgroup B. Let GrG be the affine +Grassmannian of G. T is called the small torus, in contrary to the big torus Ta of GrG. The theory of +hTa(GrG) when h is the equivariant cohomology or the K-theory, is studied by Kostant and Kumar +in [KK86, KK90]. It is dual to the so-called affine nil-Hecke algebra (equivariant cohomology case) +or the affine 0-Hecke algebra. Alternatively, the affine nil-Hecke algebra and the affine 0-Hecke +algebra can be called the equivariant homology and the equivariant K-homology theory. +The small torus equivariant homology theory HT(GrG) of the affine Grassmannian was first +studied by Peterson [P97]. Moreover, he raised a conjecture (without a proof) saying that HT(GrG) +is isomorphic to the quantum cohomology QHT (G/B)of G/B. This conjecture, together with its +partial flag variety version, is proved by Lam-Shimozono in [LS10]. One key step is the identification +of HT(G/B) with the centralizer of HT (pt) in HT (Ga/Ba) where Ga is the Kac-Moody group +associated to the affine root system and Ba is its Borel subgroup. +For K-theory, similar property was expected to hold. +In [LSS10], the authors study the K- +theoretic Peterson subalgebra, i.e., the centralizer of the ring KT (pt) in the small-torus affine +0-Hecke algebra, i.e., the equivariant K-homology KT (Ga/Ba). It is proved that this algebra is iso- +morphic to KT (GrG). One of the main tools is the small-torus GKM condition of the T-equivariant +K-cohomology. In [LLMS18], some evidence was provided in supporting the K-theory Peterson +Conjecture. In [K18], using the study of semi-infinite flag variety, Kato proves this conjecture. +More precisely, he embeds quantum K-theory of flag variety and certain localization of the Peter- +son subalgebra into T-equivariant K-theory of semi-infinite flag variety, and proves that their image +coincide. +In all the work mentioned above, the Peterson subalgebra plays key roles. In this paper, we +generalize the construction of the Peterson subalgebra into general oriented cohomology theory +h. Associated to such theory, there is a formal group law F over the coefficient ring R = h(pt). +Associated to F and a Kac-Moody root system, in [CZZ16, CZZ19, CZZ15, CZZ20], the author +generalized Kostant-Kumar’s construction and defined the formal affine Demazure algebra (FADA). +It is a non-commutative algebra generated by the divided difference operators. Its dual give an +algebraic model for hTa(Ga/Ba). Since Levine-Morel’s oriented cohomology theory is only defined +1 + +2 +C. ZHONG +for smooth projective varieties, in this paper we do not intend to generalize the geometric theory. +Instead, we only work with the algebraic model, i.e., the FADA associated to h. +Following the same idea as the work mentioned above in cohomology and K-theory, we look at +the small-torus (the torus T) version, which is very similar as the big torus case Ta. We define +the small torus FADA, DWa. In this paper, our first main result (Theorem 4.3) shows that the +algebraic models for hT (Ga/Ba) and hT (GrG), i.e., D∗ +Wa and (D∗ +Wa)W , satisfy the small torus GKM +condition. Based on that, we prove the second main result (Theorem 5.5), which shows that the +dual of hT (GrG), denoted by DQ∨ (Q∨ being the coroot lattice), coincides with the centralizer of +hT (pt) in the FADA DWa. This defines the Peterson subalgebra associated to h. +Our result generalizes and extends properties for equivariant cohomology and K-theory. More- +over, our method is uniform and does not reply on the specific oriented cohomology theory. As +an application of this construction, we define actions of the FADA (of the big and small torus) on +the algebraic models fo hTa(GrG) and hT (GrG). This is called the left Hecke action. For finite flag +varieties case it is studied in [MNS22] using geometric arguments (see also [B97, K03, T09, LZZ20]). +For connective K-theory (which specializes to cohomology and K-theory), we compute the recursive +formulas for certain basis in hT (GrG) (Theorem 2.3). +It is natural to consider generalizing Kato’s construction to this case, that is, invert Schubert +classes in DQ∨ corresponding to tλ ∈ Q∨ +<. This localization for K-theory was proved to be isomor- +phic to QKT (G/B). For h beyong singular cohomology and K-theory, however, the first obstruction +is that there is no ‘quantum’ oriented cohomology theory defined. The other obstruction is that +the divided difference operators do not satisfy braid relations. +This was a key step in Kato’s +construction (see [K18, Theorem 1.7]). The author plans to investigate this in a future paper. +This paper is organized as follows: In §1 we recall the construction of the FADA for the big torus +Ta, and in §2 we compute the recursive formulas via the left Hecke action. In §3 we we repeat the +construction for the small torus and indicates the difference from the big torus case. In §4 we prove +that dual of the small torus FADA satisfies the small torus GKM condition, and in §5 we define +the Peterson subalgebra and show that it coincides with the centralizer of hT (pt). In the appendix +we provide some computational result in the ˆA1 case. +Notations. Let G ⊃ B ⊃ T be such that G is simple, simply connected algebraic group over C +with a Borel subgroup B and a torus T. Let Ga ⊃ Ba ⊃ Ta where Ga is the affine Kac-Moody group +with Borel subgroup Ba and the affine torus Ta. Let P be the maximal parabolic group scheme so +that Ga/P = GrG is the affine Grassmannian. Let T ∗ (resp. T ∗ +a ) be the group of characters of T +(resp. Ta), then T ∗ +a = T ∗ ⊕ Zδ. +Let W be the Weyl group of G, I = {α1, ..., αn} be the simple roots, Q = ⊕iZαi ⊂ T ∗ be the root +lattice, Q∨ = ⊕iZα∨ +i be the coroot lattice, θ be the longest element, δ is the null root, α0 = −θ + δ +be the extra simple root. Denote Ia = {α0, ..., αn}. For each λ ∈ Q∨, let tλ be the translation +acting on Q. We then have tλ1tλ2 = tλ1+λ2, and wtλw−1 = tw(λ), w ∈ W. Let Q∨ +≤ be the set of +antidominant coroots, Q∨ +< be the set of strictly antidominant coroots (i.e., (λ, αi) < 0 ∀i ∈ I). Let +Wa = W ⋉ Q∨ be the affine Weyl group, ℓ be the length function on Wa, and w0 ∈ W be the +longest element. +Let Φ be the set of roots for W, Φa = Zδ + Φ be the set of real affine roots, and Φ± +a , Φ± +be the corresponding set of positive/negative roots for the corresponding systems. Let inv(w) = +w−1Φ+ +a ∩ Φ− +a . We have +Φ+ +a = {α + kδ|α ∈ Φ+, k = 0 or α ∈ Φ, k > 0}. + +AFFINE GRASSMANNIAN +3 +Let W − +a be the minimal length representatives of Wa/W. There is a bijection +W − +a → Q∨, +w �→ λ, if wW = tλW. +Moreover, W − +a ∩ Q∨ = {tλ|λ ∈ Q≤}. The action of α + kδ on µ + mδ ∈ Q ⊕ Zδ is given by +sα+kδ(µ + mδ) = µ + mδ − ⟨µ, α∨⟩(α + kδ). +In particular, for λ ∈ Q∨, w ∈ W, µ ∈ Q, we have sα+kδ = sαtkα∨, wtλ(µ) = w(µ). +We say the set of reduced sequences Iw, w ∈ Wa is W-compatible if Iw = Iu ∪ Iv for w = uv, u ∈ +W − +a , v ∈ W. +1. FADA for the big torus +In this section, we recall the construction of the formal affine Demazure algebra (FADA) for the +affine root system. All the construction can be found in [CZZ20]. +1.1. +Let F be a one dimensional formal group law over a domain R with characteristic 0. Following +from [LM07] that there is an oriented cohomology h whose associated formal group law is F. In +this paper we won’t need any geometric property of this h, since our treatment is pure algebraic +and self-contained. +Example 1.1. Let F = Fc = x + y − cxy be the connective formal group law (for connective +K-theory) over R = Z[c]. Specializing to c = 0 or c = 1, one obtains the additive or multiplicative +formal group law. One of the simplest formal group laws beyond Fc is the hyperbolic formal group +law considered in [LZZ20]: +F(x, y) = x + y − cxy +1 + axy +, R = Z[c, a]. +Let ˆS be the formal group algebra of T ∗ +a defined in [CPZ13]. That is, +ˆS = R[[xµ|µ ∈ T ∗ +a ]]/JF , +where JF is the closure of the ideal generated x0 and xµ1+µ2 − F(xµ1, xµ2), µ1, µ2 ∈ T ∗ +a . Indeed, +after fixing a basis of T ∗ +a ∼= Zn+1, ˆS is isomorphic to the power series ring R[[x1, ..., xn+1]]. +Remark 1.2. If F = Fc is the connective formal group law, one can just replace ˆS by R[xµ|µ ∈ +T ∗ +a ]/JF . In other words, in this case one can use the polynomial ring instead of the power series ring. +For instance, if c = 0, then ˆS ∼= SymR(T ∗ +a ), xµ �→ µ. If c ∈ R×, then ˆS ∼= R[T ∗ +a ], xµ �→ c−1(1 − e−µ). +Throughout this paper, whenever we specializes to Fc, we assume that ˆS is the polynomial version. +1.2. +Define ˆQ = ˆS[ 1 +xα , α ∈ Φa]. The Weyl groups Wa acts on ˆQ, so we can define the twisted group +algebra ˆQWa := ˆQ ⋊ R[Wa],, which is a free left ˆQ-module with basis denoted by ηw, w ∈ Wa and +the product cηwc′ηw′ = cw(c′)ηww′, c, c′ ∈ ˆQ. +For each α ∈ Φa, define κα = +1 +xα + +1 +x−α ∈ ˆS. If F = Fc, then κα = c. For each simple root +αi, we define the Demazure element ˆXαi = +1 +xαi (1 − ηsi). It is easy to check that ˆX2 +α = κα ˆXα. For +simplicity, denote ηi = ηαi = ηsi, x±i = x±αi, ˆXi = ˆXαi, i ∈ Ia. If Iw = (i1, ..., ik), ij ∈ Ia is a +reduced sequence of w ∈ Wa, we define ˆXIw correspondingly. It is well known that they depends +on the choice of Iw, unless F = Fc. +Write +(1) +ˆXIw = +� +v≤w +ˆaIw,vηv, +ηw = +� +v≤w +ˆbw,Iv ˆXIv, +ˆaIw,v ∈ ˆQ, ˆbw,Iv ∈ ˆS, + +4 +C. ZHONG +then we have ˆbw,Iw = � +α∈inv(w) xα = +1 +ˆaIw,w . +Let ˆDWa be the subalgebra of ˆQWa generated by ˆS and ˆXi, i ∈ Ia. This is called the formal +affine Demazure algebra (FADA) for the big torus. It is easy to see that ˆXIw, w ∈ Wa is a ˆQ-basis +of ˆQWa, and it is proved in [CZZ20] that it is also a basis of the left ˆS-module ˆDWa. Note that +W ⊂ ˆDWa via the map si �→ ηi = 1 − xi ˆXi ∈ ˆDWa. +Remark 1.3. It is not difficult to derive that there is a residue description of the coefficients in +the expression of elements of ˆDWa as linear combinations of ηw. Such description was first given in +[GKV97]. See [ZZ17] for more details. +1.3. +We define the duals of left modules: +ˆQ∗ +Wa = Hom ˆ +Q( ˆQWa, ˆQ) = Hom(Wa, ˆQ), +ˆD∗ +Wa = Hom ˆS( ˆDWa, ˆS). +Dual to the elements ηw, ˆXIw ∈ ˆDWa ⊂ ˆQWa, we have ˆfw, ˆX∗ +Iw ∈ ˆD∗ +Wa ⊂ ˆQ∗ +Wa. +The product +structure on ˆQ∗ +Wa is defined by ˆfw ˆfv = δw,v ˆfw, with the unit given by 1 = � +w∈Wa ˆfw. Note that +here we usually use � to denote a sum of (possibly) infinitely many terms, and � to denote a +finite sum. +Lemma 1.4. We have +ˆD∗ +Wa = { ˆf ∈ ˆQ∗ +Wa| ˆf( ˆDWa) ⊂ ˆS}. +Proof. Denote the RHS by Z1. It is clear that ˆD∗ +Wa is contained in Z1 since ˆXIv generate ˆDWa, +ˆX∗ +Iw generate ˆD∗ +Wa, and ˆX∗ +Iw( ˆXIv) = δw,v. Conversely, let ˆf = � +ℓ(w)≥k cw ˆfw ∈ Z1. If ℓ(u) = k, +then from (1), we have +ˆf( ˆXIu) = +� +ℓ(w)≥k +cw ˆfw( +� +v≤u +ˆaIu,vηv) = cuˆaIu,u ∈ ˆS. +Denote ˆf ′ := ˆf −� +ℓ(u)=k cuˆaIu,u ˆX∗ +Iu. Note that ˆX∗ +Iu = � +w∈Wa ˆbw,Iufw and ˆbu,IuˆaIu,u = 1, so for any +u with ℓ(u) = k, we have ˆf ′(ηu) = cu − cuˆaIu,u ˆX∗ +Iu(ηu) = cu − cu = 0, so ˆf ′ is a linear combination +of ˆfw, ℓ(w) ≥ k + 1. Repeating this process, we get that ˆf ∈ ˆD∗ +Wa. +□ +1.4. +There is an ˆQ-linear action of ˆQWa on ˆQ∗ +Wa, defined by +(z • ˆf)(z′) = ˆf(z′z), +z, z′ ∈ ˆQWa, ˆf ∈ ˆQ∗ +Wa. +This is called the right Hecke action. We have +cηw • c′ ˆfw′ = c′w′w−1(c) ˆfw′w−1, c, c′ ∈ ˆQ. +It follows from Lemma 1.4 and similar reason as in [CZZ19, §10] that this induces an action of ˆDWa +on ˆD∗ +Wa. Moreover, it induces an action of W ⊂ ˆDWa on ˆQ∗ +Wa and ˆD∗ +Wa. By definition it is easy to +get +ˆXα • +� +w∈Wa +cw ˆfw = +� +w∈Wa +cw − csw(α)w +xw(α) +ˆfw. +(2) +The following proposition is proved in the finite case in [CZZ19, Lemma 10.2, Theorem 10.7]. +Proposition 1.5. The subset ˆD∗ +Wa ⊂ ˆQ∗ +Wa satisfies the following (big-torus) GKM condition: +ˆD∗ +Wa = { ˆf ∈ ˆQ∗ +Wa| ˆf(ηw) ∈ ˆS and ˆf(ηw − ηsαw) ∈ xα ˆS, ∀α ∈ Φa}. + +AFFINE GRASSMANNIAN +5 +Proof. Denote the RHS by Z2. Let ˆf ∈ ˆD∗ +Wa, we know ˆXα • ˆf ∈ ˆD∗ +Wa. Then (2) implies that ˆf +satisfies the condition defining Z2, so ˆD∗ +Wa ⊂ Z2. +For the other direction, we first show that ˆD∗ +Wa is a maximal ˆDWa-submodule of ˆS∗ +Wa := +Hom(Wa, ˆS). +This can be proved as follows: if M ⊂ ˆS∗ +Wa is a ˆDWa-module, for any ˆf ∈ M, +we have ˆXI • ˆf ∈ M ⊂ ˆS∗ +Wa, so ˆXI • ˆf(ηe) = ˆf( ˆXI) ∈ ˆS, so f ∈ ˆD∗ +Wa. One then can show that +the subset Z2 is a ˆDWa-module, which follows from the same proof as in the finite case in [CZZ19, +Theorem 10.2]. Since ˆD∗ +Wa is a maximal submodule, we have Z2 ⊂ ˆD∗ +Wa. The proof is finished. +□ +1.5. +We can similarly define the non-commutative ring ˆQQ∨ = ˆQ ⋊ R[Q∨] with a ˆQ-basis ηtλ, λ ∈ +Q∨. Then there is a canonical map of left ˆQ-modules: +pr : ˆQWa → ˆQQ∨, +cηtλw �→ cηtλ, +w ∈ W, λ ∈ Q∨, c ∈ ˆQ. +Define ˆDWa/W = pr( ˆDWa) ⊂ ˆQQ∨. Indeed, this is the same as the relative Demazure module +defined in [CZZ19, §11]. +We can also consider the ˆQ-dual ˆQ∗ +Q∨ and the ˆS-dual ˆD∗ +Wa/W. The elements dual to ηtλ ∈ ˆQQ∨ +are denoted by ˆftλ. +The projection pr then induces embeddings pr∗ : ˆQ∗ +Q∨ ֒→ ˆQ∗ +Wa and pr∗ : +ˆDWa/W ֒→ ˆD∗ +Wa. It is easy to see that +pr∗( ˆftλ) = +� +v∈W +ˆftλv. +Moreover, similar as in the finite case [CZZ19, Lemma 11.7], we have +pr∗( ˆQ∗ +Q∨) = ( ˆQ∗ +Wa)W , +pr∗( ˆD∗ +Wa/W) = ( ˆD∗ +Wa)W . +Indeed, elements of pr∗( ˆQ∗ +Q∨) = ( ˆQ∗ +Wa)W are precisely the elements ˆf ∈ ˆQ∗ +Wa satisfying ˆf(ηtλw − +ηtλ) = 0 for any λ ∈ Q∨, w ∈ W. It follows from similar reason as [CZZ19, Corollary 8.5, Lemma +11.5] that if Iw, w ∈ Wa is W-compatible, then ˆbuv,Iw = ˆbu,Iw for any v ∈ W. We then have +Lemma 1.6. Assume the sequences Iw, w ∈ Wa is W-compatible, then pr(XIw), w ∈ W − +a is a basis +of ˆDWa/W , and { ˆX∗ +Iw, w ∈ W − +a } is a ˆQ-basis of ( ˆQ∗ +Wa)W and a ˆS-basis of ( ˆD∗ +Wa)W . +Note that ( ˆD∗ +Wa)W is the algebraic model for hTa(GrG) and the embedding ( ˆD∗ +Wa)W ⊂ ˆD∗ +Wa is +the algebraic model for the pull-back hTa(GrG) → hTa(Ga/Ba). +1.6. +Similar as the finite case in [LZZ20, §3], there is another action of ˆQWa on ˆQ∗ +Wa by +aηv ⊙ b ˆfw = av(b) ˆfvw, a, b ∈ ˆQ, w, v ∈ Wa. +This is called the left Hecke action. It is easy to see that it commutes with the •-action. Note +however that the ⊙-action is not ˆQ-linear. +Lemma 1.7. The ⊙ action of ˆQWa on ˆQ∗ +Wa induces an action of ˆDWa on ˆD∗ +Wa. +Proof. We have +ˆXα ⊙ +� +w +cw ˆfw = 1 +xα +(1 − ηα) ⊙ +� +w +cw ˆfw = +� +w +cw − sα(csαw) +xα +ˆfw. +let dw,α = cw−sα(csαw) +xα +. We show that dw,α satisfy the big-torus GKM condition, that is, dw,α − +dsβw,α ∈ xβ ˆS for any β. + +6 +C. ZHONG +Denote cw − csαw = xαp, p ∈ ˆS and x−α = −xα + x2 +αq, q ∈ ˆS. If β = α, then we have +dw,α − dsβw,α += +cw − sα(csαw) − csαw + sα(cw) +xα += xαp + sα(cw) − sα(cw − xαp) +xα += +p + x−αsα(p) +xα += p − sα(p) + xαq, +which is clearly a multiple of xα. If β ̸= α, then +dw,α − dsβw,α = cw − sα(csαw) − (csβw − sα(csαsβw)) +xα += cw − sα(csαw) − csβw + sα(csαsβw) +xα +. +Since xα, xβ are coprime [CZZ20, Lemma 2.2], it suffices to prove the numerator is divisible by +xβ. Note cw − csβw is already divisible by xβ. Furthermore, cw − csαsβw = cw − cssα(β)sαw, so it is +divisible by ssα(β). Therefore, −sα(csαw) + sα(csαsβw) is divisible by sα(xsα(β)) = xβ. The proof is +finished. +□ +Consequently, the ⊙-action of ˆDWa on ˆD∗ +Wa restricts to an action on ( ˆD∗ +Wa)W . +1.7. +Indeed, there is a characteristic map +c : ˆS → ˆD∗ +Wa, z �→ z • 1, +whose geometric model is the map sending a character of the torus to the first Chern class of the +associated line bundle over the flag variety [CZZ15, §10]. We then have a map +φ : ˆS ⊗ ˆSWa ˆS → ˆD∗ +Wa, a ⊗ b �→ ac(b) = +� +w +aw(b) ˆfw. +This is proved to be an isomorphism in some cases. It is easy to see that for any z ∈ DWa, there +are the following commutative diagrams +ˆS ⊗ ˆSWa ˆS +φ +� +z· ⊗id +� +ˆD∗ +Wa +z⊙ +� +ˆS ⊗ ˆSWa ˆS +φ +� ˆD∗ +Wa +, +ˆS ⊗ ˆSWa ˆS +φ +� +id ⊗z· +� +ˆD∗ +Wa +z• +� +ˆS ⊗ ˆSWa ˆS +φ +� ˆD∗ +Wa +. +2. Equivariant connective K-theory of the affine Grassmannian +As an application of the left Hecke action, we derive the recursive formulas for this action on +bases in connective K-theory of GrG. In this section only, assume F = Fc. Our results specialize +to equivariant K-theory (resp. equivariant cohomology) by letting c = 1 (resp. c = 0). In both +cases, our results are only known for flag varieties of finite root systems. Since ˆXi do not satisfy +the braid relations, the result of this section do not generalize to general F. +2.1. +Denote ǫw = (−1)ℓ(w) and cw = cℓ(w). We have x−α = +xα +cxα−1 and κα = c for any α, and ˆXIw +can be denoted by ˆXw. +Note that there is another operator ˆYi = ˆYαi = c − ˆXαi such that ˆY 2 +αi = c ˆYαi and braid rela- +tions are satisfied. This is the algebraic model of the composition hTa(Ga/Ba) → hTa(Ga/Pi) → +hTa(Ga/Ba) where Pi is the minimal parabolic subgroup corresponding to αi ∈ Ia. Moreover, we +have +ˆXw = +� +v≤w +ǫvcwc−1 +v +ˆYv. + +AFFINE GRASSMANNIAN +7 +Most properties of ˆXw are also satisfied by ˆYw, except for Lemma 1.6. Indeed, ˆY ∗ +w, w ∈ W − +a is not +W-invariant. +Denote xΦ = � +α∈Φ− xα. It is well known that Yw0 = � +w∈W ηw 1 +xΦ . Moreover, the map Yw0 • +: +ˆD∗ +Wa → ( ˆD∗ +Wa)W is the algebraic model for the map hTa(Ga/Ba) → hTa(GrG). We first compute +the image of the two bases via this map. +Lemma 2.1. Let F = Fc. For any w ∈ Wa and u = u1u2, u1 ∈ W − +a , u2 ∈ W, we have +Yw0 • ˆX∗ +u1u2 = ǫu2cw0c−1 +u2 ˆX∗ +u1, Yw0 • ˆY ∗ +w = +� +v1v2≥w,v1∈W − +a ,v2∈W +ǫwǫv2cv1w0c−1 +w ˆX∗ +v1. +In particular, Yw0 • ˆY ∗ +w, w ∈ W − +a is a basis of ( ˆD∗ +Wa)W if and only if c ∈ R×. +Proof. For each v ∈ Wa, write v = v1v2, v1 ∈ W − +a , v2 ∈ W. From ˆXwYw0 = 0, w ∈ W, we have +(Yw0• ˆX∗ +u1u2)( ˆXv1v2) = ˆX∗ +u1u2( ˆXv1v2Yw0) = δv2,e ˆX∗ +u1u2( ˆXv1 +� +w′≤w0 +ǫw′cw0c−1 +w′ ˆXw′) = δv2,eδv1,u1ǫu2cw0c−1 +u2 . +This proves the first identity. For the second one, it is easy to see that ˆY ∗ +w = � +v≥w ǫwcvc−1 +w ˆX∗ +v. So +Yw0 • ˆY ∗ +w = Yw0 • +� +v≥w +ǫwcvc−1 +w ˆX∗ +v = +� +v1v2≥w,v1∈W − +a ,v2∈W +ǫwǫv2cv1w0c−1 +w ˆX∗ +v1. +This proves the second identity. +The transition matrix between ˆX∗ +v, v ∈ W − +a +and Yw0 • ˆY ∗ +w, w ∈ W − +a +is upper triangular with +diagonal entries ǫwcw0, so the last statement follows. +□ +2.2. +Before computing the ⊙-action, we need to prove some identities in ˆDWa. +Lemma 2.2. Let F = Fc. Writing ηu = � +v≤u ˆbu,v ˆXv = � +v≤u ˆbY +u,v ˆYv, then +ˆbsiu,v += +� +si(ˆbu,v), +siv > v; +(1 − cxi)si(ˆbu,v) − xisi(ˆbu,siv), +siv < v. +ˆbY +siu,v += +� +(1 − cxi)si(ˆbY +u,v), +siv > v; +xisi(ˆbY +u,siv) + si(ˆbY +u,v), +siv < v. +Proof. We prove the first one, and the second one follows similarly. Denote iWa = {v ∈ Wa|siv > v}. +We have +ηsiu += +ηiηu = ηi +� +v∈iWa +ˆbu,v ˆXv + ˆbu,siv ˆXsiv = +� +v∈iWa +si(ˆbu,v)ηi ˆXv + si(ˆbu,siv)ηi ˆXsiv += +� +v∈iWa +si(ˆbu,v)(1 − xi ˆXi) ˆXv + si(ˆbu,siv)(1 − xi ˆXi) ˆXsiv += +� +v∈iWa +si(ˆbu,v)( ˆXv − xi ˆXsiv) + si(ˆbu,siv) ˆXsiv − cxisi(ˆbu,siv) ˆXv += +� +v∈iWa +si(ˆbu,v) ˆXv + (si(ˆbu,siv)(1 − cxi) − xisi(ˆbu,v)) ˆXsiv. +The conclusion then follows. +□ +Note that if v ∈ W − +a +and siv < v, then siv ∈ W − +a . We have the following recursive formula, +whose proof follows from the definition and Lemma 2.2. + +8 +C. ZHONG +Theorem 2.3. For F = Fc, with i ∈ Ia, we have +ˆX−i ⊙ ˆX∗ +v += +� +0, +siv > v, +c ˆX∗ +v + ˆX∗ +siv, +siv < v, +ˆY−i ⊙ ˆY ∗ +v += +� +0, +siv > v, +c ˆY ∗ +v + ˆY ∗ +siv, +siv < v. +Here +ˆX−i = ηw0 ˆXiηw0 = +1 +x−i +(1 − ηi), ˆY−i = ηw0 ˆYiηw0 = 1 +xi ++ +1 +x−i +ηi. +Consequently, if v ∈ W − +a , we have +ˆY−i ⊙ (Yw0 • Y ∗ +v ) = +� +0, +siv > v, +c(Yw0 • ˆY ∗ +v ) + (Yw0 • ˆY ∗ +siv), +siv < v. +Proof. We have +ˆX−i ⊙ ˆX∗ +v = ( 1 +x−i +− +1 +x−i +ηi) ⊙ +� +u≥v +ˆbu,v ˆfu = +� +u +ˆbu,v +x−i +ˆfu − +� +u +si(ˆbu,v) +x−i +ˆfsiu = +� +u +ˆbu,v − si(ˆbsiu,v) +x−i +ˆfu. +Plugging the formula in Lemma 2.2, we obtain the formula. +The formula for ˆY−i ⊙ ˆY ∗ +v follows similarly. From the commutativity of the two actions • and ⊙, +one obtains the last statement. +□ +3. FADA for the small torus +We repeat the construction of FADA for the small torus, which is very similar as above. +3.1. +Let S be the formal group algebra associated to T ∗, that is, it is (non-canonically) isomorphic +a power series ring of rank n. When the formal group law F = Fc, we can again take the polynomial +version, i.e., see Remark 1.2. Let Q = S[ 1 +xα , α ∈ Φ], QWa = Q ⋊ R[Wa], QQ∨ = Q ⋊ R[Q∨]. For +any α ∈ Φ, let κα = +1 +xα + +1 +x−α and κα0 = +1 +x−θ + 1 +xθ . We have the projection +pr : QWa → QQ∨, ηtλw �→ ηtλ, +w ∈ W. +Define +Xα = 1 +xα +(1 − ηα), +Xα0 = +1 +x−θ +(1 − ηs0), +α ∈ Φ. +For simplicity, denote x±i = x±αi, Xi = Xαi, ηi = ηsi, X0 = Xα0. They satisfy relations similar +as that of ˆXi. One can define XIw for any reduced sequence Iw of w, which depends only on w if +F = Fc. +Remark 3.1. Consider K-theory, in which case F = Fc with c = 1. Our −X−αi is the Ti in +[LSS10, LLMS18]. Our 1 − Xαi coincides with the Di in [K18]. For cohomology, c = 0, κα = 0, +and our Xi is the Ai in [P97, Proposition 2.11] and [L06]. +Lemma 3.2. We have pr(zXi) = 0 if z ∈ QWa, i ∈ I. +Proof. Let z = pηw, p ∈ Q, w ∈ Wa , then +pr(zXi) = pr(pηwXi) = pr( +p +w(xi)(ηw − ηwsi)) = +p +w(xi)(pr(ηw) − pr(ηwsi)) = 0. +□ + +AFFINE GRASSMANNIAN +9 +Define DWa to be the subalgebra of QWa generated by S and Xi, i ∈ Ia, and DWa/W = pr(DWa). +Then DWa is a free left S-module with basis XIw, w ∈ Wa. Denote XIw = pr(XIw), w ∈ W − +a . +Lemma 3.3. If Iw, w ∈ Wa are W-compatible, then the set {XIw|w ∈ W − +a } is a basis of the left +S-module DWa/W . +Proof. They follow easily from Lemma 3.2. See [CZZ19, Lemma 11.3]. +□ +The projection p : T ∗ +a → T ∗, µ + kδ �→ µ induces projections ˆS → S, ˆQ → Q and ˆQWa → QWa. +Clearly p( ˆXαi) = Xαi and p( ˆXIw) = XIw, so p( ˆDWa) = DWa. More explicitly, we have +ˆXIw = +� +v≤w +ˆaIw,vηv ∈ ˆQWa, +XIw = +� +v≤w +aIw,vηv ∈ QWa, +p(ˆaIw,v) = aIw,v ∈ Q, +ηw = +� +v≤w +ˆbw,Iv ˆXIv ∈ ˆQWa, +ηw = +� +v≤w +bw,IvXIv ∈ QWa, +p(ˆbw,Iv) = bw,Iv ∈ S. +Note that the embedding i : Q → ˆQ induces a section QWa → ˆQWa of p. However, it does not map +DWa to ˆDWa. For example, X0 is mapped to x−θ+δ +x−θ +ˆX0 which does not belong to ˆDWa. +3.2. +As before, we can take the duals, which will give us Q-modules Q∗ +Wa, Q∗ +Q∨, and S-modules +D∗ +Wa, D∗ +Wa/W . The elements dual to +ηw, XIw ∈ DWa ⊂ QWa, ηtλ, XIw ∈ DWa/W ⊂ QQ∨, +are denoted by +fw, X∗ +Iw ∈ D∗ +Wa ⊂ Q∗ +Wa, ftλ, X∗ +Iw ∈ D∗ +Wa/W ⊂ Q∗ +Q∨, +correspondingly. Note that the notation ftλ can be thought as in Q∗ +Wa and Q∗ +Q∨, just like ηtλ can +be thought as in QWa and QQ∨. Similar as Proposition 1.5, we have +(3) +D∗ +Wa = {f ∈ Q∗ +Wa|f(DWa) ⊂ S}. +Moreover, by definition, the dual map pr∗ : Q∗ +Q∨ → Q∗ +Wa satisfies +pr∗(ftλ) = +� +w∈W +ftλw. +Following from the definition, we have +ˆX∗ +Iw = +� +v≥w +ˆbv,Iw ˆfv ∈ ˆD∗ +Wa, +X∗ +Iw = +� +v≥w +bv,Iwfv ∈ D∗ +Wa. +Since p(ˆbv,Iw) = bv,Iw, so the map q : +ˆQ∗ +Wa → Q∗ +Wa, � +w aw ˆfw �→ � +w p(aw)fw induces a map +q : ˆD∗ +Wa → D∗ +Wa such that q( ˆX∗ +Iw) = X∗ +Iw. Moreover, since +p∗(X∗ +Iw)( ˆXIv) = X∗ +Iw(p( ˆXIv)) = X∗ +Iw(XIv) = δw,v, +so p∗(X∗ +Iw) = ˆX∗ +Iw. Note that neither q nor p∗ are isomorphisms, since the domains and targets are +modules over different rings. +Similar as Lemma 1.6, we have +Lemma 3.4. If Iw, w ∈ Wa are W-compatible, then the set X∗ +Iw, w ∈ W − +a form a basis of (Q∗ +Wa)W +and of (D∗ +Wa)W , respectively. + +10 +C. ZHONG +Lemma 3.5. Assume that {Iw, w ∈ Wa} is W-compatible. For any w ∈ W, u ∈ W − +a , we have +X∗ +Iu = +� +λ∈Q∨ +btλw,Iuftλ ∈ Q∗ +Q∨. +Proof. For any λ ∈ Q∨, write +ηtλw = +� +u∈W − +a ,v∈W +btλw,Iu∪IvXIu∪Iv. +By Lemma 3.2, we have +ηtλ = pr(ηtλw) = +� +u∈W − +a ,v∈W +btλw,Iu∪Iv pr(XIu∪Iv) = +� +u∈W − +a +btλw,Iu pr(XIu) = +� +u∈W − +a +btλw,IuXIu. +Therefore, +X∗ +Iu = +� +λ∈Q∨ +btλw,Iuftλ ∈ Q∗ +Q∨. +□ +This lemma implies that we have pr∗(X∗ +Iu) = X∗ +Iu, u ∈ W − +a . +3.3. +There is a •-action of QWa on Q∗ +Wa, defined similar as the big torus case. +Lemma 3.6. The •-action of QWa on Q∗ +Warestricts to an action of DWa on D∗ +Wa. +Proof. Since DWa is a S-module with basis XIu, u ∈ Wa, so for any w, v ∈ Wa, i ∈ Ia, we have +XIvXi = � +u cIv∪si,IuXIu with cIv∪si,Iu ∈ S. We have +(Xi • X∗ +Iw)(XIv) = X∗ +Iw(XIvXi) = cIv∪si,Iw ∈ S. +By (3), Xi • X∗ +Iw ∈ D∗ +Wa. +□ +Lemma 3.7. We have +D∗ +Wa ⊂ {f ∈ Q∗ +Wa|f(ηw) ∈ S, and f(ηw − ηsαw) ∈ xαS, ∀α ∈ Φ, w ∈ Wa}. +One of the main results of this paper is to study how different the two sets are, that is, to derive +the small torus GKM condition. +Proof. Since ηw ∈ DWa, then it follows from (3) that f(ηw) ∈ S. Let i ∈ I and f = � +w∈Wa awfw ∈ +D∗ +Wa with aw = f(ηw) ∈ S. We have +Xi•f = 1 +xi +(1−ηi)• +� +w +awfw = +� +w +aw +w(xi)fw− +� +w +aw +wsi(xi)fwsi = +� +w +aw − awsi +w(xi) += +� +w +aw − asw(αi)w +xw(αi) +fw. +By Lemma 3.6, Xi • f ∈ D∗ +Wa, so f(ηw − ηsβw) = +aw−asβw +xβ +∈ S for any β ∈ Φ. +□ +3.4. +We can similarly define the ⊙ action +aηw ⊙ bfv = aw(b)fwv, w, v ∈ Wa, a, b ∈ Q. +It is easy to see that the ⊙ and the • actions commute with each other. +Lemma 3.8. For any ˆz ∈ ˆQWa, ˆf ∈ ˆQ∗ +Wa, we have +p(ˆz) ⊙ q( ˆf) = q(ˆz ⊙ ˆf). +In particular, the ⊙-action of QWa on Q∗ +Wa induces an action of DWa on D∗ +Wa. + +AFFINE GRASSMANNIAN +11 +Proof. Write ˆz = ˆaηv, ˆf = ˆb ˆfw, ˆa,ˆb ∈ ˆQ, w, v ∈ Wa and suppose p(ˆa) = a, p(ˆb) = b, then +p(ˆz) ⊙ q( ˆf) = aηv ⊙ bfw = av(b)fvw = q(ˆav(ˆb) ˆfvw) = q(ˆaηv ⊙ ˆb ˆfw) = q(ˆz ⊙ ˆf). +For the second part, note that p : ˆDWa → DWa and q : ˆD∗ +Wa → D∗ +Wa are both surjective. Given +z ∈ DWa and f ∈ D∗ +Wa, suppose z = p(ˆz) and f = q( ˆf) for some ˆz ∈ ˆDWa and ˆf ∈ ˆD∗ +Wa, then +z ⊙ f = p(ˆz) ⊙ q( ˆf) = q(ˆz ⊙ ˆf) ∈ q( ˆD∗ +Wa) = D∗ +Wa. +□ +Remark 3.9. If F = Fc, then all results in §2 holds for X∗ +w and the corresponding Y ∗ +w. +4. The small-torus GKM condition +In this section, we study the small-torus GKM condition on the equivariant oriented cohomology +of the affine flag variety and of the affine Grassmannian. +4.1. +For each α ∈ Φ, we define +Zα = +1 +x−α +(1 − ηtα∨) ∈ QWa. +Lemma 4.1. For each α ∈ Φ, we have Zα ∈ DWa. +Proof. It suffices to show that Zα is contained in the subalgebra of DWa generated by S and Xα. So +we assume the root system is the affine root system of SL2 with simple roots α1 = α, α0 = −α + δ. +Then tα∨ = s0s1. We have ηs1 = 1 − xαX1, ηs0 = 1 − x−αX0, so ηs0s1 = 1 − x−αX0 − x−αX1 + +x2 +−αX0X1. Therefore, +Zα = +1 +x−α +(1 − ηs0s1) = X0 + X1 − x−αX0X1 ∈ DWa. +□ +Example 4.2. Suppose the root system is ˆA1 with two simple roots α1 = α, α0 = −α + δ. +(1) If F = Fc with c = 0, then we have Zα = X0 + X1 + αX0X1. +(2) If F = Fc with c = 1, then we have Zα = X0 + X1 + (eα − 1)X0X1. +Since DWa acts on D∗ +Wa, so we know that Zα acts on D∗ +Wa. Note that +Zk +α = 1 +xkα +(1 − ηtα∨)k. +4.2. +We are now ready to prove the first main result of this paper. +Theorem 4.3. +(1) The subset D∗ +Wa ⊂ Q∗ +Wa consists of elements satisfying the following small- +torus GKM condition: +f +� +(1 − ηtα∨)dηw +� +∈ xd +αS, and f +� +(1 − ηtα∨)d−1(1 − ηsα)ηw +� +∈ xd +αS, ∀α ∈ Φ, w ∈ Wa, d ≥ 1. +(2) The subset (D∗ +Wa)W ⊂ (Q∗ +Wa)W consists of elements satisfying the following small-torus +Grassmannian condition: +f +� +(1 − ηtα∨)dηw +� +∈ xd +αS, ∀α ∈ Φ, w ∈ Wa, d ≥ 1. + +12 +C. ZHONG +Our proof follows similarly as that of [LSS10, Theorem 4.3]. The key improvement is that we +don’t need to prove Propositions 4.4 and 4.5 of loc.it., since we can use the operators Zα. However, +for the convenience of the readers, we include an appendix, which gives all coefficients of bw,Iv in +the ˆA1 case. They can be used to show that X∗ +Iw satisfy the small torus GKM condition. +Proof. (1). +We prove that elements of D∗ +Wa satisfy the small-torus GKM condition. +Let f = +� +w cwfw ∈ D∗ +Wa, we have +Zα • +� +w +cwfw = +� +w +( +cw +w(x−α)fw − +cw +wt−α∨(x−α)fwt−α∨) = +� +w +cw − ctw(α∨)w +x−w(α) +fw ∈ D∗ +Wa. +Note that +xα +x−α is invertible in S. Therefore, denoting w(α) = β, by (3), we have f((1 − ηtβ∨)ηw) ∈ +xβS for any β ∈ Φ. +Moreover, denote dw = +cw−cwtα∨ +x−w(α) , then dwtα∨ = +cwtα∨ −cwtα∨ tα∨ +x−wtα∨ (α) += +cwtα∨ −cwt2α∨ +x−w(α) +. Therefore, We +have +Z2 +α • f += +Zα • Zα • +� +w +cwfw = +� +w +(dw − dwtα∨ +w(x−α) +)fw = +� +w +cw − 2cwtα∨ + cwt2α∨ +w(x−α)2 +fw += +� +w +cw − 2ctw(α∨)w + ct2w(α∨)w +x2 +−w(α) +fw = +� +w +1 +x2 +−w(α) +f((1 − ηtw(α∨))2ηw)fw. +Denoting w(α) = β, we see that f((1−ηtβ∨)2ηw) ∈ x2 +βS. Inductively, we see that f((1−ηtα∨ )dηw) ∈ +xd +αS for all d ≥ 1. +Similarly, if one applies Zd−1 +α +Xα ∈ DWa on f, which gives Zd−1 +α +Xα • f ∈ D∗ +Wa, one will see that +f satisfies the second condition. +For the rest of the proof and for that of (2), it is identical to that of [LSS10, Theorem 4.3], so it +is skipped. +□ +Corollary 4.4. The subset D∗ +Wa/W ⊂ Q∗ +Q∨ consists of elements satisfying the following small torus +Grassmannian condition: +f((1 − ηt∨ +α)dηtλ) ∈ xd +αS, ∀α ∈ Φ, d ≥ 1, λ ∈ Q∨. +Proof. This follows from the identity pr∗(ftλ) = � +v∈W ftλv. +□ +5. The Peterson subalgebra +In this section, we embed DWa/W into DWa and show that it coincides with the centralizer of S +in DWa. This is called the Peterson subalgebra, which gives the algebraic model for the equivariant +oriented ‘homology’ of the affine Grassmannian. +5.1. +We have a canonical ring embedding (and also a Q-module embedding) +k : QQ∨ → QWa, +pηtλ �→ pηtλ, +such that pr ◦k = idQWa. It is easy to see that the dual map k∗ : Q∗ +Wa → Q∗ +Q∨ satisfies +(4) +k∗(ftλu) = δu,eftλ, +u ∈ W. +For K-theory, our map k is the map k : KT (GrG) → K in [LSS10, §5.2], and k∗ is the wrong-way +map ̟ of [LSS10, $4.4]. +The following lemma generalizes [P97], [L06, Theorem 4.4] for the cohomology case, and [LSS10, +Lemma 4.6] for the K-theory case. + +AFFINE GRASSMANNIAN +13 +Lemma 5.1. The map k∗ induces a map k∗ : D∗ +Wa → D∗ +Wa/W . Consequently, the map k induces a +map k : DWa/W → DWa. +Proof. Given f ∈ D∗ +Wa, then f satisfies the small-torus GKM condition Theorem 4.3, that is, +f((1 − ηtα∨)dηtλu) ∈ xd +αS, +∀u ∈ W, λ ∈ Q∨, α ∈ Φ, d ≥ 1. +Therefore, +k∗(f)((1 − ηtα∨)dηtλ) = f +� +k((1 − ηtα∨)dηtλ) +� += f((1 − ηt∨ +α)dηtλ) ∈ xd +αS. +Therefore, by Corollary 4.4, k∗(f) ∈ D∗ +Wa/W . +□ +Remark 5.2. It would be interesting to find a direct proof of the fact that k maps DWa/W to +DWa. One possible choice is to find the small torus residue condition of DWa similar to the residue +condition of [GKV97] (see [ZZ17]). +Example 5.3. Note that this result is not true for the big torus case, that is, k( ˆDWa/W ) is not +contained in ˆDWa. For example, in the ˆA1 case, we have +pr(X0) = pr( +1 +x−α+δ +(1 − ηtα∨s1)) = +1 +x−α+δ +(1 − ηtα∨) ∈ ˆDWa/W , +and +k(pr(X0)) = +1 +x−α+δ +(1 − ηtα) = +1 +x−α+δ +(1 − ηs0s1) ̸∈ ˆDWa. +Lemma 5.4. If Iw, w ∈ W is W-compatible, then k∗(X∗ +Iu) = X∗ +Iu for any u ∈ W − +a . +Proof. By (4) and Lemma 3.5, we have +k∗(X∗ +Iu) = k∗( +� +λ∈Q∨,w∈W +btλw,Iuftλw) = +� +λ∈Q∨ +btλ,Iuftλ = X∗ +Iu. +□ +5.2. +Let CDWa(S) be the centralizer of S in DWa. Our second main result is the following, which +generalizes [LSS10, Lemma 5.2] in the K-theory case and [P97, §9.3] in the cohomology case (proved +in [LS10, Theorem 6.2]). +Theorem 5.5. We have CDWa(S) = k(QQ∨) ∩ DWa = k(DWa/W ). +Proof. We look at the first identity. Since tλ(p) = p for any p ∈ S, so it is clear that QQ∨ ∩ DWa ⊂ +CDWa(S). Conversely, let z = � +w∈Wa cwηw ∈ CDWa(S), then for any µ ∈ T ∗, we have +0 = xµz − zxµ = +� +w∈Wa +cw(xµ − xw(µ))ηw. +Therefore, for any cw ̸= 0, we have µ = w(µ) for all µ ∈ T ∗. we can take µ to be W-regular, which +shows that cw ̸= 0 only when w = tλ for some λ ∈ Q∨. So z ∈ k(QQ∨). The first identity is proved. +We now look at the second identity. It follows from Lemma 5.1 that k(DWa/W) ⊂ k(QQ∨)∩DWa. +For the other inclusion, note that ηtλ ∈ DWa is a Q-basis of k(QQ∨). Given any z = � +λ∈Q∨ pληtλ ∈ +k(QQ∨) ∩ DWa, pλ ∈ Q, then pr(z) ∈ pr(DWa) = DWa/W , and +k ◦ pr(z) = k ◦ pr( +� +λ +pληtλ) = k( +� +λ +pληtλ) = +� +λ +pληtλ = z. +Therefore, k(QQ∨) ∩ DWa ⊂ k(DWa/W ). The second identity is proved. +□ + +14 +C. ZHONG +Definition 5.6. We define the Peterson subalgebra to be DQ∨ = k(DWa/W ). +Let Iw, w ∈ Wa be W-compatible. Since DWa/W is a free S-module with basis XIw, w ∈ W − +a , so +k(XIw) form a basis of DQ∨. This is the algebraic model for the oriented homology of the affine +Grassmannian GrG. The following result generalizes [LSS10, Theorem 5.3] in K-theory. +Theorem 5.7. The ring DQ∨ is a Hopf algebra, and the embedding DQ∨ → QQ∨ is an Hopf-algebra +homomorphism. +Proof. The coproduct structure on QWa is defined as △ : QWa → QWa ⊗Q QWa, ηw �→ ηw ⊗ ηw. It is +easy to see that this induces a coproduct structure on QQ∨, and by [CZZ16], it induces a coproduct +structure on DWa. Therefore, it induces a coproduct structure on DQ∨. The product structure is +induced by that of QQ∨, The antipode is s : QQ∨ → QQ∨, ηtλ �→ ηt−λ. It is then routine to check +that DQ∨ is a Hopf algebra and the embedding to QQ∨ is an embedding of Hopf algebras. +□ +Remark 5.8. For K-theory, we know the Hecke algebra is contained DWa. +It is proved by +Berenstein-Kazhdan [BK19] that certain localization of the Hecke algbra is a Hopf algebra. +It +is not difficult to see that it is compatible with the Hopf algebra structure of DQ∨. +5.3. +The following theorem generalizes [LSS10, Theorem 5.4] in the K-theory case and [LS10, +Theorem 6.2] in the cohomology case. +Theorem 5.9. Assume Iw, w ∈ Wa is W-compatible. If u ∈ W − +a , then we have +k(XIu) = XIu + +� +v∈Wa\W − +a +cIu,IvXIv, cIu,Iv ∈ S. +Proof. If w ∈ W − +a , by Lemma 5.4, we have +X∗ +Iw(k(XIu)) = k∗(X∗ +Iw)((XIu)) = X∗ +Iw(XIu) = δw,u, +Therefore, +k(XIu) = +� +v∈Wa +X∗ +Iv(k(XIu))XIv = XIu + +� +v∈Wa\W − +a +cIu,IvXIv. +□ +Example 5.10. Consider the ˆA1 case, then there are two simple roots α1 = α, α0 = −α + δ. By +direct computation, we have +(1) k(X0) = X0 + X1 − x−αX01. +(2) k(X10) = X10 − x−α +xα X01. +(3) k(X010) = X010 + X101 − x−αX1010. +Corollary 5.11. Assume Iw, w ∈ Wa is W-compatible. Let u, v ∈ W − +a . Write +XIuXIv = +� +w∈Wa +dIw +Iu,IvXIw, XIuXIv = +� +w∈W − +a +dIw +Iu,IvXIw, +then +d +Iw3 +Iu,Iv = +� +w2∈Wa +cIu,Iw2d +Iw3 +Iw2,Iv. +Proof. We have +k( +� +w∈W − +a +dIw +Iu,IvXIw) += +k(XIuXIv) = k(XIu)k(XIv) = k(XIu) +� +w1∈Wa +cIv,Iw1XIw1 + +AFFINE GRASSMANNIAN +15 += +� +w1∈Wa +cIv,Iw1k(XIu)XIw1 = +� +w1,w2∈Wa +cIv,Iw1cIu,Iw2XIw2XIw1. +Let w3 ∈ W − +a . By [CZZ19, Theorem 8.2], we know that X∗ +Iw3(XIw2XIw1) = 0 unless w1 ∈ W − +a , in +which case cIv,Iw1 = δKr +v,w1 by Theorem 5.9. Therefore, applying X∗ +Iw3, w3 ∈ W − +a , and using Lemma +5.4, we get +d +Iw3 +Iu,Iv += +X∗ +Iw3( +� +w∈W − +a +dIw +Iu,IvXIw) = k∗(X∗ +Iw3)( +� +w∈W − +a +dIw +Iu,IvXIw) += +X∗ +Iw3(k( +� +w∈W − +a +dIw +Iu,IvXIw)) = X∗ +Iw3( +� +w1,w2∈Wa +cIv,Iw1cIu,Iw2XIw2XIw1) += +� +w2∈Wa +cIu,Iw2X∗ +Iw3(XIw2XIv) = +� +w2∈Wa +cIu,Iw2d +Iw3 +Iw2,Iv. +□ +6. Appendix: Restriction formula in the ˆA1 case +In this Appendix, we perform some computation in the ˆA1 case. +6.1. +In this case, there are two simple roots, α1 = α, α0 = −α + δ, and any w ∈ Wa has a unique +reduced decomposition, so XIw, YIw can be denoted by Xw, Yw, respectively. Moreover, X2 +i = καXi. +We use the notation as in [LSS10, §4.3]. Let +σ0 = e, σ2i = (s1s0)i = t−iα∨, σ−2i = (s0s1)i = tiα∨, σ2i+1 = s0σ2i, σ−(2i+1) = s1σ−2i, i ≥ 1, +and W − +a = {σi|i ≥ 0}. Denote µ = − x−1 +x1 . So if F = Fc with c = 0, then µ = 1, and if F = Fc with +c = 1, then µ = eα if one identifies xα with 1 − e−α. +Let S≤a be the sum h0 + h1 + · · · + ha of homogeneous symmetric functions. Denote Si +≤a to be +S≤a(x, x, · · · , x) where there are i copies of x. For instance, S3 +≤3(x) = 1 + 3x + 6x2 + 10x3. We +have the following identities: +Si +≤a(x) = xSi +≤a−1(x) + Si−1 +≤a (x), +Si +≤a(x) = +a +� +j=0 +xj +� +j + i − 1 +i − 1 +� +. +Then the following identities can be verified by direct computation for lower k and then continued +with induction: +ησ2k += +1 + x2k +1 Xσ2k + +� +1≤j≤k−1 +x2j +1 (S2j +≤k−j(µ−1)Xσ2j + S2j +≤k−j−1(µ−1)Xσ−2j) +− +� +1≤i≤k +x2i−1 +1 +S2i−1 +≤k−i(µ−1)(Xσ2i−1 + Xσ−2i+1), +ησ−2k += +1 + x2k +−1Xσ−2k + +� +1≤j≤k−1 +x2j +−1(S2j +≤k−j−1(µ)Xσ2j + S2j +≤k−j(µ)Xσ−2j) +− +� +1≤i≤k +x2i−1 +−1 S2i−1 +≤k−i(µ)(Xσ2i−1 + Xσ−2i+1), +ησ−2k−1 += +1 − x2k+1 +1 +Xσ−2k−1 + +� +1≤j≤k +x2j +1 S2j +≤k−j(µ−1)(Xσ2j + Xσ−2j) + +16 +C. ZHONG +− +� +1≤i≤k +x2i−1 +1 +� +S2i−1 +≤k−i(µ−1)Xσ2i−1 + S2i−1 +≤k−i+1(µ−1)Xσ−2i+1 +� +, +ησ2k+1 += +1 − x2k+1 +−1 +Xσ2k+1 + +� +1≤j≤k +x2j +−1S2j +≤k−j(µ)(Xσ2j + Xσ−2j) +− +� +1≤i≤k +x2i−1 +−1 +� +S2i−1 +≤k−i+1(µ)Xσ2i−1 + S2i−1 +≤k−i(µ)Xσ−2i+1 +� +. +For F = Fc with c = 1, that is, in the K-theory case, these identities specializes to the corresponding +ones in [LSS10, (4.5), (4.6)] after identifying our −X−αi with Ti in [LSS10] (see Remark 3.1). By +using these identities, following the same idea as in [LSS10, §4.3], one can prove that X∗ +Iw satisfy +the small torus GKM conditions in Theorem 4.3. +Acknowledge. The author would like to thank Cristian Lenart, Changzheng Li and Gufang Zhao +for helpful discussions. +References +[BK19] A. Berenstein, D. Kazhcan, Hecke-Hopf algebras, Advances in Mathematics, 353 (2019) 312-395. 5.8 +[CPZ13] B. Calm´es, V. Petrov, K. Zainoulline, Invariants, torsion indices and oriented cohomology of complete flags, +Annales scientifiques de l’ ´Ecole normale sup´erieure (4) 46(3), 405–448 (2013). 1.1 +[CZZ16] B. Calm`es, K. Zainoulline, and C. Zhong, A coproduct structure on the formal affine Demazure algebra, +Mathematische Zeitschrift, 282 (2016) (3), 1191-1218. 0, 5.2 +[CZZ19] B. Calm`es, K. Zainoulline, and C. Zhong, Push-pull operators on the formal affine Demazure algebra and +its dual, Manuscripta Mathematica, 160 (2019), no. 1-2, 9-50. 0, 1.4, 1.4, 1.4, 1.5, 3.1, 5.3 +[CZZ15] B. Calm`es, K. Zainoulline, and C. Zhong, Equivariant oriented cohomology of flag varieties, Documenta +Mathematica, Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday (2015), 113-144. 0, 1.7 +[CZZ20] B. Calm`es, K. Zainoulline, and C. Zhong, Formal affine Demazure and Hecke algebras associated to Kac- +Moody root systems, Algebra Representation Theory, 23 (2020), no.3, 1031-1050. 0, 1, 1.2, 1.6 +[B97] M. Brion, Equivariant Chow groups for torus actions, Transformation Groups, 2(3): 225-267, 1997. 0 +[GKV97] V. Ginzburg, M. Kapranov, and E. Vasserot, Residue construction of Hecke algebras, Advances in Mathe- +matics 128 (1997), no. 1, 1-19. 1.3, 5.2 +[K18] S. Kato, Loop structure on equivariant K-theory of semi-infinite flag manifolds, arXiv:1805.01718. 0, 3.1 +[KK86] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G∗, Advances +in Mathematics. 62 (1986), no. 3, 187-237. 0 +[KK90] B. Kostant and S. Kumar, T -equivariant K-theory of generalized flag varieties, Journal of Differential Ge- +ometry 32 (1990), 549–603. +[K03] A. Knutson, A Schubert calculus recurrence from the noncomplex W-action on G/B,arXiv:0306304. 0 +0 +[L06] T. Lam, Schubert polynomials for the affine Grassmannian, Journal of the American Mathematical Society, 21 +(1), 3.1, 5.1 +[LLMS18] T. Lam, C. Li, L. Mihalcea, M. Shimozono, A conjectural Peterson isomorphism in K-theory, Journal of +Algebra, 513:326–343, 2018. 0, 3.1 +[LSS10] T. Lam, A. Schilling, M. Shimozono, K-theory Schubert calculus of the affine Grassmannian, Compositio +Mathematica, 146 (2010), no. 4, 811–852. 0, 3.1, 4.2, 5.1, 5.2, 5.2, 5.3, 6.1 +[LS10] T. Lam, M. Shimozono, Quantum cohomology of G/P and homology of affine Grassmannian, Acta Mathe- +matica, 204(1):49–90, 2010. 0, 5.2, 5.3 +[LZZ20] C. Lenart, K. Zainoulline, C. Zhong, Parabolic Kazhdan-Lusztig basis, Schubert classes and equivariant +oriented cohomology, Journal of the Institute of Mathematics of Jussieu, 19 (2020), no. 6, 1889-1929. 0, 1.1, 1.6 +[LM07] M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics. Springer-Verlag, Berlin, +2007. 1.1 +[MNS22] L.C. Mihalcea, H. Naruse, C. Su, Left Demazure-Lusztig operators on equivariant (quantum) cohomology +and K-theory, International Mathematics Research Notices, 2022, no. 16, 12096–12147. 0 +[P97] D. Peterson, Quantum cohomology of G/P, Lecture at MIT, 1997. 0, 3.1, 5.1, 5.2 +(4) 53 (2020), no. 3, 663–711. + +AFFINE GRASSMANNIAN +17 +[T09] J. Tymoczko, Divided difference operators for partial flag varieties, arXiv:0912.2545 0 +[ZZ17] G. Zhao and C. Zhong, Geometric representations of the formal affine Hecke algebra, Advances in Mathemat- +ics, 317 (2017), 50-90. 1.3, 5.2 +State University of New York at Albany, 1400 Washington Ave, CK399, Albany, NY, 12222 +Email address: czhong@albany.edu + diff --git a/1NFPT4oBgHgl3EQfUTQQ/content/tmp_files/load_file.txt b/1NFPT4oBgHgl3EQfUTQQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5738b99ef01bb30d38fc6c7307135e7fed04c045 --- /dev/null +++ b/1NFPT4oBgHgl3EQfUTQQ/content/tmp_files/load_file.txt @@ -0,0 +1,731 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf,len=730 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='13056v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='AG] 30 Jan 2023 EQUIVARIANT ORIENTED HOMOLOGY OF THE AFFINE GRASSMANNIAN CHANGLONG ZHONG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We generalize the property of small-torus equivariant K-homology of the affine Grass- mannian to general oriented (co)homology theory in the sense of Levine and Morel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The main tool we use is the formal affine Demazure algebra associated to the affine root system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the GKM condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the the formal affine Demazure algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Introduction Let h be an oriented cohomology theory in the sense of Levine and Morel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let G be a semi-simple linear algebraic group over C with maximal torus T and a Borel subgroup B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let GrG be the affine Grassmannian of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' T is called the small torus, in contrary to the big torus Ta of GrG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The theory of hTa(GrG) when h is the equivariant cohomology or the K-theory, is studied by Kostant and Kumar in [KK86, KK90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is dual to the so-called affine nil-Hecke algebra (equivariant cohomology case) or the affine 0-Hecke algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Alternatively, the affine nil-Hecke algebra and the affine 0-Hecke algebra can be called the equivariant homology and the equivariant K-homology theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The small torus equivariant homology theory HT(GrG) of the affine Grassmannian was first studied by Peterson [P97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, he raised a conjecture (without a proof) saying that HT(GrG) is isomorphic to the quantum cohomology QHT (G/B)of G/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This conjecture, together with its partial flag variety version, is proved by Lam-Shimozono in [LS10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' One key step is the identification of HT(G/B) with the centralizer of HT (pt) in HT (Ga/Ba) where Ga is the Kac-Moody group associated to the affine root system and Ba is its Borel subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For K-theory, similar property was expected to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In [LSS10], the authors study the K- theoretic Peterson subalgebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', the centralizer of the ring KT (pt) in the small-torus affine 0-Hecke algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', the equivariant K-homology KT (Ga/Ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is proved that this algebra is iso- morphic to KT (GrG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' One of the main tools is the small-torus GKM condition of the T-equivariant K-cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In [LLMS18], some evidence was provided in supporting the K-theory Peterson Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In [K18], using the study of semi-infinite flag variety, Kato proves this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' More precisely, he embeds quantum K-theory of flag variety and certain localization of the Peter- son subalgebra into T-equivariant K-theory of semi-infinite flag variety, and proves that their image coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In all the work mentioned above, the Peterson subalgebra plays key roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In this paper, we generalize the construction of the Peterson subalgebra into general oriented cohomology theory h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Associated to such theory, there is a formal group law F over the coefficient ring R = h(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Associated to F and a Kac-Moody root system, in [CZZ16, CZZ19, CZZ15, CZZ20], the author generalized Kostant-Kumar’s construction and defined the formal affine Demazure algebra (FADA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is a non-commutative algebra generated by the divided difference operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Its dual give an algebraic model for hTa(Ga/Ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since Levine-Morel’s oriented cohomology theory is only defined 1 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG for smooth projective varieties, in this paper we do not intend to generalize the geometric theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Instead, we only work with the algebraic model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', the FADA associated to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Following the same idea as the work mentioned above in cohomology and K-theory, we look at the small-torus (the torus T) version, which is very similar as the big torus case Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We define the small torus FADA, DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In this paper, our first main result (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3) shows that the algebraic models for hT (Ga/Ba) and hT (GrG), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', D∗ Wa and (D∗ Wa)W , satisfy the small torus GKM condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Based on that, we prove the second main result (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5), which shows that the dual of hT (GrG), denoted by DQ∨ (Q∨ being the coroot lattice), coincides with the centralizer of hT (pt) in the FADA DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This defines the Peterson subalgebra associated to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Our result generalizes and extends properties for equivariant cohomology and K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' More- over, our method is uniform and does not reply on the specific oriented cohomology theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' As an application of this construction, we define actions of the FADA (of the big and small torus) on the algebraic models fo hTa(GrG) and hT (GrG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is called the left Hecke action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For finite flag varieties case it is studied in [MNS22] using geometric arguments (see also [B97, K03, T09, LZZ20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For connective K-theory (which specializes to cohomology and K-theory), we compute the recursive formulas for certain basis in hT (GrG) (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is natural to consider generalizing Kato’s construction to this case, that is, invert Schubert classes in DQ∨ corresponding to tλ ∈ Q∨ <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This localization for K-theory was proved to be isomor- phic to QKT (G/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For h beyong singular cohomology and K-theory, however, the first obstruction is that there is no ‘quantum’ oriented cohomology theory defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The other obstruction is that the divided difference operators do not satisfy braid relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This was a key step in Kato’s construction (see [K18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The author plans to investigate this in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This paper is organized as follows: In §1 we recall the construction of the FADA for the big torus Ta, and in §2 we compute the recursive formulas via the left Hecke action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In §3 we we repeat the construction for the small torus and indicates the difference from the big torus case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In §4 we prove that dual of the small torus FADA satisfies the small torus GKM condition, and in §5 we define the Peterson subalgebra and show that it coincides with the centralizer of hT (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In the appendix we provide some computational result in the ˆA1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let G ⊃ B ⊃ T be such that G is simple, simply connected algebraic group over C with a Borel subgroup B and a torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let Ga ⊃ Ba ⊃ Ta where Ga is the affine Kac-Moody group with Borel subgroup Ba and the affine torus Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let P be the maximal parabolic group scheme so that Ga/P = GrG is the affine Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let T ∗ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' T ∗ a ) be the group of characters of T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Ta), then T ∗ a = T ∗ ⊕ Zδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let W be the Weyl group of G, I = {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', αn} be the simple roots, Q = ⊕iZαi ⊂ T ∗ be the root lattice, Q∨ = ⊕iZα∨ i be the coroot lattice, θ be the longest element, δ is the null root, α0 = −θ + δ be the extra simple root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote Ia = {α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', αn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For each λ ∈ Q∨, let tλ be the translation acting on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We then have tλ1tλ2 = tλ1+λ2, and wtλw−1 = tw(λ), w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let Q∨ ≤ be the set of antidominant coroots, Q∨ < be the set of strictly antidominant coroots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', (λ, αi) < 0 ∀i ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let Wa = W ⋉ Q∨ be the affine Weyl group, ℓ be the length function on Wa, and w0 ∈ W be the longest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let Φ be the set of roots for W, Φa = Zδ + Φ be the set of real affine roots, and Φ± a , Φ± be the corresponding set of positive/negative roots for the corresponding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let inv(w) = w−1Φ+ a ∩ Φ− a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have Φ+ a = {α + kδ|α ∈ Φ+, k = 0 or α ∈ Φ, k > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' AFFINE GRASSMANNIAN 3 Let W − a be the minimal length representatives of Wa/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' There is a bijection W − a → Q∨, w �→ λ, if wW = tλW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, W − a ∩ Q∨ = {tλ|λ ∈ Q≤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The action of α + kδ on µ + mδ ∈ Q ⊕ Zδ is given by sα+kδ(µ + mδ) = µ + mδ − ⟨µ, α∨⟩(α + kδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In particular, for λ ∈ Q∨, w ∈ W, µ ∈ Q, we have sα+kδ = sαtkα∨, wtλ(µ) = w(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We say the set of reduced sequences Iw, w ∈ Wa is W-compatible if Iw = Iu ∪ Iv for w = uv, u ∈ W − a , v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' FADA for the big torus In this section, we recall the construction of the formal affine Demazure algebra (FADA) for the affine root system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' All the construction can be found in [CZZ20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let F be a one dimensional formal group law over a domain R with characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Following from [LM07] that there is an oriented cohomology h whose associated formal group law is F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In this paper we won’t need any geometric property of this h, since our treatment is pure algebraic and self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let F = Fc = x + y − cxy be the connective formal group law (for connective K-theory) over R = Z[c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Specializing to c = 0 or c = 1, one obtains the additive or multiplicative formal group law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' One of the simplest formal group laws beyond Fc is the hyperbolic formal group law considered in [LZZ20]: F(x, y) = x + y − cxy 1 + axy , R = Z[c, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let ˆS be the formal group algebra of T ∗ a defined in [CPZ13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' That is, ˆS = R[[xµ|µ ∈ T ∗ a ]]/JF , where JF is the closure of the ideal generated x0 and xµ1+µ2 − F(xµ1, xµ2), µ1, µ2 ∈ T ∗ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Indeed, after fixing a basis of T ∗ a ∼= Zn+1, ˆS is isomorphic to the power series ring R[[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', xn+1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If F = Fc is the connective formal group law, one can just replace ˆS by R[xµ|µ ∈ T ∗ a ]/JF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In other words, in this case one can use the polynomial ring instead of the power series ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For instance, if c = 0, then ˆS ∼= SymR(T ∗ a ), xµ �→ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If c ∈ R×, then ˆS ∼= R[T ∗ a ], xµ �→ c−1(1 − e−µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Throughout this paper, whenever we specializes to Fc, we assume that ˆS is the polynomial version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Define ˆQ = ˆS[ 1 xα , α ∈ Φa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The Weyl groups Wa acts on ˆQ, so we can define the twisted group algebra ˆQWa := ˆQ ⋊ R[Wa],, which is a free left ˆQ-module with basis denoted by ηw, w ∈ Wa and the product cηwc′ηw′ = cw(c′)ηww′, c, c′ ∈ ˆQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For each α ∈ Φa, define κα = 1 xα + 1 x−α ∈ ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If F = Fc, then κα = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For each simple root αi, we define the Demazure element ˆXαi = 1 xαi (1 − ηsi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to check that ˆX2 α = κα ˆXα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For simplicity, denote ηi = ηαi = ηsi, x±i = x±αi, ˆXi = ˆXαi, i ∈ Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If Iw = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', ik), ij ∈ Ia is a reduced sequence of w ∈ Wa, we define ˆXIw correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is well known that they depends on the choice of Iw, unless F = Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Write (1) ˆXIw = � v≤w ˆaIw,vηv, ηw = � v≤w ˆbw,Iv ˆXIv, ˆaIw,v ∈ ˆQ, ˆbw,Iv ∈ ˆS, 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG then we have ˆbw,Iw = � α∈inv(w) xα = 1 ˆaIw,w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let ˆDWa be the subalgebra of ˆQWa generated by ˆS and ˆXi, i ∈ Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is called the formal affine Demazure algebra (FADA) for the big torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to see that ˆXIw, w ∈ Wa is a ˆQ-basis of ˆQWa, and it is proved in [CZZ20] that it is also a basis of the left ˆS-module ˆDWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that W ⊂ ˆDWa via the map si �→ ηi = 1 − xi ˆXi ∈ ˆDWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is not difficult to derive that there is a residue description of the coefficients in the expression of elements of ˆDWa as linear combinations of ηw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Such description was first given in [GKV97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' See [ZZ17] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We define the duals of left modules: ˆQ∗ Wa = Hom ˆ Q( ˆQWa, ˆQ) = Hom(Wa, ˆQ), ˆD∗ Wa = Hom ˆS( ˆDWa, ˆS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Dual to the elements ηw, ˆXIw ∈ ˆDWa ⊂ ˆQWa, we have ˆfw, ˆX∗ Iw ∈ ˆD∗ Wa ⊂ ˆQ∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The product structure on ˆQ∗ Wa is defined by ˆfw ˆfv = δw,v ˆfw, with the unit given by 1 = � w∈Wa ˆfw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that here we usually use � to denote a sum of (possibly) infinitely many terms, and � to denote a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have ˆD∗ Wa = { ˆf ∈ ˆQ∗ Wa| ˆf( ˆDWa) ⊂ ˆS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote the RHS by Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is clear that ˆD∗ Wa is contained in Z1 since ˆXIv generate ˆDWa, ˆX∗ Iw generate ˆD∗ Wa, and ˆX∗ Iw( ˆXIv) = δw,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Conversely, let ˆf = � ℓ(w)≥k cw ˆfw ∈ Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If ℓ(u) = k, then from (1), we have ˆf( ˆXIu) = � ℓ(w)≥k cw ˆfw( � v≤u ˆaIu,vηv) = cuˆaIu,u ∈ ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote ˆf ′ := ˆf −� ℓ(u)=k cuˆaIu,u ˆX∗ Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that ˆX∗ Iu = � w∈Wa ˆbw,Iufw and ˆbu,IuˆaIu,u = 1, so for any u with ℓ(u) = k, we have ˆf ′(ηu) = cu − cuˆaIu,u ˆX∗ Iu(ηu) = cu − cu = 0, so ˆf ′ is a linear combination of ˆfw, ℓ(w) ≥ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Repeating this process, we get that ˆf ∈ ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' There is an ˆQ-linear action of ˆQWa on ˆQ∗ Wa, defined by (z • ˆf)(z′) = ˆf(z′z), z, z′ ∈ ˆQWa, ˆf ∈ ˆQ∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is called the right Hecke action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have cηw • c′ ˆfw′ = c′w′w−1(c) ˆfw′w−1, c, c′ ∈ ˆQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4 and similar reason as in [CZZ19, §10] that this induces an action of ˆDWa on ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, it induces an action of W ⊂ ˆDWa on ˆQ∗ Wa and ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By definition it is easy to get ˆXα • � w∈Wa cw ˆfw = � w∈Wa cw − csw(α)w xw(α) ˆfw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (2) The following proposition is proved in the finite case in [CZZ19, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The subset ˆD∗ Wa ⊂ ˆQ∗ Wa satisfies the following (big-torus) GKM condition: ˆD∗ Wa = { ˆf ∈ ˆQ∗ Wa| ˆf(ηw) ∈ ˆS and ˆf(ηw − ηsαw) ∈ xα ˆS, ∀α ∈ Φa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' AFFINE GRASSMANNIAN 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote the RHS by Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let ˆf ∈ ˆD∗ Wa, we know ˆXα • ˆf ∈ ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Then (2) implies that ˆf satisfies the condition defining Z2, so ˆD∗ Wa ⊂ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For the other direction, we first show that ˆD∗ Wa is a maximal ˆDWa-submodule of ˆS∗ Wa := Hom(Wa, ˆS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This can be proved as follows: if M ⊂ ˆS∗ Wa is a ˆDWa-module, for any ˆf ∈ M, we have ˆXI • ˆf ∈ M ⊂ ˆS∗ Wa, so ˆXI • ˆf(ηe) = ˆf( ˆXI) ∈ ˆS, so f ∈ ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' One then can show that the subset Z2 is a ˆDWa-module, which follows from the same proof as in the finite case in [CZZ19, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since ˆD∗ Wa is a maximal submodule, we have Z2 ⊂ ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We can similarly define the non-commutative ring ˆQQ∨ = ˆQ ⋊ R[Q∨] with a ˆQ-basis ηtλ, λ ∈ Q∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Then there is a canonical map of left ˆQ-modules: pr : ˆQWa → ˆQQ∨, cηtλw �→ cηtλ, w ∈ W, λ ∈ Q∨, c ∈ ˆQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Define ˆDWa/W = pr( ˆDWa) ⊂ ˆQQ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Indeed, this is the same as the relative Demazure module defined in [CZZ19, §11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We can also consider the ˆQ-dual ˆQ∗ Q∨ and the ˆS-dual ˆD∗ Wa/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The elements dual to ηtλ ∈ ˆQQ∨ are denoted by ˆftλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The projection pr then induces embeddings pr∗ : ˆQ∗ Q∨ ֒→ ˆQ∗ Wa and pr∗ : ˆDWa/W ֒→ ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to see that pr∗( ˆftλ) = � v∈W ˆftλv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, similar as in the finite case [CZZ19, Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7], we have pr∗( ˆQ∗ Q∨) = ( ˆQ∗ Wa)W , pr∗( ˆD∗ Wa/W) = ( ˆD∗ Wa)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Indeed, elements of pr∗( ˆQ∗ Q∨) = ( ˆQ∗ Wa)W are precisely the elements ˆf ∈ ˆQ∗ Wa satisfying ˆf(ηtλw − ηtλ) = 0 for any λ ∈ Q∨, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It follows from similar reason as [CZZ19, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5, Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5] that if Iw, w ∈ Wa is W-compatible, then ˆbuv,Iw = ˆbu,Iw for any v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We then have Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Assume the sequences Iw, w ∈ Wa is W-compatible, then pr(XIw), w ∈ W − a is a basis of ˆDWa/W , and { ˆX∗ Iw, w ∈ W − a } is a ˆQ-basis of ( ˆQ∗ Wa)W and a ˆS-basis of ( ˆD∗ Wa)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that ( ˆD∗ Wa)W is the algebraic model for hTa(GrG) and the embedding ( ˆD∗ Wa)W ⊂ ˆD∗ Wa is the algebraic model for the pull-back hTa(GrG) → hTa(Ga/Ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Similar as the finite case in [LZZ20, §3], there is another action of ˆQWa on ˆQ∗ Wa by aηv ⊙ b ˆfw = av(b) ˆfvw, a, b ∈ ˆQ, w, v ∈ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is called the left Hecke action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to see that it commutes with the •-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note however that the ⊙-action is not ˆQ-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The ⊙ action of ˆQWa on ˆQ∗ Wa induces an action of ˆDWa on ˆD∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have ˆXα ⊙ � w cw ˆfw = 1 xα (1 − ηα) ⊙ � w cw ˆfw = � w cw − sα(csαw) xα ˆfw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' let dw,α = cw−sα(csαw) xα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We show that dw,α satisfy the big-torus GKM condition, that is, dw,α − dsβw,α ∈ xβ ˆS for any β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG Denote cw − csαw = xαp, p ∈ ˆS and x−α = −xα + x2 αq, q ∈ ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If β = α, then we have dw,α − dsβw,α = cw − sα(csαw) − csαw + sα(cw) xα = xαp + sα(cw) − sα(cw − xαp) xα = p + x−αsα(p) xα = p − sα(p) + xαq, which is clearly a multiple of xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If β ̸= α, then dw,α − dsβw,α = cw − sα(csαw) − (csβw − sα(csαsβw)) xα = cw − sα(csαw) − csβw + sα(csαsβw) xα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since xα, xβ are coprime [CZZ20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2], it suffices to prove the numerator is divisible by xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note cw − csβw is already divisible by xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Furthermore, cw − csαsβw = cw − cssα(β)sαw, so it is divisible by ssα(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, −sα(csαw) + sα(csαsβw) is divisible by sα(xsα(β)) = xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Consequently, the ⊙-action of ˆDWa on ˆD∗ Wa restricts to an action on ( ˆD∗ Wa)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Indeed, there is a characteristic map c : ˆS → ˆD∗ Wa, z �→ z • 1, whose geometric model is the map sending a character of the torus to the first Chern class of the associated line bundle over the flag variety [CZZ15, §10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We then have a map φ : ˆS ⊗ ˆSWa ˆS → ˆD∗ Wa, a ⊗ b �→ ac(b) = � w aw(b) ˆfw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is proved to be an isomorphism in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to see that for any z ∈ DWa, there are the following commutative diagrams ˆS ⊗ ˆSWa ˆS φ � z· ⊗id � ˆD∗ Wa z⊙ � ˆS ⊗ ˆSWa ˆS φ � ˆD∗ Wa , ˆS ⊗ ˆSWa ˆS φ � id ⊗z· � ˆD∗ Wa z• � ˆS ⊗ ˆSWa ˆS φ � ˆD∗ Wa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Equivariant connective K-theory of the affine Grassmannian As an application of the left Hecke action, we derive the recursive formulas for this action on bases in connective K-theory of GrG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In this section only, assume F = Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Our results specialize to equivariant K-theory (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' equivariant cohomology) by letting c = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' c = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In both cases, our results are only known for flag varieties of finite root systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since ˆXi do not satisfy the braid relations, the result of this section do not generalize to general F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote ǫw = (−1)ℓ(w) and cw = cℓ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have x−α = xα cxα−1 and κα = c for any α, and ˆXIw can be denoted by ˆXw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that there is another operator ˆYi = ˆYαi = c − ˆXαi such that ˆY 2 αi = c ˆYαi and braid rela- tions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is the algebraic model of the composition hTa(Ga/Ba) → hTa(Ga/Pi) → hTa(Ga/Ba) where Pi is the minimal parabolic subgroup corresponding to αi ∈ Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, we have ˆXw = � v≤w ǫvcwc−1 v ˆYv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' AFFINE GRASSMANNIAN 7 Most properties of ˆXw are also satisfied by ˆYw, except for Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Indeed, ˆY ∗ w, w ∈ W − a is not W-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote xΦ = � α∈Φ− xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is well known that Yw0 = � w∈W ηw 1 xΦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, the map Yw0 • : ˆD∗ Wa → ( ˆD∗ Wa)W is the algebraic model for the map hTa(Ga/Ba) → hTa(GrG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We first compute the image of the two bases via this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let F = Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For any w ∈ Wa and u = u1u2, u1 ∈ W − a , u2 ∈ W, we have Yw0 • ˆX∗ u1u2 = ǫu2cw0c−1 u2 ˆX∗ u1, Yw0 • ˆY ∗ w = � v1v2≥w,v1∈W − a ,v2∈W ǫwǫv2cv1w0c−1 w ˆX∗ v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In particular, Yw0 • ˆY ∗ w, w ∈ W − a is a basis of ( ˆD∗ Wa)W if and only if c ∈ R×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For each v ∈ Wa, write v = v1v2, v1 ∈ W − a , v2 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' From ˆXwYw0 = 0, w ∈ W, we have (Yw0• ˆX∗ u1u2)( ˆXv1v2) = ˆX∗ u1u2( ˆXv1v2Yw0) = δv2,e ˆX∗ u1u2( ˆXv1 � w′≤w0 ǫw′cw0c−1 w′ ˆXw′) = δv2,eδv1,u1ǫu2cw0c−1 u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This proves the first identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For the second one, it is easy to see that ˆY ∗ w = � v≥w ǫwcvc−1 w ˆX∗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' So Yw0 • ˆY ∗ w = Yw0 • � v≥w ǫwcvc−1 w ˆX∗ v = � v1v2≥w,v1∈W − a ,v2∈W ǫwǫv2cv1w0c−1 w ˆX∗ v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This proves the second identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The transition matrix between ˆX∗ v, v ∈ W − a and Yw0 • ˆY ∗ w, w ∈ W − a is upper triangular with diagonal entries ǫwcw0, so the last statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Before computing the ⊙-action, we need to prove some identities in ˆDWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let F = Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Writing ηu = � v≤u ˆbu,v ˆXv = � v≤u ˆbY u,v ˆYv, then ˆbsiu,v = � si(ˆbu,v), siv > v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (1 − cxi)si(ˆbu,v) − xisi(ˆbu,siv), siv < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ˆbY siu,v = � (1 − cxi)si(ˆbY u,v), siv > v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' xisi(ˆbY u,siv) + si(ˆbY u,v), siv < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We prove the first one, and the second one follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote iWa = {v ∈ Wa|siv > v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have ηsiu = ηiηu = ηi � v∈iWa ˆbu,v ˆXv + ˆbu,siv ˆXsiv = � v∈iWa si(ˆbu,v)ηi ˆXv + si(ˆbu,siv)ηi ˆXsiv = � v∈iWa si(ˆbu,v)(1 − xi ˆXi) ˆXv + si(ˆbu,siv)(1 − xi ˆXi) ˆXsiv = � v∈iWa si(ˆbu,v)( ˆXv − xi ˆXsiv) + si(ˆbu,siv) ˆXsiv − cxisi(ˆbu,siv) ˆXv = � v∈iWa si(ˆbu,v) ˆXv + (si(ˆbu,siv)(1 − cxi) − xisi(ˆbu,v)) ˆXsiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The conclusion then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Note that if v ∈ W − a and siv < v, then siv ∈ W − a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have the following recursive formula, whose proof follows from the definition and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For F = Fc, with i ∈ Ia, we have ˆX−i ⊙ ˆX∗ v = � 0, siv > v, c ˆX∗ v + ˆX∗ siv, siv < v, ˆY−i ⊙ ˆY ∗ v = � 0, siv > v, c ˆY ∗ v + ˆY ∗ siv, siv < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Here ˆX−i = ηw0 ˆXiηw0 = 1 x−i (1 − ηi), ˆY−i = ηw0 ˆYiηw0 = 1 xi + 1 x−i ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Consequently, if v ∈ W − a , we have ˆY−i ⊙ (Yw0 • Y ∗ v ) = � 0, siv > v, c(Yw0 • ˆY ∗ v ) + (Yw0 • ˆY ∗ siv), siv < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have ˆX−i ⊙ ˆX∗ v = ( 1 x−i − 1 x−i ηi) ⊙ � u≥v ˆbu,v ˆfu = � u ˆbu,v x−i ˆfu − � u si(ˆbu,v) x−i ˆfsiu = � u ˆbu,v − si(ˆbsiu,v) x−i ˆfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Plugging the formula in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, we obtain the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The formula for ˆY−i ⊙ ˆY ∗ v follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' From the commutativity of the two actions • and ⊙, one obtains the last statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' FADA for the small torus We repeat the construction of FADA for the small torus, which is very similar as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let S be the formal group algebra associated to T ∗, that is, it is (non-canonically) isomorphic a power series ring of rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' When the formal group law F = Fc, we can again take the polynomial version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let Q = S[ 1 xα , α ∈ Φ], QWa = Q ⋊ R[Wa], QQ∨ = Q ⋊ R[Q∨].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For any α ∈ Φ, let κα = 1 xα + 1 x−α and κα0 = 1 x−θ + 1 xθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have the projection pr : QWa → QQ∨, ηtλw �→ ηtλ, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Define Xα = 1 xα (1 − ηα), Xα0 = 1 x−θ (1 − ηs0), α ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For simplicity, denote x±i = x±αi, Xi = Xαi, ηi = ηsi, X0 = Xα0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' They satisfy relations similar as that of ˆXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' One can define XIw for any reduced sequence Iw of w, which depends only on w if F = Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Consider K-theory, in which case F = Fc with c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Our −X−αi is the Ti in [LSS10, LLMS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Our 1 − Xαi coincides with the Di in [K18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For cohomology, c = 0, κα = 0, and our Xi is the Ai in [P97, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='11] and [L06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have pr(zXi) = 0 if z ∈ QWa, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let z = pηw, p ∈ Q, w ∈ Wa , then pr(zXi) = pr(pηwXi) = pr( p w(xi)(ηw − ηwsi)) = p w(xi)(pr(ηw) − pr(ηwsi)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ AFFINE GRASSMANNIAN 9 Define DWa to be the subalgebra of QWa generated by S and Xi, i ∈ Ia, and DWa/W = pr(DWa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Then DWa is a free left S-module with basis XIw, w ∈ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote XIw = pr(XIw), w ∈ W − a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If Iw, w ∈ Wa are W-compatible, then the set {XIw|w ∈ W − a } is a basis of the left S-module DWa/W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' They follow easily from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' See [CZZ19, Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ The projection p : T ∗ a → T ∗, µ + kδ �→ µ induces projections ˆS → S, ˆQ → Q and ˆQWa → QWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Clearly p( ˆXαi) = Xαi and p( ˆXIw) = XIw, so p( ˆDWa) = DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' More explicitly, we have ˆXIw = � v≤w ˆaIw,vηv ∈ ˆQWa, XIw = � v≤w aIw,vηv ∈ QWa, p(ˆaIw,v) = aIw,v ∈ Q, ηw = � v≤w ˆbw,Iv ˆXIv ∈ ˆQWa, ηw = � v≤w bw,IvXIv ∈ QWa, p(ˆbw,Iv) = bw,Iv ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that the embedding i : Q → ˆQ induces a section QWa → ˆQWa of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' However, it does not map DWa to ˆDWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For example, X0 is mapped to x−θ+δ x−θ ˆX0 which does not belong to ˆDWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' As before, we can take the duals, which will give us Q-modules Q∗ Wa, Q∗ Q∨, and S-modules D∗ Wa, D∗ Wa/W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The elements dual to ηw, XIw ∈ DWa ⊂ QWa, ηtλ, XIw ∈ DWa/W ⊂ QQ∨, are denoted by fw, X∗ Iw ∈ D∗ Wa ⊂ Q∗ Wa, ftλ, X∗ Iw ∈ D∗ Wa/W ⊂ Q∗ Q∨, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that the notation ftλ can be thought as in Q∗ Wa and Q∗ Q∨, just like ηtλ can be thought as in QWa and QQ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Similar as Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5, we have (3) D∗ Wa = {f ∈ Q∗ Wa|f(DWa) ⊂ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, by definition, the dual map pr∗ : Q∗ Q∨ → Q∗ Wa satisfies pr∗(ftλ) = � w∈W ftλw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Following from the definition, we have ˆX∗ Iw = � v≥w ˆbv,Iw ˆfv ∈ ˆD∗ Wa, X∗ Iw = � v≥w bv,Iwfv ∈ D∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since p(ˆbv,Iw) = bv,Iw, so the map q : ˆQ∗ Wa → Q∗ Wa, � w aw ˆfw �→ � w p(aw)fw induces a map q : ˆD∗ Wa → D∗ Wa such that q( ˆX∗ Iw) = X∗ Iw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, since p∗(X∗ Iw)( ˆXIv) = X∗ Iw(p( ˆXIv)) = X∗ Iw(XIv) = δw,v, so p∗(X∗ Iw) = ˆX∗ Iw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that neither q nor p∗ are isomorphisms, since the domains and targets are modules over different rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Similar as Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6, we have Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If Iw, w ∈ Wa are W-compatible, then the set X∗ Iw, w ∈ W − a form a basis of (Q∗ Wa)W and of (D∗ Wa)W , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Assume that {Iw, w ∈ Wa} is W-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For any w ∈ W, u ∈ W − a , we have X∗ Iu = � λ∈Q∨ btλw,Iuftλ ∈ Q∗ Q∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For any λ ∈ Q∨, write ηtλw = � u∈W − a ,v∈W btλw,Iu∪IvXIu∪Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, we have ηtλ = pr(ηtλw) = � u∈W − a ,v∈W btλw,Iu∪Iv pr(XIu∪Iv) = � u∈W − a btλw,Iu pr(XIu) = � u∈W − a btλw,IuXIu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, X∗ Iu = � λ∈Q∨ btλw,Iuftλ ∈ Q∗ Q∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ This lemma implies that we have pr∗(X∗ Iu) = X∗ Iu, u ∈ W − a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' There is a •-action of QWa on Q∗ Wa, defined similar as the big torus case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The •-action of QWa on Q∗ Warestricts to an action of DWa on D∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since DWa is a S-module with basis XIu, u ∈ Wa, so for any w, v ∈ Wa, i ∈ Ia, we have XIvXi = � u cIv∪si,IuXIu with cIv∪si,Iu ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have (Xi • X∗ Iw)(XIv) = X∗ Iw(XIvXi) = cIv∪si,Iw ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By (3), Xi • X∗ Iw ∈ D∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have D∗ Wa ⊂ {f ∈ Q∗ Wa|f(ηw) ∈ S, and f(ηw − ηsαw) ∈ xαS, ∀α ∈ Φ, w ∈ Wa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' One of the main results of this paper is to study how different the two sets are, that is, to derive the small torus GKM condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since ηw ∈ DWa, then it follows from (3) that f(ηw) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let i ∈ I and f = � w∈Wa awfw ∈ D∗ Wa with aw = f(ηw) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have Xi•f = 1 xi (1−ηi)• � w awfw = � w aw w(xi)fw− � w aw wsi(xi)fwsi = � w aw − awsi w(xi) = � w aw − asw(αi)w xw(αi) fw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6, Xi • f ∈ D∗ Wa, so f(ηw − ηsβw) = aw−asβw xβ ∈ S for any β ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We can similarly define the ⊙ action aηw ⊙ bfv = aw(b)fwv, w, v ∈ Wa, a, b ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to see that the ⊙ and the • actions commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For any ˆz ∈ ˆQWa, ˆf ∈ ˆQ∗ Wa, we have p(ˆz) ⊙ q( ˆf) = q(ˆz ⊙ ˆf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In particular, the ⊙-action of QWa on Q∗ Wa induces an action of DWa on D∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' AFFINE GRASSMANNIAN 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Write ˆz = ˆaηv, ˆf = ˆb ˆfw, ˆa,ˆb ∈ ˆQ, w, v ∈ Wa and suppose p(ˆa) = a, p(ˆb) = b, then p(ˆz) ⊙ q( ˆf) = aηv ⊙ bfw = av(b)fvw = q(ˆav(ˆb) ˆfvw) = q(ˆaηv ⊙ ˆb ˆfw) = q(ˆz ⊙ ˆf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For the second part, note that p : ˆDWa → DWa and q : ˆD∗ Wa → D∗ Wa are both surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Given z ∈ DWa and f ∈ D∗ Wa, suppose z = p(ˆz) and f = q( ˆf) for some ˆz ∈ ˆDWa and ˆf ∈ ˆD∗ Wa, then z ⊙ f = p(ˆz) ⊙ q( ˆf) = q(ˆz ⊙ ˆf) ∈ q( ˆD∗ Wa) = D∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If F = Fc, then all results in §2 holds for X∗ w and the corresponding Y ∗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The small-torus GKM condition In this section, we study the small-torus GKM condition on the equivariant oriented cohomology of the affine flag variety and of the affine Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For each α ∈ Φ, we define Zα = 1 x−α (1 − ηtα∨) ∈ QWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For each α ∈ Φ, we have Zα ∈ DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It suffices to show that Zα is contained in the subalgebra of DWa generated by S and Xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' So we assume the root system is the affine root system of SL2 with simple roots α1 = α, α0 = −α + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Then tα∨ = s0s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have ηs1 = 1 − xαX1, ηs0 = 1 − x−αX0, so ηs0s1 = 1 − x−αX0 − x−αX1 + x2 −αX0X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, Zα = 1 x−α (1 − ηs0s1) = X0 + X1 − x−αX0X1 ∈ DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Suppose the root system is ˆA1 with two simple roots α1 = α, α0 = −α + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (1) If F = Fc with c = 0, then we have Zα = X0 + X1 + αX0X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (2) If F = Fc with c = 1, then we have Zα = X0 + X1 + (eα − 1)X0X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since DWa acts on D∗ Wa, so we know that Zα acts on D∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that Zk α = 1 xkα (1 − ηtα∨)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We are now ready to prove the first main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (1) The subset D∗ Wa ⊂ Q∗ Wa consists of elements satisfying the following small- torus GKM condition: f � (1 − ηtα∨)dηw � ∈ xd αS, and f � (1 − ηtα∨)d−1(1 − ηsα)ηw � ∈ xd αS, ∀α ∈ Φ, w ∈ Wa, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (2) The subset (D∗ Wa)W ⊂ (Q∗ Wa)W consists of elements satisfying the following small-torus Grassmannian condition: f � (1 − ηtα∨)dηw � ∈ xd αS, ∀α ∈ Φ, w ∈ Wa, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG Our proof follows similarly as that of [LSS10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The key improvement is that we don’t need to prove Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=', since we can use the operators Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' However, for the convenience of the readers, we include an appendix, which gives all coefficients of bw,Iv in the ˆA1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' They can be used to show that X∗ Iw satisfy the small torus GKM condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We prove that elements of D∗ Wa satisfy the small-torus GKM condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let f = � w cwfw ∈ D∗ Wa, we have Zα • � w cwfw = � w ( cw w(x−α)fw − cw wt−α∨(x−α)fwt−α∨) = � w cw − ctw(α∨)w x−w(α) fw ∈ D∗ Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that xα x−α is invertible in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, denoting w(α) = β, by (3), we have f((1 − ηtβ∨)ηw) ∈ xβS for any β ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, denote dw = cw−cwtα∨ x−w(α) , then dwtα∨ = cwtα∨ −cwtα∨ tα∨ x−wtα∨ (α) = cwtα∨ −cwt2α∨ x−w(α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, We have Z2 α • f = Zα • Zα • � w cwfw = � w (dw − dwtα∨ w(x−α) )fw = � w cw − 2cwtα∨ + cwt2α∨ w(x−α)2 fw = � w cw − 2ctw(α∨)w + ct2w(α∨)w x2 −w(α) fw = � w 1 x2 −w(α) f((1 − ηtw(α∨))2ηw)fw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denoting w(α) = β, we see that f((1−ηtβ∨)2ηw) ∈ x2 βS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Inductively, we see that f((1−ηtα∨ )dηw) ∈ xd αS for all d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Similarly, if one applies Zd−1 α Xα ∈ DWa on f, which gives Zd−1 α Xα • f ∈ D∗ Wa, one will see that f satisfies the second condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For the rest of the proof and for that of (2), it is identical to that of [LSS10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3], so it is skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The subset D∗ Wa/W ⊂ Q∗ Q∨ consists of elements satisfying the following small torus Grassmannian condition: f((1 − ηt∨ α)dηtλ) ∈ xd αS, ∀α ∈ Φ, d ≥ 1, λ ∈ Q∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This follows from the identity pr∗(ftλ) = � v∈W ftλv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The Peterson subalgebra In this section, we embed DWa/W into DWa and show that it coincides with the centralizer of S in DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is called the Peterson subalgebra, which gives the algebraic model for the equivariant oriented ‘homology’ of the affine Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have a canonical ring embedding (and also a Q-module embedding) k : QQ∨ → QWa, pηtλ �→ pηtλ, such that pr ◦k = idQWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to see that the dual map k∗ : Q∗ Wa → Q∗ Q∨ satisfies (4) k∗(ftλu) = δu,eftλ, u ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For K-theory, our map k is the map k : KT (GrG) → K in [LSS10, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2], and k∗ is the wrong-way map ̟ of [LSS10, $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The following lemma generalizes [P97], [L06, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4] for the cohomology case, and [LSS10, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6] for the K-theory case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' AFFINE GRASSMANNIAN 13 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The map k∗ induces a map k∗ : D∗ Wa → D∗ Wa/W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Consequently, the map k induces a map k : DWa/W → DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Given f ∈ D∗ Wa, then f satisfies the small-torus GKM condition Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3, that is, f((1 − ηtα∨)dηtλu) ∈ xd αS, ∀u ∈ W, λ ∈ Q∨, α ∈ Φ, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, k∗(f)((1 − ηtα∨)dηtλ) = f � k((1 − ηtα∨)dηtλ) � = f((1 − ηt∨ α)dηtλ) ∈ xd αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4, k∗(f) ∈ D∗ Wa/W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It would be interesting to find a direct proof of the fact that k maps DWa/W to DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' One possible choice is to find the small torus residue condition of DWa similar to the residue condition of [GKV97] (see [ZZ17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Note that this result is not true for the big torus case, that is, k( ˆDWa/W ) is not contained in ˆDWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For example, in the ˆA1 case, we have pr(X0) = pr( 1 x−α+δ (1 − ηtα∨s1)) = 1 x−α+δ (1 − ηtα∨) ∈ ˆDWa/W , and k(pr(X0)) = 1 x−α+δ (1 − ηtα) = 1 x−α+δ (1 − ηs0s1) ̸∈ ˆDWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If Iw, w ∈ W is W-compatible, then k∗(X∗ Iu) = X∗ Iu for any u ∈ W − a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By (4) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5, we have k∗(X∗ Iu) = k∗( � λ∈Q∨,w∈W btλw,Iuftλw) = � λ∈Q∨ btλ,Iuftλ = X∗ Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let CDWa(S) be the centralizer of S in DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Our second main result is the following, which generalizes [LSS10, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2] in the K-theory case and [P97, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3] in the cohomology case (proved in [LS10, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have CDWa(S) = k(QQ∨) ∩ DWa = k(DWa/W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We look at the first identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since tλ(p) = p for any p ∈ S, so it is clear that QQ∨ ∩ DWa ⊂ CDWa(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Conversely, let z = � w∈Wa cwηw ∈ CDWa(S), then for any µ ∈ T ∗, we have 0 = xµz − zxµ = � w∈Wa cw(xµ − xw(µ))ηw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, for any cw ̸= 0, we have µ = w(µ) for all µ ∈ T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' we can take µ to be W-regular, which shows that cw ̸= 0 only when w = tλ for some λ ∈ Q∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' So z ∈ k(QQ∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The first identity is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We now look at the second identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1 that k(DWa/W) ⊂ k(QQ∨)∩DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For the other inclusion, note that ηtλ ∈ DWa is a Q-basis of k(QQ∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Given any z = � λ∈Q∨ pληtλ ∈ k(QQ∨) ∩ DWa, pλ ∈ Q, then pr(z) ∈ pr(DWa) = DWa/W , and k ◦ pr(z) = k ◦ pr( � λ pληtλ) = k( � λ pληtλ) = � λ pληtλ = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, k(QQ∨) ∩ DWa ⊂ k(DWa/W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The second identity is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We define the Peterson subalgebra to be DQ∨ = k(DWa/W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let Iw, w ∈ Wa be W-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Since DWa/W is a free S-module with basis XIw, w ∈ W − a , so k(XIw) form a basis of DQ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' This is the algebraic model for the oriented homology of the affine Grassmannian GrG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The following result generalizes [LSS10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3] in K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The ring DQ∨ is a Hopf algebra, and the embedding DQ∨ → QQ∨ is an Hopf-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The coproduct structure on QWa is defined as △ : QWa → QWa ⊗Q QWa, ηw �→ ηw ⊗ ηw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is easy to see that this induces a coproduct structure on QQ∨, and by [CZZ16], it induces a coproduct structure on DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, it induces a coproduct structure on DQ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The product structure is induced by that of QQ∨, The antipode is s : QQ∨ → QQ∨, ηtλ �→ ηt−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is then routine to check that DQ∨ is a Hopf algebra and the embedding to QQ∨ is an embedding of Hopf algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For K-theory, we know the Hecke algebra is contained DWa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is proved by Berenstein-Kazhdan [BK19] that certain localization of the Hecke algbra is a Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' It is not difficult to see that it is compatible with the Hopf algebra structure of DQ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The following theorem generalizes [LSS10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4] in the K-theory case and [LS10, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2] in the cohomology case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Assume Iw, w ∈ Wa is W-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If u ∈ W − a , then we have k(XIu) = XIu + � v∈Wa\\W − a cIu,IvXIv, cIu,Iv ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' If w ∈ W − a , by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4, we have X∗ Iw(k(XIu)) = k∗(X∗ Iw)((XIu)) = X∗ Iw(XIu) = δw,u, Therefore, k(XIu) = � v∈Wa X∗ Iv(k(XIu))XIv = XIu + � v∈Wa\\W − a cIu,IvXIv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Consider the ˆA1 case, then there are two simple roots α1 = α, α0 = −α + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By direct computation, we have (1) k(X0) = X0 + X1 − x−αX01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (2) k(X10) = X10 − x−α xα X01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' (3) k(X010) = X010 + X101 − x−αX1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Assume Iw, w ∈ Wa is W-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let u, v ∈ W − a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Write XIuXIv = � w∈Wa dIw Iu,IvXIw, XIuXIv = � w∈W − a dIw Iu,IvXIw, then d Iw3 Iu,Iv = � w2∈Wa cIu,Iw2d Iw3 Iw2,Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have k( � w∈W − a dIw Iu,IvXIw) = k(XIuXIv) = k(XIu)k(XIv) = k(XIu) � w1∈Wa cIv,Iw1XIw1 AFFINE GRASSMANNIAN 15 = � w1∈Wa cIv,Iw1k(XIu)XIw1 = � w1,w2∈Wa cIv,Iw1cIu,Iw2XIw2XIw1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let w3 ∈ W − a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By [CZZ19, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2], we know that X∗ Iw3(XIw2XIw1) = 0 unless w1 ∈ W − a , in which case cIv,Iw1 = δKr v,w1 by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Therefore, applying X∗ Iw3, w3 ∈ W − a , and using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4, we get d Iw3 Iu,Iv = X∗ Iw3( � w∈W − a dIw Iu,IvXIw) = k∗(X∗ Iw3)( � w∈W − a dIw Iu,IvXIw) = X∗ Iw3(k( � w∈W − a dIw Iu,IvXIw)) = X∗ Iw3( � w1,w2∈Wa cIv,Iw1cIu,Iw2XIw2XIw1) = � w2∈Wa cIu,Iw2X∗ Iw3(XIw2XIv) = � w2∈Wa cIu,Iw2d Iw3 Iw2,Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Appendix: Restriction formula in the ˆA1 case In this Appendix, we perform some computation in the ˆA1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' In this case, there are two simple roots, α1 = α, α0 = −α + δ, and any w ∈ Wa has a unique reduced decomposition, so XIw, YIw can be denoted by Xw, Yw, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Moreover, X2 i = καXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We use the notation as in [LSS10, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let σ0 = e, σ2i = (s1s0)i = t−iα∨, σ−2i = (s0s1)i = tiα∨, σ2i+1 = s0σ2i, σ−(2i+1) = s1σ−2i, i ≥ 1, and W − a = {σi|i ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote µ = − x−1 x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' So if F = Fc with c = 0, then µ = 1, and if F = Fc with c = 1, then µ = eα if one identifies xα with 1 − e−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Let S≤a be the sum h0 + h1 + · · · + ha of homogeneous symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Denote Si ≤a to be S≤a(x, x, · · · , x) where there are i copies of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For instance, S3 ≤3(x) = 1 + 3x + 6x2 + 10x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' We have the following identities: Si ≤a(x) = xSi ≤a−1(x) + Si−1 ≤a (x), Si ≤a(x) = a � j=0 xj � j + i − 1 i − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Then the following identities can be verified by direct computation for lower k and then continued with induction: ησ2k = 1 + x2k 1 Xσ2k + � 1≤j≤k−1 x2j 1 (S2j ≤k−j(µ−1)Xσ2j + S2j ≤k−j−1(µ−1)Xσ−2j) − � 1≤i≤k x2i−1 1 S2i−1 ≤k−i(µ−1)(Xσ2i−1 + Xσ−2i+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ησ−2k = 1 + x2k −1Xσ−2k + � 1≤j≤k−1 x2j −1(S2j ≤k−j−1(µ)Xσ2j + S2j ≤k−j(µ)Xσ−2j) − � 1≤i≤k x2i−1 −1 S2i−1 ≤k−i(µ)(Xσ2i−1 + Xσ−2i+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ησ−2k−1 = 1 − x2k+1 1 Xσ−2k−1 + � 1≤j≤k x2j 1 S2j ≤k−j(µ−1)(Xσ2j + Xσ−2j) 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' ZHONG − � 1≤i≤k x2i−1 1 � S2i−1 ≤k−i(µ−1)Xσ2i−1 + S2i−1 ≤k−i+1(µ−1)Xσ−2i+1 � , ησ2k+1 = 1 − x2k+1 −1 Xσ2k+1 + � 1≤j≤k x2j −1S2j ≤k−j(µ)(Xσ2j + Xσ−2j) − � 1≤i≤k x2i−1 −1 � S2i−1 ≤k−i+1(µ)Xσ2i−1 + S2i−1 ≤k−i(µ)Xσ−2i+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' For F = Fc with c = 1, that is, in the K-theory case, these identities specializes to the corresponding ones in [LSS10, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6)] after identifying our −X−αi with Ti in [LSS10] (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' By using these identities, following the same idea as in [LSS10, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3], one can prove that X∗ Iw satisfy the small torus GKM conditions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Acknowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' The author would like to thank Cristian Lenart, Changzheng Li and Gufang Zhao for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' References [BK19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Berenstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Kazhcan, Hecke-Hopf algebras, Advances in Mathematics, 353 (2019) 312-395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='8 [CPZ13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Calm´es, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Petrov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zainoulline, Invariants, torsion indices and oriented cohomology of complete flags, Annales scientifiques de l’ ´Ecole normale sup´erieure (4) 46(3), 405–448 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1 [CZZ16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Calm`es, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zainoulline, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zhong, A coproduct structure on the formal affine Demazure algebra, Mathematische Zeitschrift, 282 (2016) (3), 1191-1218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2 [CZZ19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Calm`es, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zainoulline, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zhong, Push-pull operators on the formal affine Demazure algebra and its dual, Manuscripta Mathematica, 160 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1-2, 9-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3 [CZZ15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Calm`es, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zainoulline, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zhong, Equivariant oriented cohomology of flag varieties, Documenta Mathematica, Extra Volume: Alexander S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Merkurjev’s Sixtieth Birthday (2015), 113-144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='7 [CZZ20] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Calm`es, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zainoulline, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zhong, Formal affine Demazure and Hecke algebras associated to Kac- Moody root systems, Algebra Representation Theory, 23 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3, 1031-1050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6 [B97] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Brion, Equivariant Chow groups for torus actions, Transformation Groups, 2(3): 225-267, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0 [GKV97] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Ginzburg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Kapranov, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Vasserot, Residue construction of Hecke algebras, Advances in Mathe- matics 128 (1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1, 1-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2 [K18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Kato, Loop structure on equivariant K-theory of semi-infinite flag manifolds, arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='01718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1 [KK86] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Kostant and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G∗, Advances in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 62 (1986), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 3, 187-237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0 [KK90] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Kostant and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Kumar, T -equivariant K-theory of generalized flag varieties, Journal of Differential Ge- ometry 32 (1990), 549–603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' [K03] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Knutson, A Schubert calculus recurrence from the noncomplex W-action on G/B,arXiv:0306304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0 0 [L06] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lam, Schubert polynomials for the affine Grassmannian, Journal of the American Mathematical Society, 21 (1), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1 [LLMS18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lam, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Mihalcea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Shimozono, A conjectural Peterson isomorphism in K-theory, Journal of Algebra, 513:326–343, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1 [LSS10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Schilling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Shimozono, K-theory Schubert calculus of the affine Grassmannian, Compositio Mathematica, 146 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 4, 811–852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1 [LS10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Shimozono, Quantum cohomology of G/P and homology of affine Grassmannian, Acta Mathe- matica, 204(1):49–90, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3 [LZZ20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Lenart, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zainoulline, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zhong, Parabolic Kazhdan-Lusztig basis, Schubert classes and equivariant oriented cohomology, Journal of the Institute of Mathematics of Jussieu, 19 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 6, 1889-1929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='6 [LM07] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Levine and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Morel, Algebraic cobordism, Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Springer-Verlag, Berlin, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1 [MNS22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Mihalcea, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Naruse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Su, Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K-theory, International Mathematics Research Notices, 2022, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 16, 12096–12147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0 [P97] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Peterson, Quantum cohomology of G/P, Lecture at MIT, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2 (4) 53 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 3, 663–711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' AFFINE GRASSMANNIAN 17 [T09] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Tymoczko, Divided difference operators for partial flag varieties, arXiv:0912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2545 0 [ZZ17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zhao and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' Zhong, Geometric representations of the formal affine Hecke algebra, Advances in Mathemat- ics, 317 (2017), 50-90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='2 State University of New York at Albany, 1400 Washington Ave, CK399, Albany, NY, 12222 Email address: czhong@albany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFPT4oBgHgl3EQfUTQQ/content/2301.13056v1.pdf'} diff --git a/1tFIT4oBgHgl3EQf4CvP/vector_store/index.faiss b/1tFIT4oBgHgl3EQf4CvP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..78e7b94a1f19b5ce71d1bb83c2ca8ea57f5ec67e --- /dev/null +++ b/1tFIT4oBgHgl3EQf4CvP/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eba1906329f5014822f7b8c4f574e98549a3da67301346ed8de5847492208f9d +size 3604525 diff --git a/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf b/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9a168f5788263f8eb9c14b3a1cef908bb818d94c --- /dev/null +++ b/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf @@ -0,0 +1,3 @@ +version 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b/59AyT4oBgHgl3EQfcfe1/content/tmp_files/2301.00285v1.pdf.txt @@ -0,0 +1,1073 @@ + + +On the Smoothness of the Solution to the Two-Dimensional Radiation +Transfer Equation + +Dean Wang + +The Ohio State University +201 West 19th Avenue, Columbus, Ohio 43210 + +wang.12239@osu.edu + + +ABSTRACT + +In this paper, we deal with the differential properties of the scalar flux 𝜙(𝑥) defined over a +two-dimensional bounded convex domain, as a solution to the integral radiation transfer +equation. Estimates for the derivatives of 𝜙(𝑥) near the boundary of the domain are given +based on Vainikko’s regularity theorem. A numerical example is presented to demonstrate the +implication of the solution smoothness on the convergence behavior of the diamond difference +method. + +KEYWORDS: Integral Equation, Radiation Transfer, Regularity, Numerical Convergence + + +1. INTRODUCTION + +Consider the integral equation of the second kind + +𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦 +! ++ 𝑓(𝑥) , +𝑥 ∈ G , + + + (1) + +where G ⊂ ℝ" , 𝑛 ≥ 1, is an open bounded domain and the kernel K(𝑥, 𝑦) is weakly singular, i.e., +|K(𝑥, 𝑦)| ≤ 𝐶|𝑥 − 𝑦|#$, 0 ≤ 𝜈 ≤ 𝑛. Weakly singular integral equations arise in many physical applications +such as elliptic boundary problems and particle transport. + +The standard integro-differential equation of radiation transfer can be reformulated as a weakly singular +integral equation. The one-group radiation transfer problem in a three-dimensional (3D) convex domain +reads as follows: find a function 𝜙: G: × Ω → ℝ% such as + +∑ +Ω& +'((*,,) +'*! +. +&/0 ++ 𝜎(𝑥)𝜙(𝑥, Ω) = +1"(*) +23 ∫ 𝑠(𝑥, Ω, Ω′)𝜙(𝑥, Ω4)𝑑Ω′ +, ++ 𝑓(𝑥, Ω) , +𝑥 ∈ G , (2) + + +𝜙(𝑥, Ω) = 𝜙56(𝑥, Ω), 𝑥 ∈ 𝜕G, Ω ∙ 𝑛D(𝑥) < 0 , + + +(3) + +where Ω denotes the direction of radiation transfer, 𝜕G is the boundary of the domain G ⊂ ℝ., 𝜎 is the +extinction coefficient (or macroscopic total cross section in neutron transport), 𝜎7 is the scattering +coefficient (or macroscopic scattering cross section), 𝑠 is the phase function of scattering with +∫ 𝑠(𝑥, Ω, Ω′)𝑑Ω′ +, += 4𝜋, 𝑓 is the external source function, and 𝑛D is the unit normal vector of the domain +surface. Note that 𝜎7(𝑥) ≤ 𝜎(𝑥) and 𝑠(𝑥, Ω, Ω4) = 𝑠(𝑥, Ω′, Ω) under physical conditions. + + +Dean Wang + +Assuming the isotropic scattering, i.e., 𝑠(𝑥, Ω, Ω4) = 1, 𝑓(𝑥, Ω) = +8(*) +23 , and 𝜙56(𝑥, Ω) = +(#$(*) +23 +, we can +obtain the so-called Peierls integral equation of radiation transfer for the scalar flux 𝜙(𝑥) as follows: + +𝜙(𝑥) = +0 +23 ∫ +1"(9):–&((,*) +|*–9|, +𝜙(𝑦)𝑑𝑦 +6 ++ +0 +23 ∫ +:–&((,*) +|*–9|, 𝑓(𝑦)𝑑𝑦 +6 + + ++ +0 +23 ∫ +:–&((,*) +|*–9|, H +*–9 +|*–9| ∙ 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9 +'6 + , + + (4) + +𝜏(𝑥, 𝑦) = ∫ +𝜎(𝑟– 𝜉Ω)𝑑𝜉 +|*–9| += + , + + + + (5) + +where 𝑑𝑆 is the differential element of the domain surface, 𝜏(𝑥, 𝑦) is the optical path between 𝑥 and 𝑦. One +can find detailed derivation in [6,7]. + +For simplicity, we assume 𝜎 and 𝜎7 are constant over the domain. Then Eq. (4) can be simplified as + +𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦 +6 ++ +0 +1" ∫ K(𝑥, 𝑦)𝑓(𝑦)𝑑𝑦 +6 + + ++ +0 +1" ∫ +K(𝑥, 𝑦) H +*–9 +|*–9| ∙ 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9 +'6 + , + + + (6) + +where the 3D radiation kernel is given as + +K(𝑥, 𝑦) = +1":–-|(–*| +23|*–9|, . + + + + (7) + +The boundary integral term in the above equation can produce singularities in the solution. We omit its +discussion in this paper. In other words, we only consider the problem with the vacuum boundary condition, +i.e., 𝜙56 = 0. Thus, Eq. (6) can be treated as the weakly integral equation of the second kind. + +Since K(𝑥, 𝑦) has a singularity at 𝑥 = 𝑦, the solution of a weakly integral equation is generally not a smooth +function and its derivatives at the boundary would become unbounded from a certain order. There was +extensive research on the smoothness (regularity) properties of the solutions to weakly integral equations +[1,2], especially those early work in neutron transport theory done in the former Soviet Union [3,4]. It is +believed that Vladimirov first proved that the scalar flux 𝜙(𝑥) possesses the property |𝜙(𝑥 + ℎ) − +𝜙(𝑥)|~ℎlogℎ for the one-group transport problem with isotropic scattering in a bounded domain [3]. +Germogenova analyzed the local regularity of the angular flux 𝜙(𝑥, Ω) in a neighborhood of the +discontinuity interface and obtained an estimate of the first derivative, which has the singularity near the +interface [4]. Pitkaranta derived a local singular resolution showing explicitly the behavior of 𝜙(𝑥) near +the smooth portion of the boundary [5]. Vainikko introduced weighted spaces and obtained sharp estimates +of pointwise derivatives near the smooth boundary for multidimensional weakly singular integral equations +[6]. + +There exists some previous research work on the regularity of the integral radiation transfer solutions [7,8]. +However, the 2D kernel used in those studies is physically incorrect. In this paper, we rederive the 2D +kernel by directly integrating the 3D kernel with respect to the third dimension. We examine the differential +properties of the new 2D kernel and provide estimates of pointwise derivatives of the scalar flux according +to Vainikko’s regularity theorem for the weakly integral equation of the second kind. + + +Smoothness of the Radiation Transfer Solution + +The remainder of the paper is organized as follows. In Sect. 2, we derive the 2D kernel for the integral +radiation transfer equation. We examine the derivatives of the kernel and show that they satisfy the +boundedness condition of Vainikko’s regularity theorem in Sect. 3. Then the estimates of local regularity +of the scalar flux near the boundary of the domain are given. Sect. 4 presents numerical results to +demonstrate that the rate of convergence of numerical methods can be affected by the smoothness of the +exact solution. Concluding remarks are given in Sect. 5. + + +2. TWO-DIMENSIONAL RADIATION TRANSFER EQUATION + +In this section, we derive the 2D integral radiation transfer equation from its 3D form, Eq. (6). In 3D, 𝑑𝑦 = +𝑑𝑦0𝑑𝑦>𝑑𝑦. and |𝑥– 𝑦| = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> + (𝑥.– 𝑦.)> . Let 𝜌 = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> , then +|𝑥– 𝑦| = S𝜌> + (𝑥.– 𝑦.)>. In a 2D domain 𝐺 ⊂ ℝ>, the solution function 𝜙(𝑥) only depends on 𝑥0 and 𝑥> +in Cartesian coordinates. Therefore, we only need to find the 2D radiation kernel, which can be obtained +by integrating out 𝑦. as follows: + +K(𝑥, 𝑦) = ∫ +1":–-|(–*| +23|*–9|, 𝑑𝑦. +? +#? + + += +1" +23 ∫ +: +/-01,2((3–*3), +@,%(*3–93), +𝑑𝑦. +? +#? + . + (8) + +To proceed, we introduce the variables 𝑡 = 𝜎S𝜌> + (𝑥.– 𝑦.)> and 𝑧 = 𝑦. − 𝑥.. Then we substitute 𝑑𝑦. = +𝑑z = +A +1BA,–1,@, 𝑑𝑡 into the above equation to have + +K(𝑥, 𝑦) = +1" +23 ∫ +:/4 +5, +-, +𝑑z +? +#? += +1" +>3 ∫ +:/4 +5, +-, +𝑑z +? += + + += +1" +>3 ∫ +:/4 +5, +-, +A +1BA,–1,@, 𝑑𝑡 +? += + + += +1"1 +>3 ∫ +:/4 +ABA,–1,@, 𝑑𝑡 +? +1@ + = +1"1 +>3 ∫ +:/4 +ABA,#1,|*–9|, 𝑑𝑡 +? +1|*–9| + . (9) + +Note that the 2D radiation kernel is always positive. By replacing the 3D kernel of Eq. (7) with the above +one, Eq. (6) becomes the 2D integral radiation transfer equation. Notice that the surface integral in the last +term on the right-hand side of Eq. (6) should be replaced with the line integral in the 2D domain. + +Now we show that the 2D kernel K(𝑥, 𝑦) has a singularity at 𝜌 = 0 (i.e., 𝑥 = 𝑦) as follows: + +K(𝑥, 𝑦) = +1"1 +>3 ∫ +:/4 +ABA,–1,@, 𝑑𝑡 +? +1@ +> +1"1 +>3 ∫ +:/4 +A, 𝑑𝑡 +? +1@ + + += +1"1 +>3 e +C/-1 +1@ − Γ(0, 𝜎𝜌)g , (10) + +where Γ(0, 𝑎) = ∫ +:/4 +A 𝑑𝑡 +? +D +, is the incomplete gamma function. The singular behavior of K(𝑥, 𝑦) near 𝜌 = +0 is dominated by the first term +C/-1 +1@ in the brackets since the gamma function tends to infinity much slower. + +Dean Wang + +Remark 2.1. It should be noted that the 2D kernel defined by Eq. (9) is equivalent to the more conventional +one defined by the Bickley-Naylor functions [9]. Johnson and Pitkaranta derived a 2D kernel for neutron +transport by reformulating the standard integro-differential equation on the 2D plane [7]. The kernel +obtained is, K(𝑥, 𝑦) = +:–|(–*| +|*–9| (assuming 𝜎 = 1), which is however mathematically correct but physically +incorrect. Hennebach et al. also used the same 2D kernel for analyzing the radiation transfer solutions [8]. +In addition, the integral equations in other geometries such as slab or sphere can be obtained by following +the same approach, and they can be found in [10]. + +Applying Banach’s fixed-point theorem, we can prove the existence and uniqueness of the solution in the +2D domain by showing that ∫ K(𝑥, 𝑦)𝑑𝑦 +! + is bounded below unity as follows. + +∫ K(𝑥, 𝑦)𝑑𝑦 +6 += ∫ i +1"1 +>3 ∫ +:/4 +ABA,–1,@, 𝑑𝑡 +? +1@ +j 𝑑𝑦 +6 + + += +1"1 +>3 ∫ 𝑑𝑦 ∫ +:/4 +ABA,–1,@, 𝑑𝑡 +? +1@ +6 + + += +1"1 +>3 ∫ 𝜌𝑑𝜑𝑑𝜌 ∫ +:/4 +ABA,#1,@, 𝑑𝑡 +? +1@ +6 + , + +(11) + +where 𝜑 is the azimuthal angle. By extending the above bounded domain to the whole space, we have + +∫ K(𝑥, 𝑦)𝑑𝑦 +6 +< +1"1 +>3 ∫ +2𝜋𝜌𝑑𝜌 +? += +∫ +:/4 +ABA,–1,@, 𝑑𝑡 +? +1@ + + += 𝜎7𝜎 ∫ +𝜌𝑑𝜌 +? += +∫ +:/4 +ABA,–1,@, 𝑑𝑡 +? +1@ + . + + (12) + +Denoting 𝜁 = 𝜎𝜌, Eq. (12) is simplified as + +∫ K(𝑥, 𝑦)𝑑𝑦 +6 +< +1" +1 ∫ +𝜁𝑑𝜁 +? += +∫ +:/4 +ABA,–E, 𝑑𝑡 +? +E + + += +1" +1 ∫ +∫ +:/4 +A +E +BA,#E, 𝑑𝑡𝑑𝜁 +? +E +? += + = +1" +1 ∫ +:/4 +A 𝑑𝑡 ∫ +E +BA,–E, 𝑑𝜁 +A += +? += + + += +1" +1 ∫ +e#F𝑑𝑡 +? += + + += +1" +1 ≤ 1 . (13) + +Notice we have changed the order of integration to solve the integral. It is apparent that for there exists a +unique solution, the physical condition, 𝜎7 ≤ 𝜎, must be satisfied. + + +3. SMOOTHNESS OF THE SOLUTIONS + +We first introduce Vainikko’s regularity theorem [6], which provides a sharp characterization of +singularities for the general weakly integral equation of the second kind. Then we analyze the +differentiational properties of the 2D radiation kernel and show that the derivatives are properly bounded. +Finally, Vainikko’s theorem is used to give the estimates of pointwise derivatives of the radiation solution. + +Smoothness of the Radiation Transfer Solution + +3.1. Vainikko’s Regularity Theorem + +Before we state the theorem, we introduce the definition of weighted spaces ℂG,$(G) [6]. + +Weighted space ℂ𝒎,𝝂(𝐆). For a 𝜆 ∈ ℝ, introduce a weight function + +𝑤J = r +1 , 𝜆 < 0 +(1 + |log𝜚(𝑥)|)#0 , 𝜆 = 0 +𝜚(𝑥)J , 𝜆 > 0 + , +𝑥 ∈ G + + (14) + +where G ⊂ ℝ" is an open bounded domain and 𝜚(𝑥) = inf +9∈'6|𝑥 − 𝑦| is the distance from 𝑥 to the boundary +𝜕G. Let 𝑚 ∈ ℕ, 𝜈 ∈ ℝ and 𝜈 < 𝑛. Define the space ℂG,$(G) as the set of all 𝑚 times continuously +differentiable functions 𝜙: G → ℝ such that + +‖𝜙‖G,$ = ∑ +sup +*∈6 +{𝑤|L|–("–$)|𝐷L𝜙(𝑥)|} +|L|MG +< ∞ . + + + (15) + +In other words, a 𝑚 times continuously differentiable function 𝜙 on G belongs to ℂG,$(G) if the growth of +its derivatives near the boundary can be estimated as follows: + +|𝐷L𝜙(𝑥)| ≤ 𝑐 r +1 , |𝛼| < 𝑛– 𝜈 +1 + |log𝜚(𝑥)| , |𝛼| = 𝑛– 𝜈 +𝜚(𝑥)"#$#|L| , |𝛼| > 𝑛– 𝜈 + , +𝑥 ∈ G, |𝛼| ≤ 𝑚 , + + (16) + +where 𝑐 is a constant. The space ℂG,$(G), equipped with the norm ‖∙‖G,$, is a complete Banach space. + +After defining the weighted space, we introduce the smoothness assumption about the kernel in the +following form: the kernel K(𝑥, 𝑦) is 𝑚 times continuously differentiable on (G × G)\{𝑥 = 𝑦} and there +exists a real number 𝜈 ∈ (−∞, 𝑛) such that the estimate + +H𝐷*L𝐷*%9 +N +K(𝑥, 𝑦)H ≤ 𝑐 r +1 , 𝜈 + |𝛼| < 0 +1 + „log|𝑥 − 𝑦|„ , 𝜈 + |𝛼| = 0 +|𝑥 − 𝑦|#$#|L| , 𝜈 + |𝛼| > 0 + , 𝑥, 𝑦 ∈ G + (17) + +where +𝐷*L = … +' +'*6† +L6 ⋯ … +' +'*7† +L7 , + + + + +(18) + +𝐷*%9 +N += … +' +'*6 + +' +'96† +N6 ⋯ … +' +'*7 + +' +'97† +N7 , + + + (19) + +holds for all multi-indices 𝛼 = (𝛼0, ⋯ , 𝛼") ∈ ℤ%" and 𝛽 = (𝛽0, ⋯ , 𝛽") ∈ ℤ%" with |𝛼| + |𝛽| ≤ 𝑚. Here the +following usual conventions are adopted: |𝛼| = 𝛼0 + ⋯ + 𝛼", and |𝑥| = S𝑥0 +> + ⋯ + 𝑥">. + +Now we present Vainikko’s theorem in characterizing the regularity properties of a solution to the weakly +integral equation of the second kind [6]. + +Theorem 3.1. Let G ⊂ ℝ" be an open bounded domain, 𝑓 ∈ ℂG,$(G) and let the kernel K(𝑥, 𝑦) satisfy the +condition (17). If the integral equation (1) has a solution, 𝜙 ∈ L?(G) then 𝜙 ∈ ℂG,$(G). + +Dean Wang + + +Remark 3.1. The solution does not improve its properties near the boundary 𝜕G, remaining only in +ℂG,$(G), even if 𝜕G is of class ℂ? and, 𝑓 ∈ ℂ?(G). A proof can be found in [6]. More precisely, for any 𝑛 +and 𝜈 (𝜈 < 𝑛) there are kernels K(𝑥, 𝑦) satisfying (17) and such that Eq. (1) is uniquely solvable and, for a +suitable 𝑓 ∈ ℂ?(G), the normal derivatives of order 𝑘 of the solution behave near 𝜕G as log𝜚(𝑥) if 𝑘 = +𝑛– 𝜈, and as 𝜚(𝑥)"#$#O for 𝑘 > 𝑛– 𝜈. + +3.2. Smoothness of the Radiation Transfer Solution + +To apply the results of Theorem 3.1 to the 2D integral radiation transfer equation, we need to analyze the +kernel K(𝑥, 𝑦) and show it satisfying the condition (17), i.e., H𝐷*L𝐷*%9 +N +K(𝑥, 𝑦)H ≤ 𝑐|𝑥 − 𝑦|#0#|L|. We can +simply set |𝛽| = 0 without loss of generality for our problem. + +|𝜶| = 𝟎: +K(𝑥, 𝑦) = +1"1 +>3 ∫ +:/4 +ABA,–1,@, 𝑑𝑡 +? +1@ +< +1"1 +>3 ∫ +:/-1 +ABA,–1,@, 𝑑𝑡 +? +1@ + + += +1"1:/-1 +>3 +∫ +0 +ABA,–1,@, 𝑑𝑡 +? +1@ + = +1"1:/-1 +>3 +3 +>1@ = +1":/-|(–*| +2|*–9| + +≤ 𝑐|𝑥– 𝑦|#0 . (20) + +|𝜶| = 𝟏: Let 𝜁 = 𝜎𝜌 = 𝜎|𝑥– 𝑦| = 𝜎S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)>, then K(𝑥, 𝑦) = +1"1 +>3 ∫ +:/4 +ABA,–E, 𝑑𝑡 +? +E +, and + +|𝐷*K(𝑥, 𝑦)| = H +' +'E K(𝑥, 𝑦) 'E +'*H = H +' +'E K(𝑥, 𝑦)H H +'E +'*H , + (21) + +where, + +H +'E +'*6H = 𝜎 • +(*6–96) +B(*6–96),%(*,–9,),• ≤ 𝜎 , + (22) + +H +'E +'*,H = 𝜎 • +(*,–9,) +B(*6–96),%(*,–9,),• ≤ 𝜎 . + (23) + +Apparently, we only need to find the upper bound of • +' +'E ∫ +:/4 +ABA,–E, 𝑑𝑡 +? +E +• ≤ 𝑐𝜁#>, which is shown in the +following. First, we simplify the integral ∫ +:/4 +ABA,–E, 𝑑𝑡 +? +E + as + +∫ +:/4 +ABA,–E, 𝑑𝑡 +? +E += +0 +E, ∫ +A:/4 +BA,–E, 𝑑𝑡 +? +E +− +0 +E, ∫ +:/4BA,–E, +A +𝑑𝑡 +? +E + + += +P6(E) +E +− +0 +E, ∫ +:/4BA,–E, +A +𝑑𝑡 +? +E + , +(24) + +where 𝐾0(𝜁) is the modified Bessel function of the second kind, and 𝐾0(𝜁)~ +0 +E when 𝜁 → 0 [11]. + +• +' +'E ∫ +:/4 +ABA,–E, 𝑑𝑡 +? +E +• + +Smoothness of the Radiation Transfer Solution + + += •− +P6(E) +E, + +P68(E) +E ++ +> +E3 ∫ +:/4BA,–E, +A +𝑑𝑡 +? +E +− +0 +E ∫ +:/4 +ABA,–E, 𝑑𝑡 +E +? +• . + + + (25) + +Notice the third term on the right-hand side of Eq. (25), ∫ +:/4BA,–E, +A +𝑑𝑡 +? +E +→ 1 as 𝜁 → 0. It is not difficult to +find that the first three terms will cancel out when 𝜁 → 0. Then we obtain + +• +' +'E ∫ +:/4 +ABA,–E, 𝑑𝑡 +? +E +• ≤ • +0 +E ∫ +:/4 +ABA,–E, 𝑑𝑡 +E +? +• + +≤ +3 +> +:/9 +E, = +3 +>1, +:/-|(–*| +|*–9|, . (26) + +Notice that here we have used the upper bound of Eq. (20). Now we arrive at the desired result for |𝛼| = 1: + +|𝐷*K(𝑥, 𝑦)| ≤ 𝑐|𝑥– 𝑦|#> . (27) + +|𝜶| = 𝟐 (and larger): we can follow the same procedure to find |𝐷*LK(𝑥, 𝑦)| ≤ 𝑐|𝑥 − 𝑦|#0#|L|. + +Finally, we conclude that the 2D radiation kernel satisfies the condition (17). Therefore, by Theorem 3.1, +the estimates of derivatives of the scalar flux 𝜙(𝑥) for radiation transfer are the same as for the general +weakly integral equation of the second kind: + +|𝐷L𝜙(𝑥)| ≤ 𝑐 r +1 , |𝛼| < 1 +1 + |log𝜚(𝑥)| , |𝛼| = 1 +𝜚(𝑥)0#|L| , |𝛼| > 1 + , +𝑥 ∈ G . + (28) + +Remark 3.2. The first derivative of the solution 𝜙(𝑥) behaves as log𝜚(𝑥) and becomes unbounded as +approaching the boundary. The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1. As mentioned in +Remark 3.1, these pointwise estimates cannot be improved by adding more strong smoothness on the data +and domain boundary. We point out that the lack of smoothness in the exact solution could adversely affect +the convergence rate of spatial discretization schemes for solving the radiation transfer equation [12-14]. +According to the regularity results, it is expected that the asymptotic convergence rate of the spatial +discretization error of finite difference methods would be around 1 in the 𝐿? or 𝐿0 norm. + + +4. NUMERICAL RESULTS + +In this section, we demonstrate how the regularity of the exact solution will impact the numerical +convergence rate by solving the SN neutron transport equation in its original integro-differential form, using +the classic second-order diamond difference (DD) method. The model problem is a 1cm × 1cm square with +the vacuum boundary condition. Thus, there will be no complication from the boundary condition. The S12 +level-symmetric quadrature set is used for angular discretization. + +We analyze the following four cases: Case 1: ΣA = 1, Σ7 = 0; Case 2: ΣA = 1, Σ7 = 0.8; Case 3: ΣA = 10, +Σ7 = 0, and Case 4: ΣA = 10, Σ7 = 0.9. For all the cases, the external source 𝑓 = 1, is infinitely +differentiable, i.e., 𝑓 ∈ ℂ?(G). Cases 1 and 3 are pure absorption problems, while Case 3 is optically +thicker. It is interesting to note that the solutions are only determined by the external source for these two +cases. Cases 2 and 4 include the scattering effects, while Case 4 is optically thicker and more diffusive. +Both the scattering and external source contribute to the solution. The flux L1 errors as a function of mesh + +Dean Wang + +size and the rates of convergences are summarized in Table I. The error distributions on the mesh +160 × 160 are plotted in Fig. 1. The reference solution for each case is obtained on a very fine mesh, +5120 × 5120. + + +Table I. Flux L1 errors and convergence rates. + +Mesh +(𝑵 × 𝑵) +Case 1 +Case 2 +Case 3 +Case 4 +Error +Rate +Error +Rate +Error +Rate +Error +Rate +10 × 10 +2.87E-03 + +3.59E-03 + +2.31E-03 + +9.29E-03 + +20 × 20 +7.95E-04 +1.85 +1.01E-03 +1.83 +8.12E-04 +1.51 +2.56E-03 +1.86 +40 × 40 +2.90E-04 +1.45 +3.73E-04 +1.44 +2.31E-04 +1.82 +5.89E-04 +2.12 +80 × 80 +1.14E-04 +1.35 +1.44E-04 +1.37 +5.19E-05 +2.15 +1.37E-04 +2.10 +160 × 160 +5.04E-05 +1.17 +6.32E-05 +1.19 +1.32E-05 +1.97 +3.53E-05 +1.96 +320 × 320 +2.46E-05 +1.03 +3.06E-05 +1.04 +3.61E-06 +1.87 +9.39E-06 +1.91 +640 × 640 +1.31E-05 +0.91 +1.63E-05 +0.91 +1.11E-06 +1.71 +2.70E-06 +1.80 +1280 × 1280 +6.26E-06 +1.07 +7.76E-06 +1.07 +3.87E-07 +1.51 +8.51E-07 +1.66 + + + Case 1 + + + + Case 2 + + + Case 3 + + + + Case 4 + +Figure 1. Flux error distribution on the mesh 𝟏𝟔𝟎 × 𝟏𝟔𝟎. + +×10-4 +4 +Flux L1 Error +3 +2 +0 +150 +100 +150 +100 +50 +50 +0 +0×10-4 +4 +Flux L1 Error +3 +2 +0 +150 +100 +150 +100 +50 +50 +0 +0×10-4 +4 +3 +Flux L1 Error +2 +0 +150 +150 +100 +100 +50 +50 +0 +0×10-4 +6 +Flux L1 Error +4 +2 +0 +150 +150 +100 +100 +50 +50 +0 +0Smoothness of the Radiation Transfer Solution + +It is evident that the convergence rate decreases as the mesh refines, and the errors are much larger at the +boundary. The “noisier” distributions in Cases 1 and 2 are due to the ray effects of the discrete ordinates +(SN) method, which are more pronounced in the optically thin problem. The convergence behavior is similar +between the cases with and without the scattering, indicating that the source term plays a significant role in +defining the irregularity of the solution. Cases 3 and 4 show the improved convergence rate as compared to +Cases 1 and 2 because the exponential function e–1|*–9| makes the kernel less singular as the total cross +section 𝜎 increases. In addition, Case 4 has a slightly better rate of convergence than Case 3 on fine meshes +(e.g., 1.84 vs. 1.75 on 640 × 640), because the transport problem becomes more like an elliptic diffusion +problem [17], and the diffusion solution in general has better regularity. It should be pointed out that in +Case 3, the convergence rate is only 1.51 on the coarse mesh. It is because for the pure absorption case, the +DD method becomes unstable when the mesh size is larger than +>Q! +1 , where 𝜇& is the direction cosine of the +radiation transfer direction. However, it is more stable for the scattering case. + +Remark 4.1. The error of the DD can be estimated by „𝜙& − 𝜙& +R„ ≤ 𝐶ℎ& +>‖𝜙′′‖?, where 𝜙& is the exact +solution at cell 𝑗, 𝜙& +R is its numerical result, and ℎ& is the mesh size [15]. Although this optimal error +estimate is obtained for the 1D slab geometry, one can expect the same to be true in two dimensions. As +given by Eq. (28), the second derivative 𝜙44 will be bounded in the interior of the domain, while it would +behave as 𝜙44~ℎ& +#0 near the boundary. Therefore, it is expected that the convergence rate of the DD would +decrease with refining the mesh, and asymptotically tend to 𝑂(ℎ). If the solution is sufficiently smooth +(e.g., a manufactured smooth solution), the DD would maintain its second order of accuracy on any mesh +size [16]. + +Remark 4.2. The scattering does not appear to play a role in defining the smoothness of the solution. For +the problem without the external source, if there exists a nonsmooth incoming flux on the boundary, then +the scattering may not be able to regularize the solution either, since the irregularity caused by the incoming +flux, which is defined by the surface integral term of Eq. (4), has nothing to do with the scattering and the +solution flux 𝜙. + + +5. CONCLUSIONS + +We have derived the two-dimensional integral radiation transfer equation and examined the differential +properties of the integral kernel for fulfilling the boundedness conditions of Vainikko’s theorem. We use +the theorem to estimate the derivatives of the radiation transfer solution near the boundary of the domain. +It is noted that the first derivative of the scalar flux 𝜙(𝑥) becomes unbounded when approaching the +boundary. The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1, where 𝜚(𝑥) is the distance to the +boundary. A numerical example is presented to demonstrate that the irregularity of the exact solution will +reduce the rate of convergence of numerical solutions. The convergence rate improves as the optically +thickness of the problem increases. It is interesting to note that the scattering does not help smoothen the +solution. However, it does play a crucial role in transforming the transport problem into an elliptic diffusion +problem in the asymptotic diffusion limit. We are currently extending the analysis to the boundary integral +transport problem in considering nonzero incoming boundary conditions and corner effects. In addition, it +would be interesting to study the convergence behavior of weak solutions. + + +REFERENCES + +1. S. G. Mikhlin, S. Prossdorf, Singular Integral Operators, Springer-Verlag (1986). + +Dean Wang + +2. S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford +(1965). +3. V. S. Vladimirov, Mathematical Problems in the One-Velocity Theory of Particle Transport, (Translated +from Transactions of the V. A. Steklov Mathematical Institute, 61, 1961), Atomic Energy of Canada +Limited (1963). +4. T. A. Germogenova, “Local properties of the solution of the transport equation,” Dokl. Akad. Nauk +SSSR, 187(5), pp. 978-981 (1969). +5. J. Pitkaranta, “Estimates for the Derivatives of Solutions to Weakly Singular Fredholm Integral +Equations,” SIAM J. Math. Anal., 11(6), pp. 952-968 (1980). +6. G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin +Heidelberg (1993). +7. C. Johnson and J. Pitkaranta, “Convergence of A Fully Discrete Scheme for Two-Dimensional Neutron +Transport,” SIAM J. Math. Anal., 20(5), pp. 951-966 (1983). +8. E. Hennebach, P. Junghanns, G. Vainikko, “Weakly Singular Integral Equations with Operator-Valued +Kernels and An Application to Radiation Transfer Problems,” Integr. Equat. Oper. Th., 22, pp. 37-64 +(1995). +9. E. E. Lewis and W. F. Miller, Jr., Computational Methods of Neutron Transport, American Nuclear +Society (1993). +10. G. J. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand Reinhold Company, New York +(1970). +11. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and +Mathematical Tables, Dover, New York (1970). +12. N. K. Madsen, “Convergence of Singular Difference Approximations for the Discrete Ordinate +Equations in 𝑥– 𝑦 Geometry,” Math. Comput., 26(117), 45-50 (1972). +13. E. W. Larsen, “Spatial Convergence Properties of the Diamond Difference Method in x, y Geometry,” +Nucl. Sci. Eng., 80, 710-713 (1982). +14. Y. Wang and J. C. Ragusa, “On the Convergence of DGFEM Applied to the Discrete Ordinates +Transport Equation for Structured and Unstructured Triangular Meshes,” Nucl. Sci. Eng., 163, 56-72 +(2009). +15. D. Wang, “Error Analysis of Numerical Methods for Thick Diffusive Neutron Transport Problems on +Shishkin Mesh,” Proceedings of International Conference on Physics of Reactors 2022 (PHYSOR +2022), Pittsburgh, PA, USA, May 15-20, 2022, pp. 977-986 (2022). +16. D. Wang, et al., “Solving the SN Transport Equation Using High Order Lax-Friedrichs WENO Fast +Sweeping Methods,” Proceedings of International Conference on Mathematics and Computational +Methods Applied to Nuclear Science and Engineering 2019 (M&C 2019), Portland, OR, USA, August +25-29, 2019, pp. 61-72 (2019). +17. D. Wang and T. Byambaakhuu, “A New Proof of the Asymptotic Diffusion Limit of the SN Neutron +Transport Equation,” Nucl. Sci. Eng., 195, 1347-1358 (2021). + + diff --git a/59AyT4oBgHgl3EQfcfe1/content/tmp_files/load_file.txt b/59AyT4oBgHgl3EQfcfe1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98c1e4e8beb4dbba07795752f278a2ad377ab56b --- /dev/null +++ b/59AyT4oBgHgl3EQfcfe1/content/tmp_files/load_file.txt @@ -0,0 +1,451 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf,len=450 +page_content='On the Smoothness of the Solution to the Two-Dimensional Radiation Transfer Equation Dean Wang The Ohio State University 201 West 19th Avenue, Columbus, Ohio 43210 wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='12239@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='edu ABSTRACT In this paper, we deal with the differential properties of the scalar flux 𝜙(𝑥) defined over a two-dimensional bounded convex domain, as a solution to the integral radiation transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Estimates for the derivatives of 𝜙(𝑥) near the boundary of the domain are given based on Vainikko’s regularity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' A numerical example is presented to demonstrate the implication of the solution smoothness on the convergence behavior of the diamond difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' KEYWORDS: Integral Equation, Radiation Transfer, Regularity, Numerical Convergence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' INTRODUCTION Consider the integral equation of the second kind 𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' + 𝑓(𝑥) , 𝑥 ∈ G , (1) where G ⊂ ℝ" , 𝑛 ≥ 1, is an open bounded domain and the kernel K(𝑥, 𝑦) is weakly singular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', |K(𝑥, 𝑦)| ≤ 𝐶|𝑥 − 𝑦|#$, 0 ≤ 𝜈 ≤ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Weakly singular integral equations arise in many physical applications such as elliptic boundary problems and particle transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The standard integro-differential equation of radiation transfer can be reformulated as a weakly singular integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=" The one-group radiation transfer problem in a three-dimensional (3D) convex domain reads as follows: find a function 𝜙: G: × Ω → ℝ% such as ∑ Ω& '((*,,) '*!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' &/0 + 𝜎(𝑥)𝜙(𝑥, Ω) = 1"(*) 23 ∫ 𝑠(𝑥, Ω, Ω′)𝜙(𝑥, Ω4)𝑑Ω′ , + 𝑓(𝑥, Ω) , 𝑥 ∈ G , (2) 𝜙(𝑥, Ω) = 𝜙56(𝑥, Ω), 𝑥 ∈ 𝜕G, Ω ∙ 𝑛D(𝑥) < 0 , (3) where Ω denotes the direction of radiation transfer, 𝜕G is the boundary of the domain G ⊂ ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 𝜎 is the extinction coefficient (or macroscopic total cross section in neutron transport), 𝜎7 is the scattering coefficient (or macroscopic scattering cross section), 𝑠 is the phase function of scattering with ∫ 𝑠(𝑥, Ω, Ω′)𝑑Ω′ , = 4𝜋, 𝑓 is the external source function, and 𝑛D is the unit normal vector of the domain surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Note that 𝜎7(𝑥) ≤ 𝜎(𝑥) and 𝑠(𝑥, Ω, Ω4) = 𝑠(𝑥, Ω′, Ω) under physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Dean Wang Assuming the isotropic scattering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 𝑠(𝑥, Ω, Ω4) = 1, 𝑓(𝑥, Ω) = 8(*) 23 , and 𝜙56(𝑥, Ω) = (#$(*) 23 , we can obtain the so-called Peierls integral equation of radiation transfer for the scalar flux 𝜙(𝑥) as follows: 𝜙(𝑥) = 0 23 ∫ 1"(9):–&((, ) | –9|, 𝜙(𝑦)𝑑𝑦 6 + 0 23 ∫ :–&((, ) | –9|, 𝑓(𝑦)𝑑𝑦 6 + 0 23 ∫ :–&((, ) | –9|, H –9 | –9| 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9 \'6 , (4) 𝜏(𝑥, 𝑦) = ∫ 𝜎(𝑟– 𝜉Ω)𝑑𝜉 | –9| = , (5) where 𝑑𝑆 is the differential element of the domain surface, 𝜏(𝑥, 𝑦) is the optical path between 𝑥 and 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' One can find detailed derivation in [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' For simplicity, we assume 𝜎 and 𝜎7 are constant over the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (4) can be simplified as 𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦 6 + 0 1" ∫ K(𝑥, 𝑦)𝑓(𝑦)𝑑𝑦 6 + 0 1" ∫ K(𝑥, 𝑦) H –9 | –9| 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9 \'6 , (6) where the 3D radiation kernel is given as K(𝑥, 𝑦) = 1":– |(– | 23| –9|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (7) The boundary integral term in the above equation can produce singularities in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We omit its discussion in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In other words, we only consider the problem with the vacuum boundary condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 𝜙56 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (6) can be treated as the weakly integral equation of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Since K(𝑥, 𝑦) has a singularity at 𝑥 = 𝑦, the solution of a weakly integral equation is generally not a smooth function and its derivatives at the boundary would become unbounded from a certain order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' There was extensive research on the smoothness (regularity) properties of the solutions to weakly integral equations [1,2], especially those early work in neutron transport theory done in the former Soviet Union [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It is believed that Vladimirov first proved that the scalar flux 𝜙(𝑥) possesses the property |𝜙(𝑥 + ℎ) − 𝜙(𝑥)|~ℎlogℎ for the one-group transport problem with isotropic scattering in a bounded domain [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Germogenova analyzed the local regularity of the angular flux 𝜙(𝑥, Ω) in a neighborhood of the discontinuity interface and obtained an estimate of the first derivative, which has the singularity near the interface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Pitkaranta derived a local singular resolution showing explicitly the behavior of 𝜙(𝑥) near the smooth portion of the boundary [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Vainikko introduced weighted spaces and obtained sharp estimates of pointwise derivatives near the smooth boundary for multidimensional weakly singular integral equations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' There exists some previous research work on the regularity of the integral radiation transfer solutions [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' However, the 2D kernel used in those studies is physically incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In this paper, we rederive the 2D kernel by directly integrating the 3D kernel with respect to the third dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We examine the differential properties of the new 2D kernel and provide estimates of pointwise derivatives of the scalar flux according to Vainikko’s regularity theorem for the weakly integral equation of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Smoothness of the Radiation Transfer Solution The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 2, we derive the 2D kernel for the integral radiation transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We examine the derivatives of the kernel and show that they satisfy the boundedness condition of Vainikko’s regularity theorem in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Then the estimates of local regularity of the scalar flux near the boundary of the domain are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 4 presents numerical results to demonstrate that the rate of convergence of numerical methods can be affected by the smoothness of the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Concluding remarks are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' TWO DIMENSIONAL RADIATION TRANSFER EQUATION In this section, we derive the 2D integral radiation transfer equation from its 3D form, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In 3D, 𝑑𝑦 = 𝑑𝑦0𝑑𝑦>𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' and |𝑥– 𝑦| = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> + (𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='– 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' )> .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Let 𝜌 = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> , then |𝑥– 𝑦| = S𝜌> + (𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='– 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=')>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In a 2D domain 𝐺 ⊂ ℝ>, the solution function 𝜙(𝑥) only depends on 𝑥0 and 𝑥> in Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Therefore, we only need to find the 2D radiation kernel, which can be obtained by integrating out 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' as follows: K(𝑥, 𝑦) = ∫ 1":– |(– | 23| –9|, 𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' #?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = 1" 23 ∫ : / 01,2((3– 3), @,%( 3–93), 𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' #?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (8) To proceed, we introduce the variables 𝑡 = 𝜎S𝜌> + (𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='– 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' )> and 𝑧 = 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' − 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='. Then we substitute 𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = 𝑑z = A 1BA,–1,@, 𝑑𝑡 into the above equation to have K(𝑥, 𝑦) = 1" 23 ∫ :/4 5, , 𝑑z ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' #?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = 1" >3 ∫ :/4 5, , 𝑑z ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = = 1" >3 ∫ :/4 5, , A 1BA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = = 1"1 >3 ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ = 1"1 >3 ∫ :/4 ABA,#1,|*–9|, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1|*–9| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (9) Note that the 2D radiation kernel is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' By replacing the 3D kernel of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (7) with the above one, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (6) becomes the 2D integral radiation transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Notice that the surface integral in the last term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (6) should be replaced with the line integral in the 2D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Now we show that the 2D kernel K(𝑥, 𝑦) has a singularity at 𝜌 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 𝑥 = 𝑦) as follows: K(𝑥, 𝑦) = 1"1 >3 ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ > 1"1 >3 ∫ :/4 A, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ = 1"1 >3 e C/-1 1@ − Γ(0, 𝜎𝜌)g , (10) where Γ(0, 𝑎) = ∫ :/4 A 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' D , is the incomplete gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The singular behavior of K(𝑥, 𝑦) near 𝜌 = 0 is dominated by the first term C/-1 1@ in the brackets since the gamma function tends to infinity much slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Dean Wang Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It should be noted that the 2D kernel defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (9) is equivalent to the more conventional one defined by the Bickley-Naylor functions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Johnson and Pitkaranta derived a 2D kernel for neutron transport by reformulating the standard integro-differential equation on the 2D plane [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The kernel obtained is, K(𝑥, 𝑦) = :–|(–*| |*–9| (assuming 𝜎 = 1), which is however mathematically correct but physically incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Hennebach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' also used the same 2D kernel for analyzing the radiation transfer solutions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In addition, the integral equations in other geometries such as slab or sphere can be obtained by following the same approach, and they can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Applying Banach’s fixed-point theorem, we can prove the existence and uniqueness of the solution in the 2D domain by showing that ∫ K(𝑥, 𝑦)𝑑𝑦 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' is bounded below unity as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' ∫ K(𝑥, 𝑦)𝑑𝑦 6 = ∫ i 1"1 >3 ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ j 𝑑𝑦 6 = 1"1 >3 ∫ 𝑑𝑦 ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ 6 = 1"1 >3 ∫ 𝜌𝑑𝜑𝑑𝜌 ∫ :/4 ABA,#1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ 6 , (11) where 𝜑 is the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' By extending the above bounded domain to the whole space, we have ∫ K(𝑥, 𝑦)𝑑𝑦 6 < 1"1 >3 ∫ 2𝜋𝜌𝑑𝜌 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ = 𝜎7𝜎 ∫ 𝜌𝑑𝜌 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (12) Denoting 𝜁 = 𝜎𝜌, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (12) is simplified as ∫ K(𝑥, 𝑦)𝑑𝑦 6 < 1" 1 ∫ 𝜁𝑑𝜁 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = ∫ :/4 ABA,–E, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E = 1" 1 ∫ ∫ :/4 A E BA,#E, 𝑑𝑡𝑑𝜁 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = = 1" 1 ∫ :/4 A 𝑑𝑡 ∫ E BA,–E, 𝑑𝜁 A = ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = = 1" 1 ∫ e#F𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' = = 1" 1 ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (13) Notice we have changed the order of integration to solve the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It is apparent that for there exists a unique solution, the physical condition, 𝜎7 ≤ 𝜎, must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' SMOOTHNESS OF THE SOLUTIONS We first introduce Vainikko’s regularity theorem [6], which provides a sharp characterization of singularities for the general weakly integral equation of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Then we analyze the differentiational properties of the 2D radiation kernel and show that the derivatives are properly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Finally, Vainikko’s theorem is used to give the estimates of pointwise derivatives of the radiation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Smoothness of the Radiation Transfer Solution 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Vainikko’s Regularity Theorem Before we state the theorem, we introduce the definition of weighted spaces ℂG,$(G) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Weighted space ℂ𝒎,𝝂(𝐆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' For a 𝜆 ∈ ℝ, introduce a weight function 𝑤J = r 1 , 𝜆 < 0 (1 + |log𝜚(𝑥)|)#0 , 𝜆 = 0 𝜚(𝑥)J , 𝜆 > 0 , 𝑥 ∈ G (14) where G ⊂ ℝ" is an open bounded domain and 𝜚(𝑥) = inf 9∈\'6|𝑥 − 𝑦| is the distance from 𝑥 to the boundary 𝜕G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Let 𝑚 ∈ ℕ, 𝜈 ∈ ℝ and 𝜈 < 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Define the space ℂG,$(G) as the set of all 𝑚 times continuously differentiable functions 𝜙: G → ℝ such that ‖𝜙‖G,$ = ∑ sup ∈6 {𝑤|L|–("–$)|𝐷L𝜙(𝑥)|} |L|MG < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (15) In other words, a 𝑚 times continuously differentiable function 𝜙 on G belongs to ℂG,$(G) if the growth of its derivatives near the boundary can be estimated as follows: |𝐷L𝜙(𝑥)| ≤ 𝑐 r 1 , |𝛼| < 𝑛– 𝜈 1 + |log𝜚(𝑥)| , |𝛼| = 𝑛– 𝜈 𝜚(𝑥)"#$#|L| , |𝛼| > 𝑛– 𝜈 , 𝑥 ∈ G, |𝛼| ≤ 𝑚 , (16) where 𝑐 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The space ℂG,$(G), equipped with the norm ‖∙‖G,$, is a complete Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' After defining the weighted space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' we introduce the smoothness assumption about the kernel in the following form: the kernel K(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝑦) is 𝑚 times continuously differentiable on (G × G)\\{𝑥 = 𝑦} and there exists a real number 𝜈 ∈ (−∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝑛) such that the estimate H𝐷*L𝐷*%9 N K(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝑦)H ≤ 𝑐 r 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝜈 + |𝛼| < 0 1 + „log|𝑥 − 𝑦|„ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝜈 + |𝛼| = 0 |𝑥 − 𝑦|#$#|L| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝜈 + |𝛼| > 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=" 𝑦 ∈ G (17) where 𝐷 L = … ' ' 6† L6 ⋯ … ' ' 7† L7 ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=" (18) 𝐷 %9 N = … ' ' 6 + ' '96† N6 ⋯ … ' ' 7 + ' '97† N7 ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (19) holds for all multi-indices 𝛼 = (𝛼0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝛼") ∈ ℤ%" and 𝛽 = (𝛽0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 𝛽") ∈ ℤ%" with |𝛼| + |𝛽| ≤ 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Here the following usual conventions are adopted: |𝛼| = 𝛼0 + ⋯ + 𝛼", and |𝑥| = S𝑥0 > + ⋯ + 𝑥">.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Now we present Vainikko’s theorem in characterizing the regularity properties of a solution to the weakly integral equation of the second kind [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Let G ⊂ ℝ" be an open bounded domain, 𝑓 ∈ ℂG,$(G) and let the kernel K(𝑥, 𝑦) satisfy the condition (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' If the integral equation (1) has a solution, 𝜙 ∈ L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (G) then 𝜙 ∈ ℂG,$(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Dean Wang Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The solution does not improve its properties near the boundary 𝜕G, remaining only in ℂG,$(G), even if 𝜕G is of class ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' and, 𝑓 ∈ ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' A proof can be found in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' More precisely, for any 𝑛 and 𝜈 (𝜈 < 𝑛) there are kernels K(𝑥, 𝑦) satisfying (17) and such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (1) is uniquely solvable and, for a suitable 𝑓 ∈ ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (G), the normal derivatives of order 𝑘 of the solution behave near 𝜕G as log𝜚(𝑥) if 𝑘 = 𝑛– 𝜈, and as 𝜚(𝑥)"#$#O for 𝑘 > 𝑛– 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Smoothness of the Radiation Transfer Solution To apply the results of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1 to the 2D integral radiation transfer equation, we need to analyze the kernel K(𝑥, 𝑦) and show it satisfying the condition (17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', H𝐷*L𝐷*%9 N K(𝑥, 𝑦)H ≤ 𝑐|𝑥 − 𝑦|#0#|L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We can simply set |𝛽| = 0 without loss of generality for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' |𝜶| = 𝟎: K(𝑥, 𝑦) = 1"1 >3 ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ < 1"1 >3 ∫ :/ 1 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ = 1"1:/ 1 >3 ∫ 0 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1@ = 1"1:/ 1 >3 3 >1@ = 1":/ |(– | 2| –9| ≤ 𝑐|𝑥– 𝑦|#0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (20) |𝜶| = 𝟏: Let 𝜁 = 𝜎𝜌 = 𝜎|𝑥– 𝑦| = 𝜎S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)>, then K(𝑥, 𝑦) = 1"1 >3 ∫ :/4 ABA,–E, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=" E , and |𝐷 K(𝑥, 𝑦)| = H ' 'E K(𝑥, 𝑦) 'E ' H = H ' 'E K(𝑥, 𝑦)H H 'E ' H , (21) where, H 'E ' 6H = 𝜎 ( 6–96) B( 6–96),%( ,–9,), ≤ 𝜎 , (22) H 'E ' ,H = 𝜎 ( ,–9,) B( 6–96),%( ,–9,), ≤ 𝜎 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=" (23) Apparently, we only need to find the upper bound of • ' 'E ∫ :/4 ABA,–E, 𝑑𝑡 ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E • ≤ 𝑐𝜁#>, which is shown in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' First, we simplify the integral ∫ :/4 ABA,–E, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E as ∫ :/4 ABA,–E, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E = 0 E, ∫ A:/4 BA,–E, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E − 0 E, ∫ :/4BA,–E, A 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E = P6(E) E − 0 E, ∫ :/4BA,–E, A 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E , (24) where 𝐾0(𝜁) is the modified Bessel function of the second kind, and 𝐾0(𝜁)~ 0 E when 𝜁 → 0 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=" ' 'E ∫ :/4 ABA,–E, 𝑑𝑡 ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E Smoothness of the Radiation Transfer Solution = − P6(E) E, + P68(E) E + > E3 ∫ :/4BA,–E, A 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E − 0 E ∫ :/4 ABA,–E, 𝑑𝑡 E ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (25) Notice the third term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (25), ∫ :/4BA,–E, A 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E → 1 as 𝜁 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It is not difficult to find that the first three terms will cancel out when 𝜁 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=" Then we obtain ' 'E ∫ :/4 ABA,–E, 𝑑𝑡 ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E ≤ 0 E ∫ :/4 ABA,–E, 𝑑𝑡 E ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' ≤ 3 > :/9 E, = 3 >1, :/-|(–*| |*–9|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (26) Notice that here we have used the upper bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Now we arrive at the desired result for |𝛼| = 1: |𝐷*K(𝑥, 𝑦)| ≤ 𝑐|𝑥– 𝑦|#> .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (27) |𝜶| = 𝟐 (and larger): we can follow the same procedure to find |𝐷*LK(𝑥, 𝑦)| ≤ 𝑐|𝑥 − 𝑦|#0#|L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Finally, we conclude that the 2D radiation kernel satisfies the condition (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Therefore, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1, the estimates of derivatives of the scalar flux 𝜙(𝑥) for radiation transfer are the same as for the general weakly integral equation of the second kind: |𝐷L𝜙(𝑥)| ≤ 𝑐 r 1 , |𝛼| < 1 1 + |log𝜚(𝑥)| , |𝛼| = 1 𝜚(𝑥)0#|L| , |𝛼| > 1 , 𝑥 ∈ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (28) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The first derivative of the solution 𝜙(𝑥) behaves as log𝜚(𝑥) and becomes unbounded as approaching the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' As mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1, these pointwise estimates cannot be improved by adding more strong smoothness on the data and domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We point out that the lack of smoothness in the exact solution could adversely affect the convergence rate of spatial discretization schemes for solving the radiation transfer equation [12-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' According to the regularity results, it is expected that the asymptotic convergence rate of the spatial discretization error of finite difference methods would be around 1 in the 𝐿?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' or 𝐿0 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we demonstrate how the regularity of the exact solution will impact the numerical convergence rate by solving the SN neutron transport equation in its original integro-differential form, using the classic second-order diamond difference (DD) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The model problem is a 1cm × 1cm square with the vacuum boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Thus, there will be no complication from the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The S12 level-symmetric quadrature set is used for angular discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We analyze the following four cases: Case 1: ΣA = 1, Σ7 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Case 2: ΣA = 1, Σ7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Case 3: ΣA = 10, Σ7 = 0, and Case 4: ΣA = 10, Σ7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' For all the cases, the external source 𝑓 = 1, is infinitely differentiable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 𝑓 ∈ ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Cases 1 and 3 are pure absorption problems, while Case 3 is optically thicker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It is interesting to note that the solutions are only determined by the external source for these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Cases 2 and 4 include the scattering effects, while Case 4 is optically thicker and more diffusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Both the scattering and external source contribute to the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The flux L1 errors as a function of mesh Dean Wang size and the rates of convergences are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The error distributions on the mesh 160 × 160 are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The reference solution for each case is obtained on a very fine mesh, 5120 × 5120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Flux L1 errors and convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Mesh (𝑵 × 𝑵) Case 1 Case 2 Case 3 Case 4 Error Rate Error Rate Error Rate Error Rate 10 × 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='87E 03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='59E 03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='31E 03 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='29E 03 20 × 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='95E 04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='01E 03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='83 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='12E 04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='56E 03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='86 40 × 40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='90E 04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='73E 04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='31E 04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='82 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='89E 04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='12 80 × 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='14E 04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='44E 04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='19E 05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='37E 04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='10 160 × 160 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='04E 05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='32E 05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='32E 05 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='61E 06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='87 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='39E 06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='91 640 × 640 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='31E 05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='63E 05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='11E 06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='70E 06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='80 1280 × 1280 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='26E 06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='07 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='76E 06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='87E 07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='51 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='51E 07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='66 Case 1 Case 2 Case 3 Case 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Flux error distribution on the mesh 𝟏𝟔𝟎 × 𝟏𝟔𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' ×10-4 4 Flux L1 Error 3 2 0 150 100 150 100 50 50 0 0×10-4 4 Flux L1 Error 3 2 0 150 100 150 100 50 50 0 0×10-4 4 3 Flux L1 Error 2 0 150 150 100 100 50 50 0 0×10-4 6 Flux L1 Error 4 2 0 150 150 100 100 50 50 0 0Smoothness of the Radiation Transfer Solution It is evident that the convergence rate decreases as the mesh refines, and the errors are much larger at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The “noisier” distributions in Cases 1 and 2 are due to the ray effects of the discrete ordinates (SN) method, which are more pronounced in the optically thin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The convergence behavior is similar between the cases with and without the scattering, indicating that the source term plays a significant role in defining the irregularity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Cases 3 and 4 show the improved convergence rate as compared to Cases 1 and 2 because the exponential function e–1|*–9| makes the kernel less singular as the total cross section 𝜎 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In addition, Case 4 has a slightly better rate of convergence than Case 3 on fine meshes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='84 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='75 on 640 × 640), because the transport problem becomes more like an elliptic diffusion problem [17], and the diffusion solution in general has better regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It should be pointed out that in Case 3, the convergence rate is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='51 on the coarse mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It is because for the pure absorption case, the DD method becomes unstable when the mesh size is larger than >Q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 1 , where 𝜇& is the direction cosine of the radiation transfer direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' However, it is more stable for the scattering case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The error of the DD can be estimated by „𝜙& − 𝜙& R„ ≤ 𝐶ℎ& >‖𝜙′′‖?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', where 𝜙& is the exact solution at cell 𝑗, 𝜙& R is its numerical result, and ℎ& is the mesh size [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Although this optimal error estimate is obtained for the 1D slab geometry, one can expect the same to be true in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' As given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (28), the second derivative 𝜙44 will be bounded in the interior of the domain, while it would behave as 𝜙44~ℎ& #0 near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Therefore, it is expected that the convergence rate of the DD would decrease with refining the mesh, and asymptotically tend to 𝑂(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' If the solution is sufficiently smooth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', a manufactured smooth solution), the DD would maintain its second order of accuracy on any mesh size [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The scattering does not appear to play a role in defining the smoothness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' For the problem without the external source, if there exists a nonsmooth incoming flux on the boundary, then the scattering may not be able to regularize the solution either, since the irregularity caused by the incoming flux, which is defined by the surface integral term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' (4), has nothing to do with the scattering and the solution flux 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' CONCLUSIONS We have derived the two-dimensional integral radiation transfer equation and examined the differential properties of the integral kernel for fulfilling the boundedness conditions of Vainikko’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We use the theorem to estimate the derivatives of the radiation transfer solution near the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It is noted that the first derivative of the scalar flux 𝜙(𝑥) becomes unbounded when approaching the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1, where 𝜚(𝑥) is the distance to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' A numerical example is presented to demonstrate that the irregularity of the exact solution will reduce the rate of convergence of numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' The convergence rate improves as the optically thickness of the problem increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' It is interesting to note that the scattering does not help smoothen the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' However, it does play a crucial role in transforming the transport problem into an elliptic diffusion problem in the asymptotic diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' We are currently extending the analysis to the boundary integral transport problem in considering nonzero incoming boundary conditions and corner effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' In addition, it would be interesting to study the convergence behavior of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Mikhlin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Prossdorf, Singular Integral Operators, Springer-Verlag (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Dean Wang 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Vladimirov, Mathematical Problems in the One-Velocity Theory of Particle Transport, (Translated from Transactions of the V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Steklov Mathematical Institute, 61, 1961), Atomic Energy of Canada Limited (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Germogenova, “Local properties of the solution of the transport equation,” Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Nauk SSSR, 187(5), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 978-981 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Pitkaranta, “Estimates for the Derivatives of Solutions to Weakly Singular Fredholm Integral Equations,” SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 11(6), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 952-968 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin Heidelberg (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Johnson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Pitkaranta, “Convergence of A Fully Discrete Scheme for Two-Dimensional Neutron Transport,” SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 20(5), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 951-966 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Hennebach, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Junghanns, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Vainikko, “Weakly Singular Integral Equations with Operator-Valued Kernels and An Application to Radiation Transfer Problems,” Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Equat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 37-64 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Lewis and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Miller, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', Computational Methods of Neutron Transport, American Nuclear Society (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Bell and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Glasstone, Nuclear Reactor Theory, Van Nostrand Reinhold Company, New York (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Abramowitz and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover, New York (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Madsen, “Convergence of Singular Difference Approximations for the Discrete Ordinate Equations in 𝑥– 𝑦 Geometry,” Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 26(117), 45-50 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Larsen, “Spatial Convergence Properties of the Diamond Difference Method in x, y Geometry,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 80, 710-713 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Ragusa, “On the Convergence of DGFEM Applied to the Discrete Ordinates Transport Equation for Structured and Unstructured Triangular Meshes,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 163, 56-72 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Wang, “Error Analysis of Numerical Methods for Thick Diffusive Neutron Transport Problems on Shishkin Mesh,” Proceedings of International Conference on Physics of Reactors 2022 (PHYSOR 2022), Pittsburgh, PA, USA, May 15-20, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 977-986 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', “Solving the SN Transport Equation Using High Order Lax-Friedrichs WENO Fast Sweeping Methods,” Proceedings of International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering 2019 (M&C 2019), Portland, OR, USA, August 25-29, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 61-72 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Wang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Byambaakhuu, “A New Proof of the Asymptotic Diffusion Limit of the SN Neutron Transport Equation,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} +page_content=', 195, 1347-1358 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'} diff --git a/59E1T4oBgHgl3EQfTAPi/content/tmp_files/2301.03074v1.pdf.txt b/59E1T4oBgHgl3EQfTAPi/content/tmp_files/2301.03074v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..37580b8768cd6c9adc16d4ce11586984b8c448ad --- /dev/null +++ b/59E1T4oBgHgl3EQfTAPi/content/tmp_files/2301.03074v1.pdf.txt @@ -0,0 +1,1442 @@ +SeedTree: A Dynamically Optimal and +Local Self-Adjusting Tree +Arash Pourdamghani1, Chen Avin2, Robert Sama3, Stefan Schmid1,4 +1TU Berlin, Germany 2School of Electrical and Computer Engineering, Ben Gurion University of the Negev, Israel +3Faculty of Computer Science, University of Vienna, Austria 4Fraunhofer SIT, Germany +Abstract—We consider the fundamental problem of design- +ing a self-adjusting tree, which efficiently and locally adapts +itself towards the demand it serves (namely accesses to the +items stored by the tree nodes), striking a balance between +the benefits of such adjustments (enabling faster access) and +their costs (reconfigurations). This problem finds applications, +among others, in the context of emerging demand-aware and +reconfigurable datacenter networks and features connections to +self-adjusting data structures. Our main contribution is SeedTree, +a dynamically optimal self-adjusting tree which supports local +(i.e., greedy) routing, which is particularly attractive under highly +dynamic demands. SeedTree relies on an innovative approach +which defines a set of unique paths based on randomized item +addresses, and uses a small constant number of items per node. +We complement our analytical results by showing the benefits +of SeedTree empirically, evaluating it on various synthetic and +real-world communication traces. +Index Terms—Reconfigurable datacenters, Online algorithms, +Self-adjusting data structure +I. INTRODUCTION +This paper considers the fundamental problem of designing +self-adjusting trees: trees which adapt themselves towards the +demand they serve. Such self-adjusting trees need to strike +an efficient tradeoff between the benefits of such adjustments +(better performance in the future) and their costs (reconfigura- +tion overheads now). The problem is motivated by the fact that +workloads in practice often feature much temporal and spatial +structure, which may be exploited by self-adjusting optimiza- +tions [1], [2]. Furthermore, such adjustments are increasingly +available, as researchers and practitioners are currently making +great efforts to render networked and distributed systems more +flexible, supporting dynamic reconfigurations, e.g., by leverag- +ing programmability (via software-defined networks) [3], [4], +network virtualization [5], or reconfigurable optical commu- +nication technologies [6]. +In particular, we study the following abstract model (appli- +cations will follow): we consider a binary tree which serves +access requests, issued at the root of the tree, to the items +stored by the nodes. Each node (e.g., server) stores up to +c items (e.g., virtual machines), where c is a parameter +indicating the capacity of a node. We consider an online +perspective where items are requested over time. An online +algorithm aims to optimize the tree in order to minimize +the cost of future access requests (defined as the path length +This project has received funding from the European Research Council +(ERC) under grant agreement No. 864228 (AdjustNet), 2020-2025. +between root and accessed item), while minimizing the number +of items moving up or down in the tree: the reconfigurations. +We call each movement a reconfiguration, and keep track of +its cost. In particular, the online algorithm which does not +know the future access requests, aims to be competitive with +an optimal offline algorithm that knows the entire request +sequence ahead of time. In other words, we are interested in +an online algorithm with minimum competitive ratio [7] over +any (even worst-case) request sequence. +Self-adjusting trees are not only one of the most fundamen- +tal topological structures of their own merit, they also have +interesting applications. For example, such trees are a crucial +building block for more general self-adjusting networks: Avin +et al. [8] recently showed that multiple trees optimized indi- +vidually for a single root, can be combined to build general +communication networks which provide low degree and low +distortion. The design of a competitive self-adjusting tree as +studied in this paper, is hence a stepping stone. +Self-adjusting trees also feature interesting connections to +self-adjusting data structures (see §VI for a detailed discus- +sion), for some of which designing and proving constant- +competitive online algorithms is still an open question [9]. +Interestingly, a recent result shows that constant-competitive +online algorithms exist for self-adjusting balanced binary trees +if one maintains a global map of the items in the tree; it was +proposed to store such a map centrally, at a logical root [10]. +In this paper, we are interested in the question whether +this limitation can be overcome, and whether a competitive +decentralized solution exist. +Our main contribution is a dynamically optimal self- +adjusting tree, SeedTree*, which achieves a constant compet- +itive ratio by keeping recently accessed items closer to the +root, ensuring a working set theorem [9]. Our result also im- +plies weaker notions such as key independent optimality [11] +(details will follow). SeedTree further supports local (that is, +greedy and hence decentralized) routing, which is particularly +attractive in dynamic networks, by relying on an innovative +and simple routing approach that enables nodes to take local +forwarding decisions: SeedTree hashes items to i.i.d. random +addresses and defines a set of greedy paths based on these +addresses. A main insight from our work is that a constant +competitive ratio with locality property can be achieved if +*The name is due to the additional capacity in nodes of the tree, which +resembles seeds in fruits of a tree. +arXiv:2301.03074v1 [cs.DS] 8 Jan 2023 + +Fig. 1: A depiction of SeedTree with capacity 2. Large circles +represent nodes (nodes) of the system, and small circles +represent items. The number inside each small circle is the +hash of the corresponding item. +nodes feature small constant capacities, that is, by allowing +nodes to store a small constant number of items. Storing more +than a single item on a node is often practical, e.g., on a server +or a peer [12], and it is common in hashing data structures with +collision [13], [14]. We also evaluate SeedTree empirically, +both on synthetic traces with ranging temporal locality and +also data derived from Facebook datacenter networks [1], +showing how tuning parameters of the SeedTree can lower +the total (and access) cost for various scenarios. +The remainder of the paper is organized as follows. §II in- +troduces our model and preliminaries. We present and analyze +our online algorithm in §III, and transform it to the matching +model of datacenter networks in §IV. After discussing our +empirical evaluation results in §V, we review related works +in §VI and conclude our contributions in §VII. +II. MODEL AND PRELIMINARIES +This section presents our model and introduces preliminar- +ies used in the design of SeedTree. +Items and nodes. We assume a set of items V += +(v1, . . . , vn), and a set of nodes S = (s1, . . . )† arranged as a +binary tree T. We call the node s1 the root, which is at depth +0 in the tree T, and a node sj is at depth ⌊log j⌋. +Each node can store c items, where c is a parameter +indicating the capacity of a node. In our model, we assume +that c is a constant. The assignment of items to nodes can +change over time. We say a node is full if it contains c items, +and empty if it contains no item (See an example in Figure 1). +We define the level of item v at time t, levelt(v), as the +depth of the node containing v. For example, if item v is at +node sj at time t, we have levelt(v) = ⌊log j⌋. +Request Sequence and Working Set. Items are requested +over time in an online manner, modeled as a request sequence +σ = (σ1, . . . , σm), where σt = v ∈ V means item v is +requested at time t. We are sometimes interested in the recency +of item requests, particularly the size of the working set. +Formally, we define wst(σ, v) as the working set of item v in +at time t in the request sequence σ. The working set wst(σ, v) +is a set of unique items requested since the last request to the +item v before time t. We define a rank of item v at time t, +rankt(v), as the size of working set of the item v at time t. +†We assume the set of nodes to be arbitrarily large, as the exact number +of nodes will be determined based on their used capacity. +Costs and Competitive Ratio. We partition costs incurred +by an algorithm, ALG, into two parts, the cost of finding an +item: the access cost, and the cost of reconfigurations: the +reconfiguration cost. The search for any item starts at the root +node and ends at the node containing the item. Based on our +assumption of constant capacity, we assume the cost of search +inside a node to be negligible. Furthermore, assuming the local +routing property, we find an item by traversing a single path +in our tree; hence the access cost for an access request σi, +CA +ALG(σi), equals the level at which the item is stored. +In our model, a reconfiguration consists of moving an item +one level up or one level down in the tree, plus potentially +additional lookups inside a node. We denote the total recon- +figuration cost after an access request σi by CR +ALG(σi). Hence, +the total cost of each access request is CA +ALG(σi)+CR +ALG(σi), +and the total cost of the algorithm on the whole request +sequence is: CALG(σ) = �m +i=1 CA +ALG(σi) + CR +ALG(σi). The +objective of SeedTree is to operate at the lowest possible cost, +or more specifically, as close as possible to the cost of an +optimal offline algorithm, OPT. +Definition 1 (Competitive ratio). Given an online algorithm +ALG and an optimal offline algorithm OPT, the (strict) +competitive ratio is defined as: ρALG = maxσ +CALG(σ) +COP T (σ) +Furthermore, we say an algorithm has (strict) access com- +petitive ratio considering only the access cost of the online +algorithm ALG (not including the reconfiguration cost). +In this paper, we prove that SeedTree is dynamically optimal. +It means that the cost of our algorithm matches the cost of the +optimal offline algorithm asymptotically. +Definition 2 (Dynamic optimality). Algorithm ALG is dy- +namically optimal if it has constant competitive ratio, i.e., +ρALG = O(1). +MRU trees. We define a specific class of self-adjusting +trees, MRU trees. An algorithm maintains a MRU tree if it +keeps items at a similar level to their ranks. +Definition 3 (MRU tree). An algorithm has the MRU(0) +property if for any item v inside its tree and at any given time +t, the equality levelt(v) = ⌊log ⌈ rankt(v) +c +⌉⌋ holds. +Similarly, we say an algorithm maintains an MRU(β) if it +ensures the relaxed bound of levelt(v) ≤ ⌊log ⌈ rankt(v) +c +⌉⌋ + β +for any item v in the tree. +III. ONLINE SEEDTREE +This section presents SeedTree, an online algorithm that +is dynamically optimal in expectation. This algorithm build +upon uniformly random generated addresses, and allows for +local routing, while ensuring dynamic optimality. Details of +the algorithm are as follows: Algorithm 1 always starts from +the root node. Upon receiving an access request to an item v +it performs a local routing (described in Procedure LocalRout- +ing) based on the uniformly random binary address generated +for the node v, which uniquely determines the path of v in the +tree. We call the i-th bit of the address of v by H(v, i). Let +us assume that the local routing for node v ends in level ℓ. + +100 +001 +101 +010 +010 +011 +0 +111) +(110 +011 +001 +100 +110 +110 +111(a) Item 001 moves up, node-by-node, until it +reaches the root. +(b) The first try of push-down failed, because +node 100 is full. +(c) After finding non-full node, items are +pushed down node-by-node. +Fig. 2: An example of steps taken in Algorithm 1, starting from the state of SeedTree in Figure 1, which has a capacity equal +to 2. In this example, the request is an access request to the item with the hash value 001 (the purple circle). Subfigure 2a +shows the move-to-the-root phase, and Subfigures 2b and 2c depict the push-down phase. +Procedure LocalRouting(s,v) +1 if H(v, level(s)) equals 0 then +2 +Return the left child of s. +3 else +4 +Return the right child of s. +Then SeedTree performs the following two-phase reconfig- +uration. These two phases are designed to ensure the level of +items remains in the same range as their rank (details will +follow), and the number of items remains the same at each +level. +1) Move-to-the-root: This phase moves the accessed item to +the node at the lowest level possible, the root of the tree. +The movement of the item is step-by-step, and it keeps +all the other items in their previous node (we keep the +item in a temporary buffer if a node on the path was full). +This phase is depicted in Figure 2a by zig-zagged purple +arrows. +2) Push-down: In this phase, our algorithm starts from the +root node, selects an item in the node (including the +item that has just moved to this node) uniformly at +random, and moves this item one level down to the new +node selected in the LocalRouting procedure. The same +procedure is continued for the new node until reaching +level ℓ, the level of the accessed item. If the node at +level ℓ was non-full, the re-establishment of balance +was successful. Otherwise, if this attempt is failed, the +algorithm reverses the previous push downs back to the +root, and starts again, until an attempt is successful. As +an example, the failed attempt of this phase is depicted +by dashed red edges in Figure 2b and the last successful +one by curved blue arrows in Figure 2c. +Algorithm 1 always terminates, as there is always the chance +that the item which has been moved to root is selected among +all candidates, and we know that the node which that item is +taken from is not full. We now state the main theorem of the +paper that proves the dynamic optimality of SeedTree. +Theorem 1. SeedTree is dynamically optimal for any given +capacity c ≥ 1. +Algorithm 1: Online SeedTree +Input: Accessed item v. +1 Set s as the root. +2 while s does not contain v do +3 +s = LocalRouting(s,v). +4 Call the current level of v as ℓ. +5 Set s as the root, and move item v to s. +6 while balance is not fixed do +7 +Call the current node s. +8 +while level of s is less than ℓ do +9 +Take an item in node s, uniformly at random, call it +v. +10 +s = LocalRouting(s,v). +11 +Add item v to the node s. +12 +if the last chosen node is full then +13 +Reverse the push-down back to the root. +The proof of Theorem 1 is at the end of the section. The first +step towards the proof is showing that the number of items in +each level remains the same. It is true because after removing +an item at a certain level, the algorithm adds an item to the +same level as a result of the push-down phase. +Observation 1. SeedTree keeps the number of items the same +at each level. +The rest of the analysis is based on the assumption that the +algorithm was initialized with a fixed fractional occupancy +0 < f < 1 of the capacity of each level, i.e., in level i, the +initial tree has exactly ⌊c · f · 2i⌋ items. At the end of this +section, we will see that f = 1 +2 works best for our analysis. +However, we emphasize that having 0 < f < 1 suffices for +SeedTree to run properly. +The second observation is a result of Observation 1. As the +number of items remains the same in each level (based on +Observation 1) at most a fraction f of all nodes are full. In +the lowest level, the number of full nodes might be even lower; +hence the probability of a uniformly random node being full +is at most f when we go to the next request. +Observation 2. Algorithm 1 ensures that the probability of +any uniformly random chosen node in SeedTree to be full, +after serving each access request, is at most f. + +100 +001 +001 +010 +010 +011 +111 +110 +011 +100 +110 +110100 +001 +001 +010 +010 +011 +111 +110 +011 +100 +110 +110001 +100 +001 +010 +010 +011 +101 +111 +110 +011 +100 +110 +110According to Algorithm 1, items are selected uniformly at +random inside a node. In the following lemma, we show that +a node in a certain level is also selected uniformly at random, +which enables the rest of the proof. +Lemma 1. Nodes selected on the final path of the push-down +phase with a level lower than ℓ are selected uniformly at +random. +Proof. Let us denote the probability of ℓ′-th node on the path +(the node at level ℓ′, denoted by sℓ′) being the selected node +is +1 +2ℓ′ . Our proof goes by induction. For the basis, ℓ′ = 0, it +is true since we only have one node, the root. Now assume +that in the final path of push down, we want to see the +probability of reaching the current node, sℓ′. Based on the +induction assumption, we know that the parent of sℓ′, the node +sℓ′−1, has been selected uniformly at random, with probability +1 +2ℓ′−1 . Based on Line 9 of Algorithm 1, an item is selected +from those inside sℓ′−1 uniformly at random, plus having the +independence guarantee of our hash function that generated +address of the selected item, we can conclude the decision to +go to left or right from sℓ′−1 was also uniformly at random, +hence the probability of reach sℓ′ is +1 +2ℓ′−1 · 1 +2 = +1 +2ℓ′ . Note +that the above-mentioned choices are independent of whether +or not the descents sℓ′−1 are full or not. Hence the choice +is independent of (possible) previous failed attempts of the +push-down phase (which might happen due to having a full +node at level ℓ), i.e., the previous attempts do not affect the +probability of choosing the node sℓ′. +An essential element of the proof of Theorem 1 is that the +rank and level of items are related to each other. Lemma 2 +describes one of the aspects of this relation. +Lemma 2. During the execution of the SeedTree, for items v +and u at time t, if rankt(v) > rankt(u) then E[levelt(v)] > +E[levelt(u)]. +Proof. Having rankt(v) > rankt(u), we know that u was +accessed more recently than v. Let us consider time t′, the +last time u was accessed. Since the rank of v is strictly larger +than the rank of u, and as u was moved to the root at time t′, +we know that levelt′(v) > levelt′(u). +Items u and v might reach the same level after time t′, but +it is not a must. We consider the level that they first met as +a random variable, Luv. We denote Luv = −1 if u and v +never appear on the same level after time t′. Let us quantify +the difference in the expected level of u and v, using the law +of total expectation: +E[levelt(v)] − E[levelt(u)] += +⌊log ⌈ n +c ⌉⌋ +� +k=−1 +Pr(Luv = k) · (E[levelt(v)|Luv = k] +−E[levelt(u)|Luv = k]) +For the case Luv = −1, we know that u and v never reached +the same level, and the following is always true: +E[levelt(v)|Lv,u = −1] > E[levelt(u)|Lv,u = −1] +For k ≥ 0, let us consider the time t′′ when u and v meet +at the same level, i.e levelt′′(u) = levelt′′(v). After items u +and v meet for the first time, their expected progress is the +same. More precisely, consider the current subtree of the node +containing v at time t′′, and call it T ′. Since the item addresses +are chosen uniformly at random, the expected number of times +that T ′ is a subtree of a node containing v, equals the number +of times that T ′ might be a subtree of node containing u in +the same level. Hence the expected increase in the level for +both items u and v stays the same from time t′′ onward. +Next, we explain why the number of items accessed at a +higher level is limited in expectation for any given item. +Lemma 3. For a given item v at time t, there are at most +2 · rankws +t (v) items accessed at a higher level since the last +time v was accessed, in expectation. +Proof. Given Lemma 2, the proof is along the lines of the +proof of Lemma 4 from [10]. We removed the details of the +proof due to space constraints. +Now we prove the items in the tree maintained by the online +SeedTree are not placed much farther from their position in +a tree that realizes the exact working set property. This in +turn allows us to approximate the total cost of the online +SeedTree in comparison to the optimal offline algorithm with +the same capacity. The approximation factor, 2 − log(f), is +intuitive: with less capacity in each level (lower values of +levels’ fractional occupancy), we need to put items further +down. +Lemma 4. SeedTree is MRU(2 − log(f)) in expectation. +Proof. For any given item v and time t, we show that +E[levelt(v)] ≤ ⌊log ⌈ rankt(v) +c +⌉⌋ + 2 − log(f) remains true, +considering move-to-the-root and push-down phases. As can +be seen in Line 8 of Algorithm 1, the item v might move down +if the current level of v is lower than the level of the accessed +item. +Let us denote the increase in the level from time t′ to time +t by a random variable D(t′, t). We express this increase in +terms of an indicator random variable I(t′, t, ℓ) which denotes +whether item v went down from level ℓ during [t′, t] or not. +We know that: +D(t′, t) = +� +ℓ +I(t′, t, ℓ) +Let K denote the number of items accessed from a higher +level, and let us write K = k1 + · · · + k⌈ n +c ⌉, where kℓ means +that kℓ such accesses happened when item v was at level ℓ. +For the level ℓ, based on the Observation 1 and Lemma 1 and +the fact that each level contains f · c · 2ℓ items, we conclude +v is being selected after kℓ − 1 accesses with probability (1 − +1 +f·c·2ℓ )k−1 · ( +1 +f·c·2ℓ ). +I(t′, t, ℓ) = min(1, +K +� +kℓ=0 +(1 − +1 +f · c · 2ℓ )kℓ−1 · ( +1 +f · c · 2ℓ )) + += min(1, ( +1 +f · c · 2ℓ ) · +K +� +kℓ=0 +(1 − +1 +f · c · 2ℓ )kℓ−1) +≤ min(1, ( +K +f · c · 2ℓ )) +Going back to our original goal of finding how many levels +an item goes down during a time period [t′, t], we have: +E[D(t′, t)] ≤ +� +ℓ +E[min(1, ( +K +f · c · 2ℓ ))] += log(E[K] +f · c ) + 1 = log(E[K]) − log(c) − log(f) + 1 +The last equality comes from the fact that for ℓ += +log( E[K] +f·c ), we have +K +f·c·2ℓ ≤ 1, and for all larger values of +ℓ, the value will decrease exponentially with factors of two. +From Lemma 3 we know that the expected value of K is +less than equal to 2·rankt(v); therefore, the expected increase +is: +E[D(t′, t)] ≤ log(2 · rankt(v)) − log(c) − log(f) + 1 += log(rankt(v)) − log(c) + 2 − log(f) +The following lemma shows the relation between the total +cost of the online SeedTree and fractional occupancy f. The +relation is natural: as f becomes smaller, the chance of finding +a non-full node becomes larger, and thus fewer attempts are +needed to find a non-full node. +Lemma 5. The expected cost of SeedTree is less than equal +to 2 · (⌈ +1 +(1−f)⌉ + 1) times the access cost. +Proof. Let us consider the accessed item v at level ℓ. In the +first part of the algorithm, the move-to-the-root phase costs +the same as the access, which is equal to traversing ℓ edges. +As the probability of a node being non-full is 1 − f based on +Observation 2, and as the choice of nodes is uniform based +on Observation 1, only ⌈ +1 +1−f ⌉ iterations are needed during the +push-down phase for finding a non-full node, each at cost 2·ℓ. +Hence, given the linearity of expectation, we have: +E[CALG] = E[CAccess +ALG + CMove-to-the-root +ALG ++ CPush-down +ALG +] +≤ 2 · (1 + ⌈ +1 +1 − f ⌉) · ℓ = 2 · (1 + ⌈ +1 +1 − f ⌉) · CAccess +ALG +We now describe why working set optimality is enough for +dynamic optimality, given that reconfigurations do not cost +much (which is proved in Lemma 5). Hence, any other form +of optimality, such as key independent optimality or finger +optimality is guaranteed automatically [11]. +Lemma 6. For any given c, an MRU(0) algorithm is (1+e) +access competitive. +Proof. The proof relies on the potential function argument. We +describe a potential function at time t by φt, and show that +the change in the potential from time t to t + 1 is ∆φt→t+1. +Our potential function at time t, counts the number of +items that are misplaced in the tree of the optimal offline +algorithm OPT with regard to their rank. (As the definition +of MRU(0) indicates, there exists no inversion in such a tree, +that is why we only focus on the number of inversions in +OPT.) Concretely, we say a pair (v, u) is an inversion if +rankt(v) < rankt(u) but levelt(v) > levelt(u). We denote +the number of items that have an inversion with item v at time +t by invt(v), and define Bt(v) = 1 + +invt(v) +c·2levelt(v) . Furthermore, +define Bt = �n +v=1 Bt(v). We define the potential function at +time t as φt = log Bt. We assume that the online SeedTree +rearranges its required items in the tree before the optimal +algorithm’s rearrangements. Let us first describe the change +in potential due to rearrangement in the online SeedTree after +accessing item σt = v. This change has the following effects: +1) Rank of the accessed item, v, has been set to 1. +2) Rank of other items in the tree might have been increased +by at most 1. +Since the relative rank of items other than v does not change +because of the second effect, it does not affect the number +of inversions and hence the potential function. Therefore, we +focus on the first effect. Since OPT has not changed its +configuration, for all items u that are being stored in a lower +level than v in the OPT, a single inversion is created, therefore +we have Bt+1(u) = Bt(u) + +1 +c·2levelc(u) . For the accessed +item v, as its rank has changed to one, all of its inversions +get deleted. The number of inversions for other items, except +v, remains the same. Let us denote the number of items +with lower level than v at time t by Lt(v) and partition the +�n +i=1 Bt+1(i) into three parts as we discussed (v, items stored +in a lower level than v, and other items denoted by set Ot(v)): +n +� +i=1 +Bt+1(i) = Bt+1(v) · +� +i∈Lt(v) +Bt+1(i) · +� +i∈Ot(v) +Bt+1(i) +By rewriting Bt+1(i) in terms of Bt(i), we get: +n +� +i=1 +Bt+1(i) = 1 · +� +i∈Lt(v) +(Bt(i) + +1 +c · 2levelt(i) ) · +� +i∈Ot(v) +Bt(i) +Now let us look at potential due the first effect from time +t to t + 1 by ∆φ1 +t→t+1, and describe it in more detail: +∆φ1 +t→t+1 = log Bt+1 − log Bt = log Bt+1 +Bt += log +n� +i=1 +Bt+1(i) +n� +i=1 +Bt(i) += log( +1 +Bt(v) · +� +Lt(v) +(Bt(i) + +1 +c·2levelt(i) ) +� +Lt(v) +Bt(i) +) +≤ log( +1 +Bt(v) · e|Lt(v)|) + +in which the last inequality comes from the fact that |Lt(v)| = +c · 2levelt(i) and also the inequality that: +|Lt(v)| +� +i=1 +(Bt(i) + +1 +|Lt(v)|) ≤ +|Lt(v)| +� +i=1 +(Bt(i) + Bt(i) +|Lt(v)|) += (1 + +1 +|Lt(v)|)|Lt(v)| · +|Lt(v)| +� +i=1 +Bt(i) ≤ e|Lt(v)| · +|Lt(v)| +� +i=1 +Bt(i) +Now +let +us +focus +on +Bt(v), +and +first +assume +that +⌊log ⌈ rankt(v) +c +⌉⌋ > levelt(v). We want to find the maximum +number of items that might cause inversion with the accessed +item v. +Among all c · 2⌊log ⌈ rankt(v) +c +⌉⌋ − 1 items that v might have +higher rank them, at most c · 2levelt(v) − 1 have lower level in +the OPT tree. Hence we have: +Bt(v) = (c · 2⌊log ⌈ rankt(v) +c +⌉⌋ − 1) − (c · 2levelt(v) − 1) +c · 2levelt(v) +≥ (2⌊log ⌈ rankt(v) +c +⌉⌋ − 1) +2levelt(v) +− 1 +≥ 2⌊log ⌈ rankt(v) +c +⌉⌋ +2levelt(v)+1 += 2⌊log ⌈ rankt(v) +c +⌉⌋−levelt(v)−1 +hence the change in potential due to the first effect is: +∆φ1 +t→t+1 ≤ log( +1 +2⌊log ⌈ rankt(v) +c +⌉⌋−levelt(v)−1 · elevelt(v)) += log(2(1+log e)·levelt(v)−⌊log ⌈ rankt(v) +c +⌉⌋) += (1 + log e) · levelt(v) − ⌊log ⌈rankt(v) +c +⌉⌋ +For the case ⌊log ⌈ rankt(v) +c +⌉⌋ < levelt(v), we use the fact that +Bt +v > 1, from the first inequality below: +∆φt→t+1 = log( 1 +Btv +· elevelt(v)) +≤ log(2log e·levelt(v)) = log e · levelt(v) += (1 + log e) · levelt(v) − ⌊log ⌈rankt(v) +c +⌉⌋ +Hence, in both cases of ⌊log ⌈ rankt(v) +c +⌉⌋ being larger or smaller +than levelt(v), we have ∆φt→t+1 ≤ (1 + log e) · levelt(v) − +⌊log ⌈ rankt(v) +c +⌉⌋. +We then show changes in the potential because of OPT’s +reconfiguration. Details of the computations are omitted due +to space constraints, but they are similar to the changes in +potential due to rearrangements in the ON’s algorithm, and +the result is that each OPT’s movement costs less than log e. +Summing up changes in the potential after ON’s and +OPT’s reconfiguration, assuming OPT has done wt move- +ments at time t, we end up with: +∆φt→t+1 = (1+log e)·levelt(v)−⌊log ⌈rankt(v) +c +⌉⌋+w·log e +And hence the cost of the online algorithm MRU(0) at time +t is at most: +Ct +MRU(0) = Ct +Amortized + ∆φt += ⌊log ⌈rankt(v) +c +⌉⌋ + (1 + log e) · levelt(v) +−⌊log ⌈rankt(v) +c +⌉⌋+wt ·log e ≤ (1+log e)·(levelt(v)+wt) +And then summing up the cost of the MRU(0) and OPT for +the whole request sequence, we will get: +CON = +� +t +Ct +ON ≤ +� +t +(1 + log e) · (levelt(v) + wt) += (1 + log e) · COP T +In which the last equality comes from the fact that OPT also +needs to access the item, and as we assumed an additional wt +reconfigurations. +As the first application of Lemma 6 we prove a lower bound +on the cost of any online algorithm that only depends on the +size of the working set of accessed items in the sequence. +Theorem 2. Any online algorithm maintaining a self-adjusting +complete binary tree with capacity c > 1 on a request +sequence σ = σ1, . . . σm, requires an access cost of at least +�m +i=1⌊log ⌈ rankt(σi) +c +⌉⌋ +(1+e) +. +Proof. This proof is an extension and improvement of the +proof from [10] for any values of c > 2. A result of Lemma 6 +is that even an optimal algorithm cannot be better than +1 +(1+e) +the MRU(0), otherwise contradicting Lemma 6. As the cost +of each access to the item σi is ⌊log ⌈ rankt(σi) +c +⌉⌋ in MRU(0), +we can conclude the total cost of any algorithm should be +larger than +�m +i=1⌊log ⌈ rankt(σi) +c +⌉⌋ +(1+e) +. +Lemma 7. Any MRU(β) tree is β·(1+e)-access competitive. +Proof. Lemma 6 shows that an MRU(0) is (1 + e)-access +competitive. Any item which was in level k in MRU(0), is +in level k + β in MRU(β). As an MRU(β) algorithm keeps +items with rankc(0) at level(0), and because for any k ≥ 1, +we have k +β ≤ βk, we obtain that MRU(β) is (β)·(1+e)- +access competitive. +We conclude this section by proving our main theorem, +dynamic optimality of online SeedTree. +proof of Theorem 1. Combining Lemma 4, Lemma 5 and +Lemma 7 yields that the upper bound for competitiveness is +(1+e)·(2·(1+⌈ +1 +1−f ⌉))·(2−log(f)). The fractional occupancy +f = 1/2 in the above formula is the optimal value for f, which +gives us the 43-competitive ratio. +We need to point out that the above calculation is just an +upper bound on the competitive ratio. As we will discuss in +§V, the best results are usually achieved with a slightly higher +value of f, which we hypothesize might be because of an +overestimation of items’ depth in our theoretical analysis. + +IV. APPLICATION IN RECONFIGURABLE DATACENTERS +SeedTree provides a fundamental self-adjusting structure +which is useful in different settings. For example, it may +be used to adapt the placement of containers in virtualized +settings, in order to reduce communication costs. However, +SeedTree can also be applied in reconfigurable networks in +which links can be adapted. In the following, we describe +how to use SeedTree in such a use case in more detail. In +particular, we consider reconfigurable datacenters in which the +connectivity between racks, or more specifically Top-of-the- +Rack (ToR) switches, can be adjusted dynamically, e.g., based +on optical circuit switches [6]. An optical switch provides a +matching between racks, and accordingly, the model is known +as a matching model in the literature [15]. In the following, +we will show how a SeedTree with capacity c and fractional +occupancy of f = 1 +c can be seen in terms of 2 + c matchings, +and how reconfigurations can be transformed to the matching +model‡. We group these matchings into two sets: +• Topological matchings: consists of 2 static matchings, +embedding the underlying binary tree of SeedTree. The +first matching represents edges between a node and its left +child (with the ID twice the ID of the node), and similarly +the second matching for the right children (with the ID +twice plus one of the ID of their parents). An example is +depicted with solid edges in Figure 3. +• Membership matchings: has c dynamic matchings, con- +necting nodes to items inside them. If a node has more +than one item, the corresponding order of items to match- +ings is arbitrary. An example is shown with dotted edges +in Figure 3. +Having the matchings in place, let us briefly discuss how +search and reconfiguration operations are implemented. A +search for an item starts at the node with ID 001, the root +node. We then check membership matchings of this node. If +they map to the item, we have found the node which contains +the item, and our search was successful. Otherwise, we follow +the edge determined by the hash of the item, going to the +new possible node hosting the item. We repeat the process of +checking membership matchings and going along topological +matchings until we find the item. The item will be found, as +it is stored in one of the nodes in the path determined by its +hash value. Each step of moving an item can be implemented +in the matching mode with only one edge removal and one +edge addition in membership matchings. +V. EXPERIMENTAL EVALUATION +We +complement +our +analytical +results +by +evaluating +SeedTree on multiple datasets. Concretely, we are interested +in answering the following questions: +Q1 How does the access cost of our algorithm compare +to the statically-optimal algorithm (optimized based on +frequencies) and a demand-oblivious algorithm? +‡The matching model considers perfect matchings only, however, in +practice imperfect matchings can be enforced by ignore rules in switches. +Fig. 3: A transformation from the example SeedTree shown in +Figure 1, which has capacity c = 2 and fractional occupancy +of f = 1 +2. The disco balls on top represent the reconfigurable +switches, and below are datacenter racks. Solid edges show +structural matchings, and dotted edges represent membership +matchings. +Q2 How does additional capacity improve the performance +of the online SeedTree, given fixed fractional occupancy +of each level? +Q3 What is the best initial fractional occupancy for the online +SeedTree, given a fixed capacity? +Answers to these questions would help developers tune pa- +rameters of the SeedTree based on their requirements and +needs. Before going through results, we describe the setup +that we used: Our code is written in Python 3.6 and we +used seaborn 0.11 [16] and Matplotlib 3.5 [17] libraries for +visualization. Our programs were executed on a machine with +2x Intel Xeons E5-2697V3 SR1XF with 2.6 GHz, 14 cores +each, and a total of 128 GB DDR4 RAM. +A. Input +• Real-world dataset: Our real-world dataset is communi- +cations between servers inside three different Facebook +clusters, obtained from [1]. We post-processed this dataset +for single-source communications. Among all possible +sources, we chose the most frequent source. +• Synthetic dataset: We use the Markovian model dis- +cussed in [1], [18] for generating sequences based on a +temporal locality parameter which ranges from 0 (uni- +form distribution, no locality) to 0.9 (high temporal +locality). Our synthetic input consists of 65, 535 items +and 1 million requests. For generating such a dataset, we +start from a random sample of items. We post-process +this sequence, overwriting each request with the previous +request with the probability determined by our temporal +locality parameter. After that, we execute the second post- +processing to ensure that exactly 65, 535 items are in the +final trace. +B. Algorithm setup +We use SHA-512 [19] from the hashlib-library as the hash +function in our implementation, approximating the uniform +distribution for generating addresses of items. In order to store +items in a node we used a linked list, and when we move an +item to a node that is already full with other items, items +are stored in a temporary buffer. We assume starting from a +pre-filled tree with items, a tree which respects the fractional +occupancy parameter. + +001 +010 +011 +100 +101 +110 +111 +001 +010 +011 +100 +101 +110 +111(a) +(b) +(c) +Fig. 4: Improvements in the performance of SeedTree by fine-tuning parameters. Figures are generated using the synthetic +dataset with various locality values. (4a) Comparing the access cost of the SeedTree with fractional occupancy f = 1 +2 to the +best possible static algorithm and the demand-oblivious algorithm, all given capacity c = 4. Access costs are divided by 100 +thousands. (4b) The effect of increasing capacity of nodes and temporal locality of input on the total cost of the algorithm. +The fractional occupancy is set to f = 1 +2 for all capacities. Total costs are divided by 1 million for this plot. (4c) Tradeoff +between the total cost and the fractional occupancy, given a range of temporal localities. The capacity of nodes is set to 12. +The number in each cell represents the cost, which are divided by 1 million. +(a) +(b) +Fig. 5: Improvements in the normalized access cost of the +algorithm by changing SeedTree parameters. These results +are obtained based on communications of the most frequent +source from three clusters of the real-world dataset. Costs are +normalized by the cost of the demand-oblivious algorithm. (5a) +Changes in the normalized cost by varying capacity. Fractional +occupancy is set to f = 1 +2. (5a) Changes in the normalized cost +by varying fractional occupancy. Gray dots show the minimum +values. Capacity of nodes is set to 12. +In our experiments, we range the capacities (c) from 2 to 16, +and the fractional occupancies (f) from 0.16 to 0.83. Due to +the random nature of our algorithms and input generations, we +repeat each experiment up to 100 times to ensure consistency +in our results. +C. Results +The performance of SeedTree improves significantly with +the increased temporal locality, as can be seen in Figure 4. +Furthermore, we have the following empirical answers to +questions proposed at the beginning of this section: +A1: The SeedTree improves the access cost significantly, with +increased temporal locality, as shown in Figures 4a, +which compares the access cost of SeedTree to static and +demand-oblivious algorithms. +A2: As the Figures 4b and 5a show, increasing capacity +reduces the cost of the algorithm. However, as we can +see, this increase slows down beyond capacity to 8, and +hence this value can be considered as the best option for +practical purposes. +A3: As discussed at the end of the §III and can be seen +in Figures 4c and 5b, the lowest cost can be achieved +with fractions higher or lower than 1 +2, but f = 1 +2 is near +optimal in most scenarios. +VI. ADDITIONAL RELATED WORK +Self-adjusting lists and trees have already been studied +intensively in the context of data structures. The pioneering +work is by Sleator and Tarjan [20], who initiated the study of +the dynamic list update problems and who also introduced the +move-to-front algorithm, inspiring many deterministic [21], +[22] and randomized [23]–[26] approaches for datastructures, +as well as other variations of the problem [27]. +Self-adjusting binary search trees also aim to keep recently +used elements close to the root, similarly to our approach +in this paper (a summary of results is in Table I). However, +adjustments in binary search trees are based on rotations rather +than the movement of items between different nodes. One +of the well-known self-adjusting binary search trees is the +splay tree [9], although it is still unknown whether this tree is +dynamically optimal; the problem is still open also for recent +variations such as Zipper Tree [31], Multi Splay Tree [32] +and Chain Splay [33] which improve the O(log n) competitive +ratio of the splay tree to O(log log n). For Tango Trees [29], +a matching Ω(log log n) lower bound is known. We also +know that if we allow for free rotations after access, dynamic + +Cluster C +Cluster A +Cluster B +0.95 +Cost +0.90 +Normalized +0.85 +0.80 +0.75 +0.16 +0.25 +0.33 +0.5 +0.66 +0.83 +Fractional Occupancy120 +100 +Cos +80 +Access +60 +40 +SeedTree +Oblivious Algorithm +20 +Static Algorithm +0.15 +0.3 +0.45 +0.6 +0.75 +0.9 +0 +Temporal Locality60 +Total Cost +50 +40 +30 +20 +10 +.5 +03 +汇 +6 +5 +06 +8 +9 +LO +4 +Z +S +6 +Capacity36.7 +29.0 +21.3 +5.5 +0.16 +51.7 +44.2 +13.4 +Occupancy +27.4 +5.1 +0.25 +49.2 +42.0 +34.7 +20.1 +12.6 +47.5 +40.5 +33.4 +26.4 +19.2 +12.1 +4.9 +0.33 +0.5 +45.2 +38.5 +31.8 +25.0 +18.2 +11.4 +4.7 +Fractional +24.5 +17.8 +4.5 +0.66 +44.4 +37.8 +31.1 +11.1 +0.75 +44.7 +38.0 +31.3 +24.6 +17.9 +11.2 +4.5 +46.6 +32.6 +25.7 +18.6 +11.6 +4.6 +0.83 +39.6 +0.0 +0.15 +0.3 +0.6 +0.75 +0.9 +0.45 +Temporal LocalityCluster A +Cluster C +Cluster B +0.95 +Normalized Cost +0.90 +0.85 +0.80 +0.75 +2 +6 +10 +12 +8 +14 +16 +4 +CapacityData Structure +Operation +Ratio +Search +Splay Tree [9] +Rotation +O(log n) +Yes +Greedy Future [28] +Rotation +O(log n) +Yes +Tango Tree [29] +Rotation +θ(log log n) +Yes +Adaptive Huffman [30] +Subtree swap +θ(1) +No +Push-down Tree [10] +Item swap +θ(1) +No +SeedTree +Item movement +θ(1) +Yes +TABLE I: Comparison of properties of self-adjusting tree data +structures. The best known competitive ratio (to this date) +is in terms of the data structure’s respective cost model and +optimal offline algorithm. We note that none of the above trees +considers additional capacity, except for our model. +optimally becomes possible [34]. We also point out that some +of these structures, in particular, multi splay tree and chain +splay, benefitted from additional memory as well, however, +there it is used differently, namely toward saving additional +attributes for each node. Another variation which was first +proposed by Lucas [28] in 1988 is called Greedy Future. This +tree first received attention as an offline binary search tree +algorithm [35], [36], but then an O(log n) amortized time +in online settings was suggested by Fox [37]. Greedy Future +has motivated researchers to take a geometric view of online +binary search trees [36], [38]. We note that in contrast to binary +search trees, our local tree does not require an ordering of the +items in the left and right subtrees of a node. +Self-adjusting trees have also been explored in the context +of coding, where for example adaptive Huffman coding [30], +[39]–[42] is used to minimize the depth of most frequent items. +The reconfiguration cost, however, is different: in adaptive +Huffman algorithms, two subtrees might be swapped at the +cost of one. +A few data structures have tried to achieve a better compet- +itive ratio by expanding and altering binary search trees (see +Table II for a summary): The first example, PokeTree [43], +adds extra pointers between the internal nodes of the tree and +achieves an O(log log n) competitive ratio in comparison to +an optimal binary search tree. There are also self-adjusting +data structures based on skip lists [44], [45], which have been +introduced as an alternative for balanced trees that enforce +probabilistic balancing instead. A biased version of skip lists +was considered in [46], and later on, a statically optimal +variation was given in [47] and a dynamic optimal version +in a restricted model in [48]. Another example is Iacono’s +working set structure [49] which combines a series of self- +adjusting balanced binary search trees and deques, achieving +a worst-case running time of O(log n), however, it lacks the +dynamic optimality property. We are not aware of any work +exploring augmentations to improve the competitive ratio of +these data structures. +Our work is also motivated by emerging self-adjusting +datacenter networks. Recent optical communication technolo- +gies enable datacenters to be reconfigured quickly and fre- +quently [8], [18], [50]–[58], see [59] for a recent survey. The +datacenter application mentioned in our paper is based on the +matching model proposed by [15]. Recently [60] introduced +Data Structure +Structure +Ratio +Iacono’s structure [49] +Trees & deques +O(log n) +Skip List [44] +Linked lists +O(log n) +PokeTree [43] +Tree & dynamic links +O(log log n) +SeedTree +Tree +θ(1) +TABLE II: Comparison with other self-adjusting data struc- +tures that support local-search. The best known competitive +ratio (to this date) is in terms of the data structure’s respective +cost model and optimal offline algorithm. We note that none +of the other data structures considers capacity in their design. +an online algorithm for constructing self-adjusting networks +based on this model, however the authors do not provide +dynamic optimality proof for their method. +It has been shown that demand-aware and self-adjusting +datacenter networks can be built from individual trees [61], +called ego-trees, which are used in many network designs [8], +[50], [62], [63], and also motivate our model. However, until +now it was an open problem how to design self-adjusting +and constant-competitive trees that support local routing and +adjustments, a desirable property in dynamic settings. +Last but not least, our work also features interesting con- +nections to peer-to-peer networks [12], [64]. It is known that +consistent hashing with previously assigned and fixed capaci- +ties allows for significantly improved load balancing [13], [14], +which has interesting applications and is used, e.g., in Vimeo’s +streaming service [65] and in Google’s cloud service [13]. +Although these approaches benefit from data structures with +capacity, these approaches are not demand-aware. +VII. CONCLUSION AND FUTURE WORK +This paper presented and evaluated a self-adjusting and +local tree, SeedTree, which adapts towards the workload in +an online, constant-competitive manner. SeedTree supports a +capacity augmentation approach, while providing local rout- +ing, which can be useful for other self-adjusting structures and +applications as well. We showed a transformation of our algo- +rithm into the matching model for application in reconfigurable +datacenters, and evaluated our algorithm on synthetic and real- +world communication traces. The code used for our experi- +mental evaluation is available at github.com/inet-tub/SeedTree. +We believe that our work opens several interesting avenues +for future research. In particular, while we so far focused on +randomized approaches, it would be interesting to explore de- +terministic variants of SeedTree. Furthermore, while trees are +a fundamental building block toward more complex networks +(as they, e.g., arise in datacenters today), it remains to design +and evaluate networks based on SeedTree. +REFERENCES +[1] C. Avin, M. Ghobadi, C. Griner, and S. Schmid, “On the complexity of +traffic traces and implications,” in ACM SIGMETRICS, 2020. +[2] T. Benson, A. Anand, A. Akella, and M. Zhang, “Understanding data +center traffic characteristics,” ACM SIGCOMM CCR, 2010. +[3] O. Michel, R. Bifulco, G. Retvari, and S. Schmid, “The programmable +data plane: Abstractions, architectures, algorithms, and applications,” in +ACM CSUR, 2021. + +[4] W. Kellerer, P. Kalmbach, A. Blenk, A. Basta, M. Reisslein, and +S. Schmid, “Adaptable and data-driven softwarized networks: Review, +opportunities, and challenges,” in IEEE PIEEE, 2019. +[5] A. Fischer, J. F. Botero, M. T. Beck, H. de Meer, and X. Hesselbach, +“Virtual network embedding: A survey,” IEEE Commun. Surv. Tutor., +2013. +[6] M. N. Hall, K.-T. Foerster, S. Schmid, and R. Durairajan, “A survey of +reconfigurable optical networks,” in OSN, 2021. +[7] A. Borodin and R. El-Yaniv, Online computation and competitive +analysis. +cambridge university press, 2005. +[8] C. Avin, K. Mondal, and S. Schmid, “Demand-aware network designs +of bounded degree,” in DISC, 2017. +[9] D. D. Sleator and R. E. Tarjan, “Self-adjusting binary search trees,” J. +ACM, 1985. +[10] C. Avin, K. Mondal, and S. Schmid, “Push-down trees: Optimal self- +adjusting complete trees,” in IEEE/ACM, TON, 2022. +[11] J. Iacono, “Key-independent optimality,” Algorithmica, 2005. +[12] I. Stoica, R. T. Morris, D. Liben-Nowell, D. R. Karger, M. F. Kaashoek, +F. Dabek, and H. Balakrishnan, “Chord: a scalable peer-to-peer lookup +protocol for internet applications,” IEEE/ACM Trans. Netw., 2003. +[13] V. S. Mirrokni, M. Thorup, and M. Zadimoghaddam, “Consistent +hashing with bounded loads,” in ACM-SIAM SODA, 2018. +[14] A. Aamand, J. B. T. Knudsen, and M. Thorup, “Load balancing with +dynamic set of balls and bins,” in ACM SIGACT STOC, 2021. +[15] C. Griner, J. Zerwas, A. Blenk, S. Schmid, M. Ghobadi, and C. Avin, +“Cerberus: The power of choices in datacenter topology design (a +throughput perspective),” in ACM SIGMETRICS, 2021. +[16] M. L. Waskom, “seaborn: statistical data visualization,” J. of Open +Source Softw., 2021. +[17] J. D. Hunter, “Matplotlib: A 2d graphics environment,” Comput. Sci. +Eng., 2007. +[18] C. Avin, M. Bienkowski, I. Salem, R. Sama, S. Schmid, and P. Schmidt, +“Deterministic self-adjusting tree networks using rotor walks,” in IEEE +ICDCS, 2022. +[19] C. Dobraunig, M. Eichlseder, and F. Mendel, “Analysis of SHA-512/224 +and SHA-512/256,” IACR Cryptol. ePrint Arch., 2016. +[20] D. D. Sleator and R. E. Tarjan, “Amortized efficiency of list update and +paging rules,” Commun. ACM, 1985. +[21] S. Albers, “A competitive analysis of the list update problem with +lookahead,” MFCS, 1994. +[22] S. Kamali and A. L´opez-Ortiz, “A survey of algorithms and models for +list update,” in LNTCS, 2013. +[23] S. Albers and M. Janke, “New bounds for randomized list update in the +paid exchange model,” in STACS, 2020. +[24] S. Albers, B. Von Stengel, and R. Werchner, “A combined bit and +timestamp algorithm for the list update problem,” Inf. Process. Lett., +1995. +[25] T. Garefalakis, “A new family of randomized algorithms for list access- +ing,” in ESA, 1997. +[26] N. Reingold, J. R. Westbrook, and D. D. Sleator, “Randomized compet- +itive algorithms for the list update problem,” Algorithmica, 1994. +[27] S. Albers and S. Lauer, “On list update with locality of reference,” in +ICALP, 2008. +[28] J. M. Lucas, Canonical forms for competitive binary search tree algo- +rithms. +Rutgers University, 1988. +[29] E. D. Demaine, D. Harmon, J. Iacono, and M. Patrascu, “Dynamic +optimality - almost,” in IEEE FOCS, 2004. +[30] G. V. Cormack and R. N. Horspool, “Algorithms for adaptive huffman +codes,” Inf. Process. Lett., 1984. +[31] P. Bose, K. Dou¨ıeb, V. Dujmovi´c, and R. Fagerberg, “An o (log log n)- +competitive binary search tree with optimal worst-case access times,” in +SWAT, 2010. +[32] C. C. Wang, J. Derryberry, and D. D. Sleator, “O (log log n)-competitive +dynamic binary search trees,” in ACM-SIAM SODA, 2006. +[33] G. F. Georgakopoulos, “Chain-splay trees, or, how to achieve and prove +loglogn-competitiveness by splaying,” Inf. Process. Lett., 2008. +[34] A. Blum, S. Chawla, and A. Kalai, “Static optimality and dynamic +search-optimality in lists and trees,” in ACM-SIAM SODA, 2002. +[35] J. I. Munro, “On the competitiveness of linear search,” in ESA, 2000. +[36] E. D. Demaine, D. Harmon, J. Iacono, D. M. Kane, and M. Patrascu, +“The geometry of binary search trees,” in ACM-SIAM SODA, 2009. +[37] K. Fox, “Upper bounds for maximally greedy binary search trees,” in +WADS, 2011. +[38] J. Iacono, “In pursuit of the dynamic optimality conjecture,” in Space- +Efficient Data Structures, Streams, and Algorithms, 2013. +[39] D. E. Knuth, “Dynamic huffman coding,” J. Algorithms, 1985. +[40] R. L. Milidi´u, E. S. Laber, and A. A. Pessoa, “Bounding the compression +loss of the FGK algorithm,” J. Algorithms, 1999. +[41] A. Moffat, “Huffman coding,” ACM CSUR, 2019. +[42] J. S. Vitter, “Design and analysis of dynamic huffman codes,” J. of the +ACM, 1987. +[43] J. Kujala and T. Elomaa, “Poketree: A dynamically competitive data +structure with good worst-case performance,” in ISAAC, 2006. +[44] W. Pugh, “Skip lists: A probabilistic alternative to balanced trees,” +Commun. ACM, 1990. +[45] C. Avin, I. Salem, and S. Schmid, “Working set theorems for routing in +self-adjusting skip list networks,” in IEEE INFOCOM, 2020. +[46] A. Bagchi, A. L. Buchsbaum, and M. T. Goodrich, “Biased skip lists,” +Algorithmica, 2005. +[47] V. Ciriani, P. Ferragina, F. Luccio, and S. Muthukrishnan, “A data +structure for a sequence of string accesses in external memory,” ACM +Trans. Algorithms, 2007. +[48] P. Bose, K. Dou¨ıeb, and S. Langerman, “Dynamic optimality for skip +lists and b-trees,” in ACM-SIAM SODA, 2008. +[49] J. Iacono, “Alternatives to splay trees with o(log n) worst-case access +times,” in ACM-SIAM SODA, 2001. +[50] C. Avin, K. Mondal, and S. Schmid, “Demand-aware network design +with minimal congestion and route lengths,” in IEEE INFOCOM, 2019. +[51] H. Ballani, P. Costa, R. Behrendt, D. Cletheroe, I. Haller, K. Jozwik, +F. Karinou, S. Lange et al., “Sirius: A flat datacenter network with +nanosecond optical switching,” in ACM SIGCOMM, 2020. +[52] K. Chen, A. Singla, A. Singh, K. Ramachandran, L. Xu, Y. Zhang, +X. Wen, and Y. Chen, “Osa: An optical switching architecture for data +center networks with unprecedented flexibility,” IEEE/ACM TON, 2014. +[53] F. Douglis, S. Robertson, E. Van den Berg, J. Micallef, M. Pucci, +A. Aiken, M. Hattink, M. Seok, and K. Bergman, “Fleet—fast lanes for +expedited execution at 10 terabits: Program overview,” IEEE Internet +Comput., 2021. +[54] K.-T. Foerster, M. Ghobadi, and S. Schmid, “Characterizing the al- +gorithmic complexity of reconfigurable data center architectures,” in +ACM/IEEE ANCS, 2018. +[55] M. Ghobadi, R. Mahajan, A. Phanishayee, N. Devanur, J. Kulkarni, +G. Ranade, P.-A. Blanche, H. Rastegarfar et al., “Projector: Agile +reconfigurable data center interconnect,” in ACM SIGCOMM, 2016. +[56] J. Kulkarni, S. Schmid, and P. Schmidt, “Scheduling opportunistic links +in two-tiered reconfigurable datacenters,” in ACM SPAA, 2021. +[57] W. M. Mellette, R. Das, Y. Guo, R. McGuinness, A. C. Snoeren, and +G. Porter, “Expanding across time to deliver bandwidth efficiency and +low latency,” in USENIX NSDI, 2020. +[58] W. M. Mellette, R. McGuinness, A. Roy, A. Forencich, G. Papen, A. C. +Snoeren, and G. Porter, “Rotornet: A scalable, low-complexity, optical +datacenter network,” in ACM SIGCOMM, 2017. +[59] K.-T. Foerster and S. Schmid, “Survey of reconfigurable data center +networks: Enablers, algorithms, complexity,” in SIGACT News, 2019. +[60] E. Feder, I. Rathod, P. Shyamsukha, R. Sama, V. Aksenov, I. Salem, +and S. Schmid, “Lazy self-adjusting bounded-degree networks for the +matching model,” in IEEE INFOCOM, 2022. +[61] C. Avin and S. Schmid, “Toward demand-aware networking: a theory +for self-adjusting networks,” ACM SIGCOMM CCR, 2018. +[62] ——, “Renets: Statically-optimal demand-aware networks,” in SIAM +APOCS, 2021. +[63] B. S. Peres, O. A. de Oliveira Souza, O. Goussevskaia, C. Avin, +and S. Schmid, “Distributed self-adjusting tree networks,” in IEEE +INFOCOM, 2019. +[64] D. R. Karger, E. Lehman, F. T. Leighton, R. Panigrahy, M. S. Levine, and +D. Lewin, “Consistent hashing and random trees: Distributed caching +protocols for relieving hot spots on the world wide web,” in ACM STOC, +1997. +[65] A. Rodland, “Improving load balancing with a new consistent-hashing +algorithm,” Vimeo Engineering Blog, Medium, 2016. + diff --git a/59E1T4oBgHgl3EQfTAPi/content/tmp_files/load_file.txt b/59E1T4oBgHgl3EQfTAPi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f5b3f90c99772e2798e48c3216ee3f62c78691d --- /dev/null +++ b/59E1T4oBgHgl3EQfTAPi/content/tmp_files/load_file.txt @@ -0,0 +1,976 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf,len=975 +page_content='SeedTree: A Dynamically Optimal and Local Self-Adjusting Tree Arash Pourdamghani1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Chen Avin2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Robert Sama3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Stefan Schmid1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='4 1TU Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Germany 2School of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ben Gurion University of the Negev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Israel 3Faculty of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Austria 4Fraunhofer SIT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Germany Abstract—We consider the fundamental problem of design- ing a self-adjusting tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' which efficiently and locally adapts itself towards the demand it serves (namely accesses to the items stored by the tree nodes),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' striking a balance between the benefits of such adjustments (enabling faster access) and their costs (reconfigurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' This problem finds applications, among others, in the context of emerging demand-aware and reconfigurable datacenter networks and features connections to self-adjusting data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our main contribution is SeedTree, a dynamically optimal self-adjusting tree which supports local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', greedy) routing, which is particularly attractive under highly dynamic demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' SeedTree relies on an innovative approach which defines a set of unique paths based on randomized item addresses, and uses a small constant number of items per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We complement our analytical results by showing the benefits of SeedTree empirically, evaluating it on various synthetic and real-world communication traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Index Terms—Reconfigurable datacenters, Online algorithms, Self-adjusting data structure I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' INTRODUCTION This paper considers the fundamental problem of designing self-adjusting trees: trees which adapt themselves towards the demand they serve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Such self-adjusting trees need to strike an efficient tradeoff between the benefits of such adjustments (better performance in the future) and their costs (reconfigura- tion overheads now).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The problem is motivated by the fact that workloads in practice often feature much temporal and spatial structure, which may be exploited by self-adjusting optimiza- tions [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Furthermore, such adjustments are increasingly available, as researchers and practitioners are currently making great efforts to render networked and distributed systems more flexible, supporting dynamic reconfigurations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', by leverag- ing programmability (via software-defined networks) [3], [4], network virtualization [5], or reconfigurable optical commu- nication technologies [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In particular, we study the following abstract model (appli- cations will follow): we consider a binary tree which serves access requests, issued at the root of the tree, to the items stored by the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Each node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', server) stores up to c items (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', virtual machines), where c is a parameter indicating the capacity of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We consider an online perspective where items are requested over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' An online algorithm aims to optimize the tree in order to minimize the cost of future access requests (defined as the path length This project has received funding from the European Research Council (ERC) under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 864228 (AdjustNet), 2020-2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' between root and accessed item), while minimizing the number of items moving up or down in the tree: the reconfigurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We call each movement a reconfiguration, and keep track of its cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In particular, the online algorithm which does not know the future access requests, aims to be competitive with an optimal offline algorithm that knows the entire request sequence ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In other words, we are interested in an online algorithm with minimum competitive ratio [7] over any (even worst-case) request sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Self-adjusting trees are not only one of the most fundamen- tal topological structures of their own merit, they also have interesting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For example, such trees are a crucial building block for more general self-adjusting networks: Avin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [8] recently showed that multiple trees optimized indi- vidually for a single root, can be combined to build general communication networks which provide low degree and low distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The design of a competitive self-adjusting tree as studied in this paper, is hence a stepping stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Self-adjusting trees also feature interesting connections to self-adjusting data structures (see §VI for a detailed discus- sion), for some of which designing and proving constant- competitive online algorithms is still an open question [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Interestingly, a recent result shows that constant-competitive online algorithms exist for self-adjusting balanced binary trees if one maintains a global map of the items in the tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' it was proposed to store such a map centrally, at a logical root [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In this paper, we are interested in the question whether this limitation can be overcome, and whether a competitive decentralized solution exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our main contribution is a dynamically optimal self- adjusting tree, SeedTree*, which achieves a constant compet- itive ratio by keeping recently accessed items closer to the root, ensuring a working set theorem [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our result also im- plies weaker notions such as key independent optimality [11] (details will follow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' SeedTree further supports local (that is, greedy and hence decentralized) routing, which is particularly attractive in dynamic networks, by relying on an innovative and simple routing approach that enables nodes to take local forwarding decisions: SeedTree hashes items to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' random addresses and defines a set of greedy paths based on these addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A main insight from our work is that a constant competitive ratio with locality property can be achieved if The name is due to the additional capacity in nodes of the tree, which resembles seeds in fruits of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='03074v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='DS] 8 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 1: A depiction of SeedTree with capacity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Large circles represent nodes (nodes) of the system, and small circles represent items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The number inside each small circle is the hash of the corresponding item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' nodes feature small constant capacities, that is, by allowing nodes to store a small constant number of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Storing more than a single item on a node is often practical, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', on a server or a peer [12], and it is common in hashing data structures with collision [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We also evaluate SeedTree empirically, both on synthetic traces with ranging temporal locality and also data derived from Facebook datacenter networks [1], showing how tuning parameters of the SeedTree can lower the total (and access) cost for various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' §II in- troduces our model and preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We present and analyze our online algorithm in §III, and transform it to the matching model of datacenter networks in §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' After discussing our empirical evaluation results in §V, we review related works in §VI and conclude our contributions in §VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' MODEL AND PRELIMINARIES This section presents our model and introduces preliminar- ies used in the design of SeedTree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Items and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We assume a set of items V = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' , vn), and a set of nodes S = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' )† arranged as a binary tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We call the node s1 the root, which is at depth 0 in the tree T, and a node sj is at depth ⌊log j⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Each node can store c items, where c is a parameter indicating the capacity of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In our model, we assume that c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The assignment of items to nodes can change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We say a node is full if it contains c items, and empty if it contains no item (See an example in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We define the level of item v at time t, levelt(v), as the depth of the node containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For example, if item v is at node sj at time t, we have levelt(v) = ⌊log j⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Request Sequence and Working Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Items are requested over time in an online manner, modeled as a request sequence σ = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' , σm), where σt = v ∈ V means item v is requested at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We are sometimes interested in the recency of item requests, particularly the size of the working set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Formally, we define wst(σ, v) as the working set of item v in at time t in the request sequence σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The working set wst(σ, v) is a set of unique items requested since the last request to the item v before time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We define a rank of item v at time t, rankt(v), as the size of working set of the item v at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' †We assume the set of nodes to be arbitrarily large, as the exact number of nodes will be determined based on their used capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Costs and Competitive Ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We partition costs incurred by an algorithm, ALG, into two parts, the cost of finding an item: the access cost, and the cost of reconfigurations: the reconfiguration cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The search for any item starts at the root node and ends at the node containing the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Based on our assumption of constant capacity, we assume the cost of search inside a node to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Furthermore, assuming the local routing property, we find an item by traversing a single path in our tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' hence the access cost for an access request σi, CA ALG(σi), equals the level at which the item is stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In our model, a reconfiguration consists of moving an item one level up or one level down in the tree, plus potentially additional lookups inside a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We denote the total recon- figuration cost after an access request σi by CR ALG(σi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hence, the total cost of each access request is CA ALG(σi)+CR ALG(σi), and the total cost of the algorithm on the whole request sequence is: CALG(σ) = �m i=1 CA ALG(σi) + CR ALG(σi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The objective of SeedTree is to operate at the lowest possible cost, or more specifically, as close as possible to the cost of an optimal offline algorithm, OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Definition 1 (Competitive ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Given an online algorithm ALG and an optimal offline algorithm OPT, the (strict) competitive ratio is defined as: ρALG = maxσ CALG(σ) COP T (σ) Furthermore, we say an algorithm has (strict) access com- petitive ratio considering only the access cost of the online algorithm ALG (not including the reconfiguration cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In this paper, we prove that SeedTree is dynamically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' It means that the cost of our algorithm matches the cost of the optimal offline algorithm asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Definition 2 (Dynamic optimality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithm ALG is dy- namically optimal if it has constant competitive ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', ρALG = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' MRU trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We define a specific class of self-adjusting trees, MRU trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' An algorithm maintains a MRU tree if it keeps items at a similar level to their ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Definition 3 (MRU tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' An algorithm has the MRU(0) property if for any item v inside its tree and at any given time t, the equality levelt(v) = ⌊log ⌈ rankt(v) c ⌉⌋ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Similarly, we say an algorithm maintains an MRU(β) if it ensures the relaxed bound of levelt(v) ≤ ⌊log ⌈ rankt(v) c ⌉⌋ + β for any item v in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ONLINE SEEDTREE This section presents SeedTree, an online algorithm that is dynamically optimal in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' This algorithm build upon uniformly random generated addresses, and allows for local routing, while ensuring dynamic optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Details of the algorithm are as follows: Algorithm 1 always starts from the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Upon receiving an access request to an item v it performs a local routing (described in Procedure LocalRout- ing) based on the uniformly random binary address generated for the node v, which uniquely determines the path of v in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We call the i-th bit of the address of v by H(v, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us assume that the local routing for node v ends in level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 100 001 101 010 010 011 0 111) (110 011 001 100 110 110 111(a) Item 001 moves up, node-by-node, until it reaches the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (b) The first try of push-down failed, because node 100 is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (c) After finding non-full node, items are pushed down node-by-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 2: An example of steps taken in Algorithm 1, starting from the state of SeedTree in Figure 1, which has a capacity equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In this example, the request is an access request to the item with the hash value 001 (the purple circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Subfigure 2a shows the move-to-the-root phase, and Subfigures 2b and 2c depict the push-down phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Procedure LocalRouting(s,v) 1 if H(v, level(s)) equals 0 then 2 Return the left child of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 3 else 4 Return the right child of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Then SeedTree performs the following two-phase reconfig- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' These two phases are designed to ensure the level of items remains in the same range as their rank (details will follow), and the number of items remains the same at each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 1) Move-to-the-root: This phase moves the accessed item to the node at the lowest level possible, the root of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The movement of the item is step-by-step, and it keeps all the other items in their previous node (we keep the item in a temporary buffer if a node on the path was full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' This phase is depicted in Figure 2a by zig-zagged purple arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 2) Push-down: In this phase, our algorithm starts from the root node, selects an item in the node (including the item that has just moved to this node) uniformly at random, and moves this item one level down to the new node selected in the LocalRouting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The same procedure is continued for the new node until reaching level ℓ, the level of the accessed item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' If the node at level ℓ was non-full, the re-establishment of balance was successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Otherwise, if this attempt is failed, the algorithm reverses the previous push downs back to the root, and starts again, until an attempt is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As an example, the failed attempt of this phase is depicted by dashed red edges in Figure 2b and the last successful one by curved blue arrows in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithm 1 always terminates, as there is always the chance that the item which has been moved to root is selected among all candidates, and we know that the node which that item is taken from is not full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We now state the main theorem of the paper that proves the dynamic optimality of SeedTree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' SeedTree is dynamically optimal for any given capacity c ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithm 1: Online SeedTree Input: Accessed item v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 1 Set s as the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 2 while s does not contain v do 3 s = LocalRouting(s,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 4 Call the current level of v as ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 5 Set s as the root, and move item v to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 6 while balance is not fixed do 7 Call the current node s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 8 while level of s is less than ℓ do 9 Take an item in node s, uniformly at random, call it v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 10 s = LocalRouting(s,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 11 Add item v to the node s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 12 if the last chosen node is full then 13 Reverse the push-down back to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The proof of Theorem 1 is at the end of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The first step towards the proof is showing that the number of items in each level remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' It is true because after removing an item at a certain level, the algorithm adds an item to the same level as a result of the push-down phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' SeedTree keeps the number of items the same at each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The rest of the analysis is based on the assumption that the algorithm was initialized with a fixed fractional occupancy 0 < f < 1 of the capacity of each level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', in level i, the initial tree has exactly ⌊c · f · 2i⌋ items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' At the end of this section, we will see that f = 1 2 works best for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' However, we emphasize that having 0 < f < 1 suffices for SeedTree to run properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The second observation is a result of Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As the number of items remains the same in each level (based on Observation 1) at most a fraction f of all nodes are full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In the lowest level, the number of full nodes might be even lower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' hence the probability of a uniformly random node being full is at most f when we go to the next request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithm 1 ensures that the probability of any uniformly random chosen node in SeedTree to be full, after serving each access request, is at most f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 100 001 001 010 010 011 111 110 011 100 110 110100 001 001 010 010 011 111 110 011 100 110 110001 100 001 010 010 011 101 111 110 011 100 110 110According to Algorithm 1, items are selected uniformly at random inside a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In the following lemma, we show that a node in a certain level is also selected uniformly at random, which enables the rest of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Nodes selected on the final path of the push-down phase with a level lower than ℓ are selected uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us denote the probability of ℓ′-th node on the path (the node at level ℓ′, denoted by sℓ′) being the selected node is 1 2ℓ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our proof goes by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For the basis, ℓ′ = 0, it is true since we only have one node, the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Now assume that in the final path of push down, we want to see the probability of reaching the current node, sℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Based on the induction assumption, we know that the parent of sℓ′, the node sℓ′−1, has been selected uniformly at random, with probability 1 2ℓ′−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Based on Line 9 of Algorithm 1, an item is selected from those inside sℓ′−1 uniformly at random, plus having the independence guarantee of our hash function that generated address of the selected item, we can conclude the decision to go to left or right from sℓ′−1 was also uniformly at random, hence the probability of reach sℓ′ is 1 2ℓ′−1 · 1 2 = 1 2ℓ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Note that the above-mentioned choices are independent of whether or not the descents sℓ′−1 are full or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hence the choice is independent of (possible) previous failed attempts of the push-down phase (which might happen due to having a full node at level ℓ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', the previous attempts do not affect the probability of choosing the node sℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' An essential element of the proof of Theorem 1 is that the rank and level of items are related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 2 describes one of the aspects of this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' During the execution of the SeedTree, for items v and u at time t, if rankt(v) > rankt(u) then E[levelt(v)] > E[levelt(u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Having rankt(v) > rankt(u), we know that u was accessed more recently than v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us consider time t′, the last time u was accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Since the rank of v is strictly larger than the rank of u, and as u was moved to the root at time t′, we know that levelt′(v) > levelt′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Items u and v might reach the same level after time t′, but it is not a must.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We consider the level that they first met as a random variable, Luv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We denote Luv = −1 if u and v never appear on the same level after time t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us quantify the difference in the expected level of u and v, using the law of total expectation: E[levelt(v)] − E[levelt(u)] = ⌊log ⌈ n c ⌉⌋ � k=−1 Pr(Luv = k) · (E[levelt(v)|Luv = k] −E[levelt(u)|Luv = k]) For the case Luv = −1, we know that u and v never reached the same level, and the following is always true: E[levelt(v)|Lv,u = −1] > E[levelt(u)|Lv,u = −1] For k ≥ 0, let us consider the time t′′ when u and v meet at the same level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='e levelt′′(u) = levelt′′(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' After items u and v meet for the first time, their expected progress is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' More precisely, consider the current subtree of the node containing v at time t′′, and call it T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Since the item addresses are chosen uniformly at random, the expected number of times that T ′ is a subtree of a node containing v, equals the number of times that T ′ might be a subtree of node containing u in the same level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hence the expected increase in the level for both items u and v stays the same from time t′′ onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Next, we explain why the number of items accessed at a higher level is limited in expectation for any given item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For a given item v at time t, there are at most 2 · rankws t (v) items accessed at a higher level since the last time v was accessed, in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Given Lemma 2, the proof is along the lines of the proof of Lemma 4 from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We removed the details of the proof due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Now we prove the items in the tree maintained by the online SeedTree are not placed much farther from their position in a tree that realizes the exact working set property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' This in turn allows us to approximate the total cost of the online SeedTree in comparison to the optimal offline algorithm with the same capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The approximation factor, 2 − log(f), is intuitive: with less capacity in each level (lower values of levels’ fractional occupancy), we need to put items further down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' SeedTree is MRU(2 − log(f)) in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For any given item v and time t, we show that E[levelt(v)] ≤ ⌊log ⌈ rankt(v) c ⌉⌋ + 2 − log(f) remains true, considering move-to-the-root and push-down phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As can be seen in Line 8 of Algorithm 1, the item v might move down if the current level of v is lower than the level of the accessed item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us denote the increase in the level from time t′ to time t by a random variable D(t′, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We express this increase in terms of an indicator random variable I(t′, t, ℓ) which denotes whether item v went down from level ℓ during [t′, t] or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We know that: D(t′, t) = � ℓ I(t′, t, ℓ) Let K denote the number of items accessed from a higher level, and let us write K = k1 + · · · + k⌈ n c ⌉, where kℓ means that kℓ such accesses happened when item v was at level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For the level ℓ, based on the Observation 1 and Lemma 1 and the fact that each level contains f · c · 2ℓ items, we conclude v is being selected after kℓ − 1 accesses with probability (1 − 1 f·c·2ℓ )k−1 · ( 1 f·c·2ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' I(t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ℓ) = min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' K � kℓ=0 (1 − 1 f · c · 2ℓ )kℓ−1 · ( 1 f · c · 2ℓ )) = min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ( 1 f · c · 2ℓ ) · K � kℓ=0 (1 − 1 f · c · 2ℓ )kℓ−1) ≤ min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ( K f · c · 2ℓ )) Going back to our original goal of finding how many levels an item goes down during a time period [t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' we have: E[D(t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' t)] ≤ � ℓ E[min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ( K f · c · 2ℓ ))] = log(E[K] f · c ) + 1 = log(E[K]) − log(c) − log(f) + 1 The last equality comes from the fact that for ℓ = log( E[K] f·c ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' we have K f·c·2ℓ ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' and for all larger values of ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' the value will decrease exponentially with factors of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' From Lemma 3 we know that the expected value of K is less than equal to 2·rankt(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' therefore, the expected increase is: E[D(t′, t)] ≤ log(2 · rankt(v)) − log(c) − log(f) + 1 = log(rankt(v)) − log(c) + 2 − log(f) The following lemma shows the relation between the total cost of the online SeedTree and fractional occupancy f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The relation is natural: as f becomes smaller, the chance of finding a non-full node becomes larger, and thus fewer attempts are needed to find a non-full node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The expected cost of SeedTree is less than equal to 2 · (⌈ 1 (1−f)⌉ + 1) times the access cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us consider the accessed item v at level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In the first part of the algorithm, the move-to-the-root phase costs the same as the access, which is equal to traversing ℓ edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As the probability of a node being non-full is 1 − f based on Observation 2, and as the choice of nodes is uniform based on Observation 1, only ⌈ 1 1−f ⌉ iterations are needed during the push-down phase for finding a non-full node, each at cost 2·ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hence, given the linearity of expectation, we have: E[CALG] = E[CAccess ALG + CMove-to-the-root ALG + CPush-down ALG ] ≤ 2 · (1 + ⌈ 1 1 − f ⌉) · ℓ = 2 · (1 + ⌈ 1 1 − f ⌉) · CAccess ALG We now describe why working set optimality is enough for dynamic optimality, given that reconfigurations do not cost much (which is proved in Lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hence, any other form of optimality, such as key independent optimality or finger optimality is guaranteed automatically [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For any given c, an MRU(0) algorithm is (1+e) access competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The proof relies on the potential function argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We describe a potential function at time t by φt, and show that the change in the potential from time t to t + 1 is ∆φt→t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our potential function at time t, counts the number of items that are misplaced in the tree of the optimal offline algorithm OPT with regard to their rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (As the definition of MRU(0) indicates, there exists no inversion in such a tree, that is why we only focus on the number of inversions in OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=') Concretely, we say a pair (v, u) is an inversion if rankt(v) < rankt(u) but levelt(v) > levelt(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We denote the number of items that have an inversion with item v at time t by invt(v), and define Bt(v) = 1 + invt(v) c·2levelt(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Furthermore, define Bt = �n v=1 Bt(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We define the potential function at time t as φt = log Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We assume that the online SeedTree rearranges its required items in the tree before the optimal algorithm’s rearrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us first describe the change in potential due to rearrangement in the online SeedTree after accessing item σt = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' This change has the following effects: 1) Rank of the accessed item, v, has been set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 2) Rank of other items in the tree might have been increased by at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Since the relative rank of items other than v does not change because of the second effect, it does not affect the number of inversions and hence the potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Therefore, we focus on the first effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Since OPT has not changed its configuration, for all items u that are being stored in a lower level than v in the OPT, a single inversion is created, therefore we have Bt+1(u) = Bt(u) + 1 c·2levelc(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For the accessed item v, as its rank has changed to one, all of its inversions get deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The number of inversions for other items, except v, remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Let us denote the number of items with lower level than v at time t by Lt(v) and partition the �n i=1 Bt+1(i) into three parts as we discussed (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' items stored in a lower level than v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' and other items denoted by set Ot(v)): n � i=1 Bt+1(i) = Bt+1(v) · � i∈Lt(v) Bt+1(i) · � i∈Ot(v) Bt+1(i) By rewriting Bt+1(i) in terms of Bt(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' we get: n � i=1 Bt+1(i) = 1 · � i∈Lt(v) (Bt(i) + 1 c · 2levelt(i) ) · � i∈Ot(v) Bt(i) Now let us look at potential due the first effect from time t to t + 1 by ∆φ1 t→t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' and describe it in more detail: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='∆φ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='t→t+1 = log Bt+1 − log Bt = log Bt+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='= log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt+1(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='= log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(v) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Lt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='(Bt(i) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c·2levelt(i) ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Lt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='≤ log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(v) · e|Lt(v)|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='in which the last inequality comes from the fact that |Lt(v)| = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c · 2levelt(i) and also the inequality that: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='|Lt(v)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='(Bt(i) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='|Lt(v)|) ≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='|Lt(v)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='(Bt(i) + Bt(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='|Lt(v)|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='= (1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='|Lt(v)|)|Lt(v)| · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='|Lt(v)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(i) ≤ e|Lt(v)| · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='|Lt(v)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Now ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='let ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='us ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='focus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' and first assume that ⌊log ⌈ rankt(v) c ⌉⌋ > levelt(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We want to find the maximum number of items that might cause inversion with the accessed item v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Among all c · 2⌊log ⌈ rankt(v) c ⌉⌋ − 1 items that v might have higher rank them, at most c · 2levelt(v) − 1 have lower level in the OPT tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hence we have: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='Bt(v) = (c · 2⌊log ⌈ rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋ − 1) − (c · 2levelt(v) − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c · 2levelt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='≥ (2⌊log ⌈ rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋ − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='2levelt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='≥ 2⌊log ⌈ rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='2levelt(v)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='= 2⌊log ⌈ rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋−levelt(v)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='hence the change in potential due to the first effect is: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='∆φ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='t→t+1 ≤ log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='2⌊log ⌈ rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋−levelt(v)−1 · elevelt(v)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='= log(2(1+log e)·levelt(v)−⌊log ⌈ rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='= (1 + log e) · levelt(v) − ⌊log ⌈rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='For the case ⌊log ⌈ rankt(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='⌉⌋ < levelt(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' we use the fact that Bt v > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' from the first inequality below: ∆φt→t+1 = log( 1 Btv elevelt(v)) ≤ log(2log e·levelt(v)) = log e · levelt(v) = (1 + log e) · levelt(v) − ⌊log ⌈rankt(v) c ⌉⌋ Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' in both cases of ⌊log ⌈ rankt(v) c ⌉⌋ being larger or smaller than levelt(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' we have ∆φt→t+1 ≤ (1 + log e) · levelt(v) − ⌊log ⌈ rankt(v) c ⌉⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We then show changes in the potential because of OPT’s reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Details of the computations are omitted due to space constraints, but they are similar to the changes in potential due to rearrangements in the ON’s algorithm, and the result is that each OPT’s movement costs less than log e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Summing up changes in the potential after ON’s and OPT’s reconfiguration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' assuming OPT has done wt move- ments at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' we end up with: ∆φt→t+1 = (1+log e)·levelt(v)−⌊log ⌈rankt(v) c ⌉⌋+w·log e And hence the cost of the online algorithm MRU(0) at time t is at most: Ct MRU(0) = Ct Amortized + ∆φt = ⌊log ⌈rankt(v) c ⌉⌋ + (1 + log e) · levelt(v) −⌊log ⌈rankt(v) c ⌉⌋+wt ·log e ≤ (1+log e)·(levelt(v)+wt) And then summing up the cost of the MRU(0) and OPT for the whole request sequence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' we will get: CON = � t Ct ON ≤ � t (1 + log e) · (levelt(v) + wt) = (1 + log e) · COP T In which the last equality comes from the fact that OPT also needs to access the item,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' and as we assumed an additional wt reconfigurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As the first application of Lemma 6 we prove a lower bound on the cost of any online algorithm that only depends on the size of the working set of accessed items in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Any online algorithm maintaining a self-adjusting complete binary tree with capacity c > 1 on a request sequence σ = σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' σm, requires an access cost of at least �m i=1⌊log ⌈ rankt(σi) c ⌉⌋ (1+e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' This proof is an extension and improvement of the proof from [10] for any values of c > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A result of Lemma 6 is that even an optimal algorithm cannot be better than 1 (1+e) the MRU(0), otherwise contradicting Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As the cost of each access to the item σi is ⌊log ⌈ rankt(σi) c ⌉⌋ in MRU(0), we can conclude the total cost of any algorithm should be larger than �m i=1⌊log ⌈ rankt(σi) c ⌉⌋ (1+e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Any MRU(β) tree is β·(1+e)-access competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lemma 6 shows that an MRU(0) is (1 + e)-access competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Any item which was in level k in MRU(0), is in level k + β in MRU(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As an MRU(β) algorithm keeps items with rankc(0) at level(0), and because for any k ≥ 1, we have k +β ≤ βk, we obtain that MRU(β) is (β)·(1+e)- access competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We conclude this section by proving our main theorem, dynamic optimality of online SeedTree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Combining Lemma 4, Lemma 5 and Lemma 7 yields that the upper bound for competitiveness is (1+e)·(2·(1+⌈ 1 1−f ⌉))·(2−log(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The fractional occupancy f = 1/2 in the above formula is the optimal value for f, which gives us the 43-competitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We need to point out that the above calculation is just an upper bound on the competitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' As we will discuss in §V, the best results are usually achieved with a slightly higher value of f, which we hypothesize might be because of an overestimation of items’ depth in our theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' APPLICATION IN RECONFIGURABLE DATACENTERS SeedTree provides a fundamental self-adjusting structure which is useful in different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For example, it may be used to adapt the placement of containers in virtualized settings, in order to reduce communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' However, SeedTree can also be applied in reconfigurable networks in which links can be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In the following, we describe how to use SeedTree in such a use case in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In particular, we consider reconfigurable datacenters in which the connectivity between racks, or more specifically Top-of-the- Rack (ToR) switches, can be adjusted dynamically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', based on optical circuit switches [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' An optical switch provides a matching between racks, and accordingly, the model is known as a matching model in the literature [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In the following, we will show how a SeedTree with capacity c and fractional occupancy of f = 1 c can be seen in terms of 2 + c matchings, and how reconfigurations can be transformed to the matching model‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We group these matchings into two sets: Topological matchings: consists of 2 static matchings, embedding the underlying binary tree of SeedTree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The first matching represents edges between a node and its left child (with the ID twice the ID of the node), and similarly the second matching for the right children (with the ID twice plus one of the ID of their parents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' An example is depicted with solid edges in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Membership matchings: has c dynamic matchings, con- necting nodes to items inside them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' If a node has more than one item, the corresponding order of items to match- ings is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' An example is shown with dotted edges in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Having the matchings in place, let us briefly discuss how search and reconfiguration operations are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A search for an item starts at the node with ID 001, the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We then check membership matchings of this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' If they map to the item, we have found the node which contains the item, and our search was successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Otherwise, we follow the edge determined by the hash of the item, going to the new possible node hosting the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We repeat the process of checking membership matchings and going along topological matchings until we find the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The item will be found, as it is stored in one of the nodes in the path determined by its hash value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Each step of moving an item can be implemented in the matching mode with only one edge removal and one edge addition in membership matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' EXPERIMENTAL EVALUATION We complement our analytical results by evaluating SeedTree on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Concretely, we are interested in answering the following questions: Q1 How does the access cost of our algorithm compare to the statically-optimal algorithm (optimized based on frequencies) and a demand-oblivious algorithm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ‡The matching model considers perfect matchings only, however, in practice imperfect matchings can be enforced by ignore rules in switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 3: A transformation from the example SeedTree shown in Figure 1, which has capacity c = 2 and fractional occupancy of f = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The disco balls on top represent the reconfigurable switches, and below are datacenter racks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Solid edges show structural matchings, and dotted edges represent membership matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Q2 How does additional capacity improve the performance of the online SeedTree, given fixed fractional occupancy of each level?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Q3 What is the best initial fractional occupancy for the online SeedTree, given a fixed capacity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Answers to these questions would help developers tune pa- rameters of the SeedTree based on their requirements and needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Before going through results, we describe the setup that we used: Our code is written in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='6 and we used seaborn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='11 [16] and Matplotlib 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='5 [17] libraries for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our programs were executed on a machine with 2x Intel Xeons E5-2697V3 SR1XF with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='6 GHz, 14 cores each, and a total of 128 GB DDR4 RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Input Real-world dataset: Our real-world dataset is communi- cations between servers inside three different Facebook clusters, obtained from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We post-processed this dataset for single-source communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Among all possible sources, we chose the most frequent source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Synthetic dataset: We use the Markovian model dis- cussed in [1], [18] for generating sequences based on a temporal locality parameter which ranges from 0 (uni- form distribution, no locality) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='9 (high temporal locality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our synthetic input consists of 65, 535 items and 1 million requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For generating such a dataset, we start from a random sample of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We post-process this sequence, overwriting each request with the previous request with the probability determined by our temporal locality parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' After that, we execute the second post- processing to ensure that exactly 65, 535 items are in the final trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithm setup We use SHA-512 [19] from the hashlib-library as the hash function in our implementation, approximating the uniform distribution for generating addresses of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In order to store items in a node we used a linked list, and when we move an item to a node that is already full with other items, items are stored in a temporary buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We assume starting from a pre-filled tree with items, a tree which respects the fractional occupancy parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 001 010 011 100 101 110 111 001 010 011 100 101 110 111(a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 4: Improvements in the performance of SeedTree by fine-tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Figures are generated using the synthetic dataset with various locality values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (4a) Comparing the access cost of the SeedTree with fractional occupancy f = 1 2 to the best possible static algorithm and the demand-oblivious algorithm, all given capacity c = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Access costs are divided by 100 thousands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (4b) The effect of increasing capacity of nodes and temporal locality of input on the total cost of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The fractional occupancy is set to f = 1 2 for all capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Total costs are divided by 1 million for this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (4c) Tradeoff between the total cost and the fractional occupancy, given a range of temporal localities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The capacity of nodes is set to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The number in each cell represents the cost, which are divided by 1 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' 5: Improvements in the normalized access cost of the algorithm by changing SeedTree parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' These results are obtained based on communications of the most frequent source from three clusters of the real-world dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Costs are normalized by the cost of the demand-oblivious algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (5a) Changes in the normalized cost by varying capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Fractional occupancy is set to f = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' (5a) Changes in the normalized cost by varying fractional occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Gray dots show the minimum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Capacity of nodes is set to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In our experiments, we range the capacities (c) from 2 to 16, and the fractional occupancies (f) from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='16 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Due to the random nature of our algorithms and input generations, we repeat each experiment up to 100 times to ensure consistency in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Results The performance of SeedTree improves significantly with the increased temporal locality, as can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Furthermore, we have the following empirical answers to questions proposed at the beginning of this section: A1: The SeedTree improves the access cost significantly, with increased temporal locality, as shown in Figures 4a, which compares the access cost of SeedTree to static and demand-oblivious algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A2: As the Figures 4b and 5a show, increasing capacity reduces the cost of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' However, as we can see, this increase slows down beyond capacity to 8, and hence this value can be considered as the best option for practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A3: As discussed at the end of the §III and can be seen in Figures 4c and 5b, the lowest cost can be achieved with fractions higher or lower than 1 2, but f = 1 2 is near optimal in most scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ADDITIONAL RELATED WORK Self-adjusting lists and trees have already been studied intensively in the context of data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The pioneering work is by Sleator and Tarjan [20], who initiated the study of the dynamic list update problems and who also introduced the move-to-front algorithm, inspiring many deterministic [21], [22] and randomized [23]–[26] approaches for datastructures, as well as other variations of the problem [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Self-adjusting binary search trees also aim to keep recently used elements close to the root, similarly to our approach in this paper (a summary of results is in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' However, adjustments in binary search trees are based on rotations rather than the movement of items between different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' One of the well-known self-adjusting binary search trees is the splay tree [9], although it is still unknown whether this tree is dynamically optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' the problem is still open also for recent variations such as Zipper Tree [31], Multi Splay Tree [32] and Chain Splay [33] which improve the O(log n) competitive ratio of the splay tree to O(log log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' For Tango Trees [29], a matching Ω(log log n) lower bound is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We also know that if we allow for free rotations after access, dynamic Cluster C Cluster A Cluster B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='95 Cost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='90 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='83 Fractional Occupancy120 100 Cos 80 Access 60 40 SeedTree Oblivious Algorithm 20 Static Algorithm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='9 0 Temporal Locality60 Total Cost 50 40 30 20 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='5 03 汇 6 5 06 8 9 LO 4 Z S 6 Capacity36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='16 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='4 Occupancy 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='25 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='7 20.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='45 Temporal LocalityCluster A Cluster C Cluster B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='95 Normalized Cost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='75 2 6 10 12 8 14 16 4 CapacityData Structure Operation Ratio Search Splay Tree [9] Rotation O(log n) Yes Greedy Future [28] Rotation O(log n) Yes Tango Tree [29] Rotation θ(log log n) Yes Adaptive Huffman [30] Subtree swap θ(1) No Push-down Tree [10] Item swap θ(1) No SeedTree Item movement θ(1) Yes TABLE I: Comparison of properties of self-adjusting tree data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The best known competitive ratio (to this date) is in terms of the data structure’s respective cost model and optimal offline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We note that none of the above trees considers additional capacity, except for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' optimally becomes possible [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We also point out that some of these structures, in particular, multi splay tree and chain splay, benefitted from additional memory as well, however, there it is used differently, namely toward saving additional attributes for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Another variation which was first proposed by Lucas [28] in 1988 is called Greedy Future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' This tree first received attention as an offline binary search tree algorithm [35], [36], but then an O(log n) amortized time in online settings was suggested by Fox [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Greedy Future has motivated researchers to take a geometric view of online binary search trees [36], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We note that in contrast to binary search trees, our local tree does not require an ordering of the items in the left and right subtrees of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Self-adjusting trees have also been explored in the context of coding, where for example adaptive Huffman coding [30], [39]–[42] is used to minimize the depth of most frequent items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The reconfiguration cost, however, is different: in adaptive Huffman algorithms, two subtrees might be swapped at the cost of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A few data structures have tried to achieve a better compet- itive ratio by expanding and altering binary search trees (see Table II for a summary): The first example, PokeTree [43], adds extra pointers between the internal nodes of the tree and achieves an O(log log n) competitive ratio in comparison to an optimal binary search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' There are also self-adjusting data structures based on skip lists [44], [45], which have been introduced as an alternative for balanced trees that enforce probabilistic balancing instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A biased version of skip lists was considered in [46], and later on, a statically optimal variation was given in [47] and a dynamic optimal version in a restricted model in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Another example is Iacono’s working set structure [49] which combines a series of self- adjusting balanced binary search trees and deques, achieving a worst-case running time of O(log n), however, it lacks the dynamic optimality property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We are not aware of any work exploring augmentations to improve the competitive ratio of these data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Our work is also motivated by emerging self-adjusting datacenter networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Recent optical communication technolo- gies enable datacenters to be reconfigured quickly and fre- quently [8], [18], [50]–[58], see [59] for a recent survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The datacenter application mentioned in our paper is based on the matching model proposed by [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Recently [60] introduced Data Structure Structure Ratio Iacono’s structure [49] Trees & deques O(log n) Skip List [44] Linked lists O(log n) PokeTree [43] Tree & dynamic links O(log log n) SeedTree Tree θ(1) TABLE II: Comparison with other self-adjusting data struc- tures that support local-search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The best known competitive ratio (to this date) is in terms of the data structure’s respective cost model and optimal offline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We note that none of the other data structures considers capacity in their design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' an online algorithm for constructing self-adjusting networks based on this model, however the authors do not provide dynamic optimality proof for their method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' It has been shown that demand-aware and self-adjusting datacenter networks can be built from individual trees [61], called ego-trees, which are used in many network designs [8], [50], [62], [63], and also motivate our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' However, until now it was an open problem how to design self-adjusting and constant-competitive trees that support local routing and adjustments, a desirable property in dynamic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Last but not least, our work also features interesting con- nections to peer-to-peer networks [12], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' It is known that consistent hashing with previously assigned and fixed capaci- ties allows for significantly improved load balancing [13], [14], which has interesting applications and is used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', in Vimeo’s streaming service [65] and in Google’s cloud service [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Although these approaches benefit from data structures with capacity, these approaches are not demand-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK This paper presented and evaluated a self-adjusting and local tree, SeedTree, which adapts towards the workload in an online, constant-competitive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' SeedTree supports a capacity augmentation approach, while providing local rout- ing, which can be useful for other self-adjusting structures and applications as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We showed a transformation of our algo- rithm into the matching model for application in reconfigurable datacenters, and evaluated our algorithm on synthetic and real- world communication traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' The code used for our experi- mental evaluation is available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='com/inet-tub/SeedTree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' We believe that our work opens several interesting avenues for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' In particular, while we so far focused on randomized approaches, it would be interesting to explore de- terministic variants of SeedTree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Furthermore, while trees are a fundamental building block toward more complex networks (as they, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', arise in datacenters today), it remains to design and evaluate networks based on SeedTree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ghobadi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Griner, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “On the complexity of traffic traces and implications,” in ACM SIGMETRICS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Benson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Anand, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Akella, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Zhang, “Understanding data center traffic characteristics,” ACM SIGCOMM CCR, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [3] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Michel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Bifulco, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Retvari, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “The programmable data plane: Abstractions, architectures, algorithms, and applications,” in ACM CSUR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kellerer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kalmbach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Blenk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Basta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Reisslein, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Adaptable and data-driven softwarized networks: Review, opportunities, and challenges,” in IEEE PIEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Fischer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Botero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Beck, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' de Meer, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hesselbach, “Virtual network embedding: A survey,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Tutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hall, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Foerster, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Durairajan, “A survey of reconfigurable optical networks,” in OSN, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Borodin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' El-Yaniv, Online computation and competitive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' cambridge university press, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mondal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Demand-aware network designs of bounded degree,” in DISC, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Sleator and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Tarjan, “Self-adjusting binary search trees,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ACM, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mondal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Push-down trees: Optimal self- adjusting complete trees,” in IEEE/ACM, TON, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Iacono, “Key-independent optimality,” Algorithmica, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [12] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Stoica, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Morris, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Liben-Nowell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Karger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kaashoek, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Dabek, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Balakrishnan, “Chord: a scalable peer-to-peer lookup protocol for internet applications,” IEEE/ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mirrokni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Thorup, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Zadimoghaddam, “Consistent hashing with bounded loads,” in ACM-SIAM SODA, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Aamand, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Knudsen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Thorup, “Load balancing with dynamic set of balls and bins,” in ACM SIGACT STOC, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Griner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Zerwas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Blenk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ghobadi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, “Cerberus: The power of choices in datacenter topology design (a throughput perspective),” in ACM SIGMETRICS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Waskom, “seaborn: statistical data visualization,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' of Open Source Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hunter, “Matplotlib: A 2d graphics environment,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Bienkowski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Salem, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Sama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmidt, “Deterministic self-adjusting tree networks using rotor walks,” in IEEE ICDCS, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Dobraunig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Eichlseder, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mendel, “Analysis of SHA-512/224 and SHA-512/256,” IACR Cryptol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ePrint Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Sleator and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Tarjan, “Amortized efficiency of list update and paging rules,” Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ACM, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Albers, “A competitive analysis of the list update problem with lookahead,” MFCS, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kamali and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' L´opez-Ortiz, “A survey of algorithms and models for list update,” in LNTCS, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Albers and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Janke, “New bounds for randomized list update in the paid exchange model,” in STACS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Albers, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Von Stengel, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Werchner, “A combined bit and timestamp algorithm for the list update problem,” Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Garefalakis, “A new family of randomized algorithms for list access- ing,” in ESA, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [26] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Reingold, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Westbrook, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Sleator, “Randomized compet- itive algorithms for the list update problem,” Algorithmica, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Albers and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lauer, “On list update with locality of reference,” in ICALP, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lucas, Canonical forms for competitive binary search tree algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Rutgers University, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [29] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Demaine, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Harmon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Iacono, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Patrascu, “Dynamic optimality - almost,” in IEEE FOCS, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Cormack and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Horspool, “Algorithms for adaptive huffman codes,” Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Bose, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Dou¨ıeb, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Dujmovi´c, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Fagerberg, “An o (log log n)- competitive binary search tree with optimal worst-case access times,” in SWAT, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Derryberry, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Sleator, “O (log log n)-competitive dynamic binary search trees,” in ACM-SIAM SODA, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [33] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Georgakopoulos, “Chain-splay trees, or, how to achieve and prove loglogn-competitiveness by splaying,” Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Blum, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Chawla, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kalai, “Static optimality and dynamic search-optimality in lists and trees,” in ACM-SIAM SODA, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Munro, “On the competitiveness of linear search,” in ESA, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [36] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Demaine, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Harmon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Iacono, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kane, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Patrascu, “The geometry of binary search trees,” in ACM-SIAM SODA, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [37] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Fox, “Upper bounds for maximally greedy binary search trees,” in WADS, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Iacono, “In pursuit of the dynamic optimality conjecture,” in Space- Efficient Data Structures, Streams, and Algorithms, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Knuth, “Dynamic huffman coding,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithms, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Milidi´u, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Laber, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Pessoa, “Bounding the compression loss of the FGK algorithm,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithms, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Moffat, “Huffman coding,” ACM CSUR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Vitter, “Design and analysis of dynamic huffman codes,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' of the ACM, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kujala and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Elomaa, “Poketree: A dynamically competitive data structure with good worst-case performance,” in ISAAC, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [44] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Pugh, “Skip lists: A probabilistic alternative to balanced trees,” Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' ACM, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [45] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Salem, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Working set theorems for routing in self-adjusting skip list networks,” in IEEE INFOCOM, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Bagchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Buchsbaum, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Goodrich, “Biased skip lists,” Algorithmica, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [47] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ciriani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ferragina, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Luccio, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Muthukrishnan, “A data structure for a sequence of string accesses in external memory,” ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Algorithms, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [48] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Bose, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Dou¨ıeb, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Langerman, “Dynamic optimality for skip lists and b-trees,” in ACM-SIAM SODA, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Iacono, “Alternatives to splay trees with o(log n) worst-case access times,” in ACM-SIAM SODA, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [50] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mondal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Demand-aware network design with minimal congestion and route lengths,” in IEEE INFOCOM, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ballani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Costa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Behrendt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Cletheroe, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Haller, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Jozwik, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Karinou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', “Sirius: A flat datacenter network with nanosecond optical switching,” in ACM SIGCOMM, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [52] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Singla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Singh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ramachandran, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Wen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Chen, “Osa: An optical switching architecture for data center networks with unprecedented flexibility,” IEEE/ACM TON, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [53] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Douglis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Robertson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Van den Berg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Micallef, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Pucci, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Aiken, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Hattink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Seok, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Bergman, “Fleet—fast lanes for expedited execution at 10 terabits: Program overview,” IEEE Internet Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [54] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Foerster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ghobadi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Characterizing the al- gorithmic complexity of reconfigurable data center architectures,” in ACM/IEEE ANCS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ghobadi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mahajan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Phanishayee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Devanur, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kulkarni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Ranade, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Blanche, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Rastegarfar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=', “Projector: Agile reconfigurable data center interconnect,” in ACM SIGCOMM, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Kulkarni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmidt, “Scheduling opportunistic links in two-tiered reconfigurable datacenters,” in ACM SPAA, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [57] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mellette, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Das, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Guo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' McGuinness, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Snoeren, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Porter, “Expanding across time to deliver bandwidth efficiency and low latency,” in USENIX NSDI, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [58] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Mellette, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' McGuinness, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Forencich, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Papen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Snoeren, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Porter, “Rotornet: A scalable, low-complexity, optical datacenter network,” in ACM SIGCOMM, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [59] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Foerster and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Survey of reconfigurable data center networks: Enablers, algorithms, complexity,” in SIGACT News, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [60] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Feder, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Rathod, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Shyamsukha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Sama, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Aksenov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Salem, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Lazy self-adjusting bounded-degree networks for the matching model,” in IEEE INFOCOM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [61] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Toward demand-aware networking: a theory for self-adjusting networks,” ACM SIGCOMM CCR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [62] ——, “Renets: Statically-optimal demand-aware networks,” in SIAM APOCS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [63] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Peres, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' de Oliveira Souza, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Goussevskaia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Avin, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Schmid, “Distributed self-adjusting tree networks,” in IEEE INFOCOM, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [64] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Karger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lehman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Leighton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Panigrahy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Levine, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Lewin, “Consistent hashing and random trees: Distributed caching protocols for relieving hot spots on the world wide web,” in ACM STOC, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' [65] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} +page_content=' Rodland, “Improving load balancing with a new consistent-hashing algorithm,” Vimeo Engineering Blog, Medium, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfTAPi/content/2301.03074v1.pdf'} diff --git a/5dFIT4oBgHgl3EQf7yth/content/tmp_files/2301.11399v1.pdf.txt b/5dFIT4oBgHgl3EQf7yth/content/tmp_files/2301.11399v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0caf4a6fc6fab08132f9dceca186d690fb0c8352 --- /dev/null +++ b/5dFIT4oBgHgl3EQf7yth/content/tmp_files/2301.11399v1.pdf.txt @@ -0,0 +1,1893 @@ +Distributional outcome regression and its +application to modelling continuously +monitored heart rate and physical activity +Rahul Ghosal1, Sujit K. Ghosh2, Jennifer A. Schrack3, Vadim Zipunnikov4 +1 Department of Epidemiology and Biostatistics, University of South Carolina +2Department of Statistics, North Carolina State University +3 Department of Epidemiology, Johns Hopkins Bloomberg +School of Public Health +4 Department of Biostatistics, Johns Hopkins Bloomberg +School of Public Health +January 30, 2023 +Abstract +We propose a distributional outcome regression (DOR) with scalar and distribu- +tional predictors. Distributional observations are represented via quantile functions +and the dependence on predictors is modelled via functional regression coefficients. +DOR expands existing literature with three key contributions: handling both scalar +and distributional predictors, ensuring jointly monotone regression structure with- +out enforcing monotonicity on individual functional regression coefficients, pro- +viding a statistical inference for estimated functional coefficients. Bernstein poly- +nomial bases are employed to construct a jointly monotone regression structure +without over-restricting individual functional regression coefficients to be mono- +tone. Asymptotic projection-based joint confidence bands and a statistical test of +global significance are developed to quantify uncertainty for estimated functional +regression coefficients. Simulation studies illustrate a good performance of DOR +model in accurately estimating the distributional effects. The method is applied to +continuously monitored heart rate and physical activity data of 890 participants of +Baltimore Longitudinal Study of Aging. Daily heart rate reserve, quantified via a +subject-specific distribution of minute-level heart rate, is modelled additively as a +function of age, gender, and BMI with an adjustment for the daily distribution of +minute-level physical activity counts. Findings provide novel scientific insights in +epidemiology of heart rate reserve. +Keywords: Distributional Data Analysis; Distribution-on-distribution regression; Quan- +tile function-on-scalar Regression; BLSA; Physical Activity; Heart Rate. +1 +arXiv:2301.11399v1 [stat.ME] 26 Jan 2023 + +1 +Introduction +Distributional data analysis is an emerging area of research with diverse applications in +digital medicine and health (Augustin et al., 2017; Matabuena et al., 2021; Ghosal et al., +2021; Matabuena and Petersen, 2021), radiomics (Yang et al., 2020), neuroimaging (Tang +et al., 2020) among many others. With the advent of modern medical devices and wear- +ables, many studies collect subject-specific high frequency or high density observations +including heart rate, physical activity (steps, activity counts), continuously monitored +blood glucose, functional and structured brain images, and others. The central idea of +distributional data analysis is to capture the distributional aspect in this data and model +it within regression frameworks. Thus, distributional data analysis inherently deals with +data objects which are distributions typically represented via histograms, densities, quan- +tile functions or other distributional representations. Petersen et al. (2021) provide an +in-depth overview of recent developments in this area. +Similar to functional regression models, depending on whether the outcome or the +predictor is distributional, there are various types of distributional regression models. +Petersen and M¨uller (2016) and Hron et al. (2016) developed functional compositional +methods to analyze samples of densities. For scalar outcome and distributional predictors +represented via densities, a common idea has been to transform densities by mapping +them to a proper Hilbert space L2 and then use existing functional regression approaches +for modelling scalar outcomes. Petersen and M¨uller (2016) used a log-quantile density +transformation, whereas Talsk´a et al. (2021) used a centered log-ratio transformation. +Other approaches for modelling scalar outcomes and distributional predictors include +scalar-on-quantile function regression (Ghosal et al., 2021), kernel-based approaches using +quantile functions (Matabuena and Petersen, 2021) and many others (see Petersen et al. +(2021), Chen et al. (2021) and references therein). +In parallel, there was also a substantial work on developing models with distributional +outcome and scalar predictors. +Yang et al. (2020) developed a quantile function-on- +scalar (QFOSR) regression model, where subject-specific quantile functions of data were +modelled via scalar predictors of interest using a function-on-scalar regression approach +(Ramsay and Silverman, 2005), which make use of data-driven basis functions called +quantlets. One limitation of the approach is a no guarantee of underlying monotonicity +of the predicted quantile functions. To address this, Yang (2020) extended this approach +2 + +using I-splines (Ramsay et al., 1988) or Beta CDFs which enforce monotonicity at the +estimation step. +One important limitation of this approach is enforcement of jointly +monotone (non-decreasing) regression structure via enforcement of monotonicity on each +individual functional regression coefficients. As we demonstrate in our application, this +assumption could be too restrictive in real world. +Distribution-on-distribution regression models when both outcome and predictors are +distributions have been studied by Verde and Irpino (2010); Irpino and Verde (2013); +Chen et al. (2021); Ghodrati and Panaretos (2021); Pegoraro and Beraha (2022). These +models aim to understand the association between distributions within a pre-specified, +often linear, regression structure. +Verde and Irpino (2010); Irpino and Verde (2013) +used an ordinary least square approach based on the squared L2 Wasserstein distance +between distributions. Outcome quantile function QiY (p) was modelled as a non-negative +linear combination of other quantile functions QiXj(p)s using a multiple linear regression. +This model although useful and adequate for some applications, may not be flexible +enough as it assumes a linear association between the distribution valued response and +predictors, which are additionally assumed to be constant across all quantile levels p ∈ +(0, 1). Chen et al. (2021) used a geometric approach taking distributional valued outcome +and predictor to a tangent space, where regular tools of function-on-function regression +(Ramsay and Silverman, 2005; Yao et al., 2005) were applied. +Pegoraro and Beraha +(2022) used an approximation of the Wasserstein space using monotone B-spline and +developed methods for PCA and regression for distributional data. Recently, Ghodrati +and Panaretos (2021) developed a shape-constrained approach linking Frechet mean of +the outcome distribution to the predictor distribution via an optimal transport map that +was estimated by means of isotonic regression. +Many of above-mentioned methods mainly focused on dealing with constraints en- +forced by a specific functional representation. Developing inferential tools is somewhat +under-developed are of distributional data analysis. Chen et al. (2021) derived the asymp- +totic convergence rates for the estimated regression operator in their proposed method +for Wasserstein regression. Yang et al. (2020) developed joint credible bands for distribu- +tional effects, but monotonocity of the quantile function was not imposed. Yang (2020) +developed a global statistical test for estimated functional coefficients in the distribu- +tional outcome regression, however, no confidence bands was proposed to identify and +3 + +test local quantile effects. +In this paper, we propose a distributional outcome regression that expands exist- +ing literature in three main directions. First, our model includes both scalar and dis- +tributional predictors. +Second, it ensures jointly monotone (non-decreasing) additive +regression structure without enforcing monotonicity of individual functional regression +coefficients. +Thirdly, it provides a toolbox of statistical inference tools for estimated +functional coefficients including asymptotic projection-based joint confidence bands and +a statistical test of global significance. We capture distributional aspect in outcome and +predictors via quantile functions and construct a jointly monotone regression model via +a specific shape-restricted functional regression model. The distributional effects of the +scalar covariates are captured via functional coefficient βj(p)’s varying over quantile lev- +els and the effect of the distributional predictor is captured via a monotone function +h(·), similar to an optimal transport approach in Ghodrati and Panaretos (2021). In the +special case, when there is no distributional predictor, the model resembles a quantile +function-on-scalar regression model, but with much more flexible constraints compared +to Yang (2020). In the absence of scalar predictors the model reduces to a distribution +on distribution regression model, where the monotone function representing the optimal +transport map is estimated by a non-parametric functional regression model under shape +constraints. We use Bernstein polynomial (BP) basis functions to model the distribu- +tional effects βj(p)s and the monotone map h(·), which are known to enjoy attractive +and optimal shape-preserving properties (Lorentz, 2013; Carnicer and Pena, 1993). Ad- +ditionally, BP is instrumental in constructing and enforcing a jointly monotone regression +structure without over-restricting individual functional regression coefficients to be mono- +tone. Finally, inferential tools are developed including joint asymptotic confidence bands +for distributional functional effects and p-values for testing the distributional effects of +predictors. +As a motivating application, we study continuously monitored heart rate and physical +activity collected in Baltimore Longitudinal Study of Aging (BLSA). We aim to study the +association between the distribution of heart rate as a distributional outcome and age, sex +and body mass index (BMI) while also adjusting for a key confounder, the distribution +of minute-level physical activity aggregated over 8am-8pm time period. Figure 1 displays +daily profiles of heart rate and physical activity between 8am-8pm for a BLSA participant +4 + +along with the corresponding subject-specific quantile functions. +8 +10 +12 +14 +16 +18 +20 +40 +60 +80 +100 +120 +Time of Day +Heartrate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +40 +60 +80 +100 +120 +p +Heartrate QF +8 +10 +12 +14 +16 +18 +20 +0 +200 +600 +1000 +Time of Day +Activity +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +200 +600 +1000 +p +Activity QF +Figure 1: Diurnal profile of heart rate and physical activity between 8 a.m.- 8 p.m. and +the corresponding subject specific quantile functions for a randomly chosen subject in +the BLSA. +The rest of this article is organized as follows. We present our distributional modeling +framework and illustrate the proposed estimation method in Section 2. In Section 3, we +perform numerical simulations to evaluate the performance of the proposed method and +provide comparisons with existing methods for distributional regression. In Section 4, we +demonstrate application of the proposed method in modelling continuously monitored +heart rate reserve in BLSA study. Section 5 concludes with a brief discussion of our +proposed method and some possible extensions of this work. +2 +Methodology +2.1 +Modelling Framework and Distributional Representations +We consider the scenario, where there are repeated subject-specific measurements of a +distributional response Y along with several scalar covariates zj, j = 1, 2, . . . , q and we +5 + +also have a distributional predictor X. Let us denote the subject-specific response and +covariates as Yik, Xil, zij (k = 1, . . . , n1i, l = 1, . . . , n2i), for subject i = 1, . . . , n. Here +n1i, n2i denotes the number of repeated observations of the distributional response and +predictor respectively for subject i. Assume Yik (k = 1, . . . , n1i) ∼ FiY (y), a subject- +specific cumulative distribution function (cdf), where FiY (y) = P(Yik ≤ y). Then, the +subject-specific quantile function is defined as QiY (p) = inf{y : FiY (y) ≥ p}, p ∈ [0, 1]. +The quantile function completely characterizes the distribution of the individual obser- +vations. Given Yik s, the empirical quantile function can be calculated based on linear +interpolation of order statistics (Parzen, 2004) and serves as an estimate of the latent +subject specific quantile function QiY (p) (Yang et al., 2020; Yang, 2020). In particular, +for a sample (X1, X2, . . . , Xn), letX(1) ≤ X(2) ≤ . . . , ≤ X(n) be the corresponding order +statistics. The empirical quantile function, for p ∈ [ +1 +n+1, +n +n+1], is then given by, +ˆQ(p) = (1 − w)X([(n+1)p]) + wX([(n+1)p]+1), +(1) +where and w is a weight satisfying (n + 1)p = [(n + 1)p] + w. Based on this formulation +and observations Yik, Xil, we can obtain the subject specific quantile functions ˆQiY (p) +and ˆQiX(p) which are estimators of the true quantile functions QiY (p),QiX(p). The em- +pirical quantile functions are consistent (Parzen, 2004) and are suitable for distributional +representaion for several attractive mathematical properties (Powley, 2013; Ghosal et al., +2021) without requiring any smoothing parameter selection as in density estimation. +2.2 +Distribution-on-scalar and Distribution Regression +We assume that the scalar covariates (z1, z2, . . . , zq) ∈ [0, 1]q without any loss of gener- +ality (e.g., achievable by linear transformation). We posit the following distributional +regression model, associating the distributional response QiY (p) to the scalar covariates +zij, j = 1, 2, . . . , q, and a distributional predictor QiX(p). +We will refer to this as a +distribution-on-scalar and distribution regression (DOSDR) model. +QiY (p) = β0(p) + +q +� +j=1 +zijβj(p) + h(QiX(p)) + ϵi(p). +(2) +6 + +Here β0(p) is a distributional intercept and βj(p) s are the distributional effects of the +scalar covariates Zj at quantile level p. +The unknown nonparametric function h(·) +captures the additive effect of the distributional predictor QiX(p). +The residual er- +ror process ϵi(p) is assumed to be a mean zero stochastic process with an unknown +covariance structure. +We make the following flexible and interpretable assumptions +on the coefficient functions βj(·), j = 0, 1, . . . , q and on h(·) which ensures the pre- +dicted value of the response quantile function QiY (p) conditionally on the predictors, +E(QiY (p) | zi1, zi2, . . . , ziq, qix(p)) is non-decreasing. +Theorem 1 Let the following conditions hold in the model (2). +1. The distributional intercept β0(p) is non-decreasing. +2. Any additive combination of β0(p) with distributional slopes βj(p) is non-decreasing, +i.e., β0(p) + �r +k=1 βjk(p) is non-decreasing for any sub-sample {j1, j2, . . . , jr} ⊂ +{1, 2, . . . , q}. +3. h(·) is non-decreasing. +Then E(QY (p) | z1, z2, . . . , zq, qx(p)) is non-decreasing. +Note that E(QY (p) | z1, z2, . . . , zq, qx(p)) is the predicted quantile function under the +squared Wasserstein loss function, which is same as the squared error loss for the quan- +tile functions. The proof is illustrated in Appendix A of the Supplementary Material. +Assumptions (1) and (2) are much weaker and flexible than the monotonicity conditions +of the QFOSR model in Yang (2020), where each of the function coefficients βj(p)s is re- +quired to be monotone, whereas, we only impose monotonicity on the sum of functional +coefficients. This is not just a technical aspect but this flexibility is important from a prac- +tical perspective, as it allows for capturing possible non-monotone association between +the distributional response and individual scalar predictors zj’s while still maintaining the +required monotonicity of the predicted response quantile function. Condition (3) matches +with the monotonicity assumption of the distributional regression model in Ghodrati and +Panaretos (2021) and in the absence of any scalar predictors, essentially captures the op- +timal transport map between the two distributions. Note that this optimal transport map +is constructed after adjusting for scalars of interest - thus, it provides a model general in- +ferential framework compared to that in Ghodrati and Panaretos (2021). Thus, the above +7 + +DOSDR model extends the previous inferential framework for distributional response on +scalar and contains both the QFOSR model and the distribution-on-distribution regres- +sion model as its submodels. More succinctly, in absence of distributional predictor we +have, +QiY (p) = β0(p) + +q +� +j=1 +zijβj(p) + ϵi(p), +(3) +which is a quantile-function-on-scalar regression (QFOSR) model ensuring monotonon- +icity under conditions (1),(2). Similarly, in absence of any scalar covariates, we have a +distribution-on-distribution regression model +QiY (p) = β0(p) + h(QiX(p)) + ϵi(p). +(4) +Model (4) is a bit more general than the one considered in Ghodrati and Panaretos +(2021), including a transnational effect β0(p). As a technical note, in models (2) and +(4), function h(·) is identifiable only up to an additive constant, and in particular, the +estimable quantity is the additive effect β0(p) + h(qx(p)) for a fixed QX(p) = qx(p). +2.3 +Estimation in DOSDR +We follow a shape constrained estimation approach (Ghosal et al., 2022a) for estimating +the distributional effects βj(p) and the nonparamatric function h() which naturally in- +corporates the constraints (1)-(3) of Theorem 1 in the estimation step. The univariate +coefficient functions βj(p) (j = 0, 1, . . . , p) are modelled in terms of univariate expansions +of Bernstein basis polynomials as +βj(p) = +N +� +k=0 +βjkbk(p, N), where bk(p, N) = +�N +k +� +pk(1 − p)N−k, for 0 ≤ p ≤ 1. +(5) +The number of basis polynomials depends on the degree of the polynomial basis N (which +is assumed to be same for all βj(·) for computational tractability in this paper). The +Bernstein polynomials bk(p, N) ≥ 0 and �N +k=0 bk(p, N) = 1. Wang and Ghosh (2012) +and Ghosal et al. (2022a) illustrate that various shape constraints e.g., monotonicity, +convexity, etc. can be reduced to linear constraints on the basis coefficients of the form +ANβN +j ≥ 0, where βN +j = (βj0, βj1, . . . , βjN)T and AN is the constraint matrix chosen in +8 + +a way to guarantee a desired shape restriction. In particular, in our context of DOSDR, +we need to choose constraint matrices AN in such a way which jointly ensure conditions +(1),(2) in Theorem 1 and thus guarantee a non-decreasing predicted value of the response +quantile function. The nonparametric function h(·) is modelled similarly using univariate +Bernstein polynomial expansion as +h(x) = +N +� +k=0 +θkbk(x, N), where bk(x, N) = +�N +k +� +xk(1 − x)N−k, for 0 ≤ x ≤ 1. +(6) +Since the domain of h(·) modelled via Bernstein basis is [0, 1], the quantile functions of the +distributional predictor QX(p) are transformed to a [0, 1] scale using linear transformation +of the observed predictors. We make the assumption here that the distributional predic- +tors are bounded, which is reasonable in the applications we are interested in. Henceforth, +we assume QX(p) ∈ [0, 1] without loss of generality. Further, note that, b0(x, N) = 1, +and since β0(p) already contains this constant term in the DOSDR model (2), including +the constant basis while modelling h(·) will lead to model singularity. Hence we drop +the constant basis (i.e. the first term) while modelling h(·). In particular, h(QiX(p)) is +modelled as h(QiX(p)) = �N +k=1 θkbk(QiX(p), N). Note that this is equivalent to imposing +the constraint h(0) = θ0 = 0. The non-decreasing condition in (3) of Theorem 1 can +again be specified as a linear constraint on the basis coefficients of the form Rθ ≥ 0, +where θ = (θ1, . . . , θN)T, and R is the constraint matrix. The DOSDR model (2) can be +reformulated in terms of basis expansions as +QiY (p) += +N +� +k=0 +β0kbk(p, N) + +q +� +j=1 +zij +N +� +k=0 +βjkbk(p, N) + +N +� +k=1 +θkbk(QiX(p), N) + ϵi(p).(7) += +bN(p)Tβ0 + +q +� +j=1 +ZT +ij(p)βj + bN(QiX(p))Tθ + ϵi(p). +Here βj = (βj0, βj1, . . . , βjN)T, bN(p)T = (b0(p, N), b1(p, N), . . . , bN(p, N)), bN(QiX(p))T = +(b1(QiX(p), N), +b2(QiX(p), N), . . . , bN(QiX(p), N)) and ZT +ij(p) = zij ∗ bN(p)T. Suppose that we have the +qunatile functions QiY (p), QiX(p) evaluated on a grid P = {p1, p2, . . . , pm} ⊂ [0, 1]. De- +note the stacked value of the quantiles for ith subject as QiY = (QiY (p1), QiY (p2), . . . , QiY (pm))T. +9 + +The DOSDR model in terms of Bernstein basis expansion (7) can be reformulated as +QiY += +B0β0 + +q +� +j=1 +Wijβj + Siθ + ϵi, +(8) +where B0 = (bN(p1), bN(p2), . . . , bN(pm))T,Wij = (Zij(p1), Zij(p2), . . . , Zij(pm))T and +Si = (bN(QiX(p1)), +bN(QiX(p2)), . . . , bN(QiX(pm)))T and ϵi are the stacked residuals ϵi(p)s. The parameters +in the above model are the basis coefficients ψ = (βT +0 , βT +1 , . . . , βT +q , θT)T. For estimation +of the parameters, we use the following least square criterion, which reduces to a shape +constrained optimization problem. Namely, we obtain the estimates ˆψ by minimizing +residual sum of squares as +ˆψ = argmin +ψ +n +� +i=1 +||QiY − B0β0 − +q +� +j=1 +Wijβj − Siθ||2 +2 +s.t +Dψ ≥ 0. +(9) +The universal constraint matrix D on the basis coefficients is chosen to ensure the con- +ditions (1),(2),(3) in Theorem 1. Later in this section, we illustrate examples how the +constraint matrix is formed in practice. The above optimization problem (9) can be iden- +tified as a quadratic programming problem (Goldfarb and Idnani, 1982, 1983). R package +restriktor (Vanbrabant and Rosseel, 2019) can be used for performing the above opti- +mization. +Example 1: Single scalar covariate (q = 1) and a distributional predictor +We consider the case where there is a single scalar covariate z1 (q = 1) and a dis- +tribution predictor QX(p). In this case, the DOSDR model (2) is given by QiY (p) = +β0(p) + zi1β1(p) + h(QiX(p)) + ϵi(p). The sufficient conditions (1)-(3) for non-decreasing +quantile functions in this case reduces to: A) The distributional intercept β0(p) is non- +decreasing B) β0(p) + β1(p) is non-decreasing C) h(·) is non-decreasing. Note that the +above conditions do no enforce β1(p) to be non-decreasing. Once the coefficient func- +tions are modelled in terms of Bernstein basis expansions as in (4) and (5), conditions +(A)-(C) can be be enforced via the following linear restrictions on the basis coefficients +i.e., ANβ0 ≥ 0, [AN AN](βT +0 , βT +1 )T ≥ 0, AN−1θ ≥ 0. Here AN is a constraint matrix +which imposes monotonicity on functions fN(x) modelled with Bernstein polynomials +as fN(x) = �N +k=0 βkbk(x, N), where bk(x, N) = +�N +k +� +xk(1 − x)N−k, for 0 ≤ x ≤ 1. The +10 + +derivative is given by f ′ +N(x) = N �N−1 +k=0 (βk+1 − βk)bk(x, N − 1). Hence if βk+1 ≥ βk for +k = 0, 1, . . . , N −1, fN(x) is non decreasing, which is achieved with the constraint matrix +AN. The combined linear restrictions on the parameter ψ = (βT +0 , βT +1 , θT)T is given by +Dψ ≥ 0. The matrices AN, D are given by +AN ≡ +� +� +� +� +� +� +� +� +−1 +1 +0 +. . . +0 +0 +−1 +1 +0 +. . . +... +0 +. . . +0 +−1 +1 +� +� +� +� +� +� +� +� +, D = +� +� +� +� +� +AN +0 +0 +AN +AN +0 +0 +0 +AN−1 +� +� +� +� +� . +(10) +Similar example with two scalar covariates (q = 2) and a distributional predictor is +given in Appendix B of the Supplementary Material. Our estimation ensures that the +shape restrictions are enforced everywhere and hence the predicted quantile functions +are nondecreasing in the whole domain p ∈ [0, 1] as opposed to fixed quantile levels or +design points in Ghodrati and Panaretos (2021). The order of the Bernstein polynomial +basis N controls the smoothness of the coefficient functions βj(·) and h(·). We follow a +truncated basis approach (Ramsay and Silverman, 2005; Fan et al., 2015), by restricting +the number of BP basis to ensure the resulting coefficient functions are smooth. The +optimal order of the basis functions is chosen via V -fold cross-validation method (Wang +and Ghosh, 2012) using cross-validated residual sum of squares defined as, CV SSE = +�V +v=1 +�nv +i=1 ||QiY,v − ˆQ−v +iY,v||2 +2. Here ˆQ−v +iY is the fitted quantile values of observation i within +the v th fold obtained from the constrained optimization criterion (9) and trained on the +rest (V − 1) folds. +2.4 +Uncertainty Quantification and Joint Confidence Bands +To construct confidence intervals, we use the result that the constrained estimator ˆψ in +(9) is the projection of the corresponding unconstrained estimator (Ghosal et al., 2022a) +onto the restricted space: ˆψr = argmin +ψ∈ΘR +||ψ − ˆψur||2 +ˆΩ, for a non-singular matrix ˆΩ. The +restricted parameter space is given by ΘR = {ψ ∈ RKn : Dψ ≥ 0}. The DOSDR model +(8) can be reformulated as QiY = Tiψ + ϵi, where Ti = [B0 Wi1 Wi2, . . . , Wiq Si] . The +11 + +unrestricted and restricted estimators are given by, +ˆψur = argmin +ψ∈RKn +n +� +i=1 +||QiY − Tiψ||2 +2 +(11) +ˆψr = argmin +ψ∈ΘR +n +� +i=1 +||QiY − Tiψ||2 +2 +Let us denote QT +Y = (Q1Y , Q2Y , . . . , QnY )T and T = [TT +1 , TT +2 , . . . , TT +n]T. Then we can +write, +1 +n||QY − Tψ||2 +2 = 1 +n||QY − T ˆψur||2 +2 + 1 +n||T ˆψur − Tψ||2 +2. +Hence ˆψr = argmin +ψ∈ΘR +||ψ− ˆψur||2 +ˆΩ, where ˆΩ = 1 +n +�n +i=1 TT +i Ti and Ω = E( ˆΩ) is non-singular. +Thus, we can use the projection of the large sample distribution of √n( ˆψur − ψ0) to +approximate the distribution of √n( ˆψr − ψ0). +Now √n( ˆψur − ψ0) is asymptotically +distributed as N(0, ∆) under suitable regularity conditions (Huang et al., 2004, 2002) for +general choice of basis fucntions (holds true for finite sample sizes if ϵ(p) is Gaussian), +where ∆ can be estimated by a consistent estimator. In particular, we use a sandwich +covariance estimator corresponding to model QiY = Tiψ + ϵi, for estimating ∆ following +a functional principal component analysis (FPCA) approach (Ghosal and Maity, 2022) +for estimation of the covariance matrix of the residuals ϵi (i = 1, . . . , n). Details of this +estimation procedure is included in Appendix C of the Supplementary Material. +Let us consider the scenario with a single scalar covariate and distributional pre- +dictor for simplicity of illustration. The Bernstein polynomial approximation of β1(p) +be given by β1N(p) = �N +k=0 βkb1k(p, N) = ρKn(p) +′β1. Algorithm 1 in Appendix D is +used to obtain an asymptotic 100(1 − α)% joint confidence band for the true coefficient +function β0 +1(p), corresponding to a scalar predictor of interest. Here β0 +1(p) denotes the +true distributional coefficient β1(p). The algorithm relies on two steps i) Use the asymp- +totic distribution of √n( ˆψr − ψ0) to generate samples from the asymptotic distribution +of ˆβ1r(p) (these can be used to get point-wise confidence intervals) ii) Use the gener- +ated samples and the supremum test statistic (Meyer et al., 2015; Cui et al., 2022) to +obtain joint confidence band for β0 +1(p). Similar strategy can also be employed for ob- +taining an asymptotic joint confidence band for the additive effect β0(p) + h(qx(p)), for +a fixed value of QX(p) = qx(p). Based on the joint confidence band, it is possible to +directly test for the global distributional effects β(p) (or h(x)). The p-value for the test +12 + +H0 : β(p) = 0 for all p ∈ [0, 1] versus H1 : β(p) ̸= 0 for at least one p ∈ [0, 1], could be +obtained based on the 100(1 − α)% joint confidence band for β(p). In particular, follow- +ing Sergazinov et al. (2022), the p-value for the test can be defined as the smallest level +α for which at least one of the 100(1 − α)% confidence intervals around β(p) (p ∈ P) +does not contain zero. Alternatively, a nonparametric bootstrap procedure for testing the +global effects of scalar and distributional predictors is illustrated in Appendix E of the +Supplementary Material which could be useful for finite sample sizes and non Gaussian +error process. +3 +Simulation Studies +In this Section, we investigate the performance of the proposed estimation and testing +method for DOSDR via simulations. To this end, we consider the following data gener- +ating scenarios. +3.1 +Data Generating Scenarios +Scenario A1: DOSDR, Both distributional and scalar predictor +We consider the DOSDR model given by, +QiY (p) = β0(p) + zi1β1(p) + h(QiX(p)) + ϵi(p). +(12) +The distributional effects are taken to be β0(p) = 2+3p, β1(p) = sin( π +2p) and h(x) = ( x +10)3. +The scalar predictor zi1 is generated independently from a U(0, 1) distribution. The dis- +tributional predictor QiX(p) is generated as QiX(p) = ciQN(p, 10, 1), where QN(p, 10, 1) +denotes the pth quantile of a normal distribution N(10, 1) and ci ∼ U(1, 2). The resid- +ual error process ϵ(p) is independently sampled from N(0, 0.1) for each p. Since we do +not directly observe these quantile functions QiX(p), QiY (p) in practice we assume we +have the subject-specific observations Xi = {xi1 = QiX(ui1), xi2 = QiX(ui2), . . . , xiLi1 = +QiX(uiLi1)} and Yi = {yi1 = QiY (vi1), yi2 = QiY (vi2), . . . , yiLi2 = QiY (viLi2)}, where ui, vj +s are independently generated from U(0, 1) distribution. For simplicity, we assume that +Li1 = Li2 = L many subject specific observations are available for both the distributional +outcome and the predictor. Based on the observations Xi, Yi the subject specific quantile +13 + +functions QiX(p) and QiY (p) are estimated based on empirical quantiles as illustrated in +equation (1) on a grid of p values ∈ [0, 1]. We consider number of individual measure- +ments L = 200, 400 and training sample size n = 200, 300, 400 for this data generating +scenario. The grid P = {p1, p2, .. . . . , pm} ⊂ [0, 1] is taken to be a equi-spaced grid of +length m = 100 in [0.005, 0.995]. A separate sample of size nt = 100 is used as a test set +for each of the above cases. +Scenario A2: DOSDR, Testing the effect of scalar predictor +We consider the data generating scheme (12) in scenario A1 above and test for the distri- +butional effect of the scalar predictor z1 using the proposed joint-confidence band based +test in section 2. To this end we let β1(p) = d × sin( π +2p), where the parameter d controls +the departure from the null hypothesis H0 : β1(p) = 0 for all p ∈ [0, 1] +versus +H1 : +β1(p) ̸= 0 for some p ∈ [0, 1]. The number of subject-specific measurements L is set to +200 and sample sizes n ∈ {200, 300, 400} are considered. +Scenario B: DOSDR, Only distributional predictor +We consider the following distribution on distribution regression model +QiY (p) = h(QiX(p)) + ϵi(p). +(13) +The distributional outcome QiY (p), the distributional predictor QiX(p) and the error +process ϵi(p) are generated similarly as in Scenario A. The number of subject-specific +measurements L is set to 200 and sample sizes n ∈ {200, 300, 400} are considered. This +scenario is used to compare the performance of the proposed DOSDR method with that +of the isotonic regression approach illustrated in Ghodrati and Panaretos (2021). +We consider 100 Monte-Carlo (M.C) replications from simulation scenarios A1 and +B to assess the performance of the proposed estimation method. For scenario A2, 200 +replicated datasets are used to assess type I error and power of the proposed testing +method. +14 + +3.2 +Simulation Results +Performance under scenario A1: +We evaluate the performance of our proposed method in terms of integrated mean squared +error (MSE), integrated squared Bias (Bias2) and integrated variance (Var). +For the +distributional effect β1(p), these are defined as MSE = +1 +M +�M +j=1 +� 1 +0 {ˆβj +1(p) − β1(p)}2dp, +Bias2 = +� 1 +0 {ˆ¯β1(p) − β1(p)}2dp, V ar = +1 +M +�M +j=1 +� 1 +0 {ˆβj +1(p) − ˆ¯β1(p)}2dp. Here ˆβj +1(p) is the +estimate of β1(p) from the jth replicated dataset and ˆ¯β1(p) = +1 +M +�M +j=1 ˆβj +1(p) is the M.C +average estimate based on the M replications. Table 1 reports the squared Bias, Variance +and MSE of the estimates of β1(p) for all cases considered in scenario A1. MSE as well +as squared Bias and Variance are found to decrease and be negligible as sample size n +or number of measurements L increase, illustrating satisfactory accuracy of the proposed +estimator. +Table 1: Integrated squared bias, variance and mean square error of estimated β1(p) over +100 Monte-Carlo replications, Scenario A1. +Sample Size +L=200 +L=400 +β1(p) +Bias2 +Var +MSE +Bias2 +Var +MSE +n= 200 +0.0001 +0.0034 +0.0035 +2.8 × 10−5 +0.0019 +0.0019 +n= 300 +1.9 × 10−5 +0.0026 +0.0026 +1.7 × 10−5 +0.0016 +0.0016 +n= 400 +2.6 × 10−5 +0.0018 +0.0018 +5.5 × 10−6 +0.0010 +0.0010 +Since, h(x) is not directly estimable in the DOSDR model (12), we consider estimation +of the estimable additive effect γ(p) = β0(p) + h(qx(p)) at qx(p) = 1 +n +�n +i=1 QiX(p). The +performance of the estimates in terms of squared Bias, variance and MSE are reported +in Table 2, which again illustrates satisfactory performance of the proposed method in +capturing the distributional effect of the distributional predictor QX(p). +Table 2: Integrated squared bias, variance and mean square error of the estimated additive +effect γ(p) = β0(p)+h(qx(p)) at qx(p) = 1 +n +�n +i=1 QiX(p) over 100 Monte-Carlo replications, +Scenario A1. +Sample Size +L=200 +L=400 +β0(p) + h(qx(p)) +Bias2 +Var +MSE +Bias2 +Var +MSE +n= 200 +9.9 × 10−5 +0.023 +0.023 +4.6 × 10−5 +0.023 +0.023 +n= 300 +7.3 × 10−5 +0.017 +0.017 +3.2 × 10−5 +0.017 +0.017 +n= 400 +5.8 × 10−5 +0.013 +0.013 +4.8 × 10−5 +0.013 +0.013 +The estimated M.C mean for the distributional effect β1(p) and γ(p) along with their +15 + +respective 95% point-wise confidence intervals are displayed in Figure 2, for the case +n = 400, L = 400. +The M.C mean estimates are superimposed on the true curves +and along with the narrow confidence intervals, they illustrate low variability and high +accuracy of the estimates. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +p +β1(p) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +4 +6 +8 +10 +p +γ(p) +Figure 2: Left: True distributional effect β1(p) (solid) and estimated ˆβ1(p) averaged over +100 M.C replications (dashed) along with point-wise 95% confidence interval (dotted), +scenario A1, n = 400, L = 400. Right: Additive effect γ(p) = β0(p) + h(qx(p)) (solid) at +qx(p) = 1 +n +�n +i=1 QiX(p) and its estimate ˆγ(p) averaged over 100 M.C replications (dashed) +along with point-wise 95% confidence interval (dotted). +As a measure of out-of-sample prediction performance, we report the average Wasser- +stein distance between the true quantile functions and the predicted ones in the test set +defined as WD = +1 +nt +�nt +i=1[ +� 1 +0 {Qtest +i +(p)− ˆQi +test(p)}2dp] +1 +2. Supplementary Table S1 reports +the summary of the average Wasserstein distance across the 100 Monte-Carlo replica- +tions. The low values of the average WD metric and their M.C standard error indicate +a satisfactory prediction performance of the proposed method. The prediction accuracy +appears to be improving with an increase in the number of measurements L. The perfor- +mance of the proposed projection based joint confidence intervals for β1(p) is investigated +in Supplementary Table S2 which reports the coverage and width of the joint confidence +bands for β1(p) for various choices of N and for the case L = 200. It is observed that +the nominal coverage of 95% lies within the two standard error limit of the estimated +coverage in the all the cases, particularly for choices of N picked by our proposed cross +16 + +validation method. +Performance under scenario A2: +We assess the performance of the proposed testing method in terms of estimated type +I error and power calculated from the Monte-Carlo replications. We set the order of +the Bernstein polynomial basis N = 3 based on our results from previous section. The +estimated power curve is displayed as a function of the parameter d in Supplementary +Figure S1, using a nominal level of α = 0.05. At d = 0, the null hypothesis holds and the +power corresponds to the type I error of the test. The nominal level α = 0.05 lies within +its two standard error limit for all the sample sizes, illustrating that the test maintains +proper size. For d > 0, we see the power quickly increase to 1, showing that the proposed +test is able to capture small departures from the null hypothesis successfully. +Performance under scenario B: +We again consider estimation of the estimable additive effect we consider estimation of +the estimable additive effect γ(p) = β0(p) + h(qx(p)) at qx(p) = +1 +n +�n +i=1 QiX(p), which +can be estimated by both the proposed DOSDR (2) method and the isotonic regression +method (Ghodrati and Panaretos, 2021). Note that true β0(p) = 0, but we include a +distributional intercept in our DOSDR model, nonetheless, as this information is not +available to practitioners. For the isotonic regression method we directly fit the model +(13) without any intercept. The performance of the estimates are compared in terms +of squared Bias, variance and MSE and are reported in Table 3. We observe a similar +performance of the proposed method with the PAVA based isotnic regression method. +Table 3: Integrated squared bias, variance and mean square error of the estimated additive +effect γ(p) = β0(p)+h(qx(p)) at qx(p) = 1 +n +�n +i=1 QiX(p) over 100 Monte-Carlo replications, +Scenario B, from the DOSDR method and the isotonic regression method with PAVA +(Ghodrati and Panaretos, 2021). +Sample Size +DOSDR +PAVA +β0(p) + h(qx(p)) +Bias2 +Var +MSE +Bias2 +Var +MSE +n= 200 +0.0002 +0.022 +0.022 +2.6 × 10−5 +0.022 +0.022 +n= 300 +0.0002 +0.016 +0.016 +2.4 × 10−5 +0.016 +0.016 +n= 400 +0.0002 +0.012 +0.013 +3 × 10−5 +0.012 +0.012 +The estimated M.C mean for the distributional effect γ(p) along with their respective +17 + +95% point-wise confidence intervals are displayed in Supplementary Figure S2, for the +case n = 400. Again, both the method are observed to perform a good job in capturing +γ(p). +The proposed DOSDR method enables conditional estimation of γ(p) = β0(p) + +h(qx(p)) on the entire domain p ∈ [0, 1], where as for the isotonic regression method, +interpolation is required from grid level estimates. The PAVA based isotonic regression +method failed to converge in 5% of the cases for sample size n = 200, where as, this +issue was not faced by our proposed method. In terms of model flexibility, the isotonic +regression method do not directly accommodate scalar predictors, or a distributional +intercept, and keeping these points in mind our proposed method certainly provide a +uniform and flexible approach for modelling distributional outcome, in the presence of +both distributional and scalar predictors. +4 +Modelling Distribution of Heart Rate in Baltimore +Longitudinal Study of Aging +In this section, we apply our proposed framework to continuously monitored heart rate +and physical activity data collected in Baltimore Longitudinal Study of Aging (BLSA), +the longest-running scientific study of aging in the United States. Specifically, the distri- +bution of minute-level heart rate is modelled via age, sex and BMI and the distribution +of minute-level activity counts capturing daily composition of physical activity. We set +our study period to be 8 a.m. - 8 p.m. and calculate distributional representation of +minute-level heart rate and (log-transformed) activity counts of BLSA participants via +subject-specific quantile functions QiY (p) (heart rate) and QiX(p) (represented via log- +transformed AC). For each participant, we consider only their first BLSA visit while +obtaining the subject-specific quantile functions QiY (p), QiX(p). Our final sample con- +stitutes of n = 890 BLSA participants, who had heart rate, physical activity and other +covariates used for the analysis available. Supplementary Table S3 presents the descrip- +tive statistics of the sample. +Supplementary Figure S3 shows the subject-specific quantile functions of heart rate +and physical activity (log-transformed, during 8 a.m.-8 p.m. time period). As a starting +point, we study the dependence of mean heart rate on mean activity count and age, sex +18 + +(Male=1, Female=0) and BMI via the multiple regression model, +µH,i = θ0 + θ1agei + θ2sexi + θ3BMIi + θ4µA,i + ϵi, +where µH,i, µA,i are the subject specific means of heart rate and activity counts. Supple- +mentary Table S4 reports the results of the model fit. Mean heart rate is found to be +negatively associated with age and mean activity, and positively associated with BMI. +The above results although useful, does not paint the whole picture about how the dis- +tribution of hear rate depends on these biological factors and the distribution of physical +activity. Therefore, we use the proposed DOR model +QiY (p) = β0(p) + ageiβage(p) + BMIiβBMI(p) + sexiβsex(p) + h(QiX(p)) + ϵi(p), . (14) +The scalar covariates age, BMI as well as activity counts are transformed to be [0, 1] scale +using monotone linear transformations. The distributional effects of age, sex (Male=1, +Female=0) and BMI on heart rate are captured by βage(p), βsex(p), βBMI(p), respectively. +The monotone nonparametric function h(·) is used to link the distribution of heart rate +and the distribution of activity counts. +We use the proposed estimation method for +estimation of the distributional effects βj(p)s and h(·) (h(0) = 0 is imposed). The common +degree of the Bernstein polynomial basis used to model all the distributional coefficient +was chosen via five-fold cross-validation method that resulted in N = 5. The estimated +distributional effects along with their asymptotic 95% joint confidence bands using the +proposed projection based method are displayed in Figure 3. The p-values from the joint +confidence band based global test for the intercept and the effect of age, BMI, sex, and +distribution of activity counts are found to be 1 × 10−6, 1 × 10−6, 5 × 10−5, 3 × 10−4 and +1 × 10−6, respectively, resulting in the significance of all the predictors. +The estimated distributional intercept ˆβ0(p) is monotone and represents the baseline +distribution of heart rate. +The estimated distributional effect of age is found to be +significant for all p, in particular, ˆβage(p) is negative and appears to be decreasing and +then stabilizing in p ∈ [0, 1] illustrating moderate-high levels of heart rate decrease at an +accelerated rate with age compared to sedentary levels of activity (Antelmi et al., 2004). +The maximal levels of heart rate (p > 0.8) are found to be decreasing with age (βage(p) < +0) (Kostis et al., 1982; Tanaka et al., 2001; Gellish et al., 2007). +The distributional +19 + +effect of BMI ˆβBMI(p) is found to be positive and increasing in p (especially at higher +quantiles), indicating that a higher maximal heart rate is associated with a higher BMI +after adjusting for age, sex and the daily distribution of activity counts (Foy et al., 2018). +The estimated effect of sex (Male) ˆβsex(p) illustrates that females have higher heart rate +(Antelmi et al., 2004; Prabhavathi et al., 2014) compared to males across all quantile +levels after adjusting for age, BMI and PA. The lower heart rate in males compared +to females can be attributed to size of the heart, which is typically smaller in females +than males (Prabhavathi et al., 2014) and thus need to beat faster to provide the same +output. The estimated monotone regression map between PA and heart rate distribution +ˆh(x) (estimated under constraint h(0) = 0) is found to be highly nonlinear and convex, +illustrating a non-linear dependence of heart rate on physical activity, especially at higher +values of PA. The convex nature of the map points out an accelerated increase in the +heart rate quantiles with an increase in the corresponding quantile levels of PA (Leary +et al., 2002). The estimated distributional effects especially for age and gender in our +analysis, illustrate that the distributional effects have no reason to be non-decreasing, +as enforced in the qunatile function-on-scalar regression model in Yang (2020), which +might lead to wrong conclusions here. The proposed DOR method is more flexible in +this regard and enforces the monotonicity of the quantile functions without requiring the +distributional effects to be monotone. +We also compare the predictive performance of the proposed DOSDR model with +that of the distribution-on-distribution regression model by Ghodrati and Panaretos +(2021) based on isotonic regression (DODR-ISO). Supplementary Figure S4 displays +the leave-one-out-cross-validated (LOOCV) predicted quantile functions of heart rate +from both the methods. We define the measure LOOCV R-Squared as R2 +loocv = 1 − +�N +i=1 +� 1 +0 {Qi(p)− ˆ +Qi +loocv(p)}2dp +�N +i=1 +� 1 +0 {Qi(p)− ¯Q}2dp +, where ¯Q = +1 +N +�N +i=1 +� 1 +0 Qi(p)dp to compare the out-of-sample +prediction accuracy of the two methods. The R2 +loocv value for the DOSDR and the DODR- +ISO model are calculated to be 0.60 and 0.49 respectively. This illustrates the proposed +DOSDR method is able to predict the heart rate quantile functions more accurately with +the use of additional information from the biological scalar factors age, sex and BMI. +20 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +60 +80 +100 +120 +intercept for HR +p +beta0_intercept +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−30 +−10 +10 +30 +age effect on HR +p +beta_age +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−20 +0 +10 +30 +bmi effect on HR +p +beta_bmi +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−10 +−6 +−4 +−2 +0 +sex effect on HR +p +beta_sex(M) +0 +2 +4 +6 +8 +0 +50 +100 +150 +h function +h(x) +Figure 3: Estimated distributional effects (solid) along with their joint 95% confidence +bands (dotted) for age, BMI (both scaled to [0, 1]) and sex (Male) on heart rate along +with the estimated link function h(·) (solid) (under the constraint h(0) = 0) between the +distribution of heart rate and physical activity. +5 +Discussion +In this article, we have developed a flexible distributional outcome regression. The dis- +tributional functional effects are modelled via Bernstein polynomial basis with appro- +priate shape constraints to ensure monotonicity of the predicted quantile functions. A +novel construction of BP-based regression structure results in imposing much less restric- +tive compared to existing methods for modelling monotone quantile function outcome. +21 + +Thus, the proposed framework enables more flexible dependencies between distributional +outcome and scalar and distributional predictors. Inferential tools are developed that +include projection-based asymptotic joint confidence bands and a global test of statisti- +cal significance for estimated functional regression coefficients. Numerical analysis using +simulations illustrate an accurate performance of the estimation method. The proposed +test is also shown to maintain the nominal test size and have a satisfactory power. An +additional nonparametric bootstrap test provided in the supplementary material could +be particularly useful in finite sample sizes. +Application of DOR is demonstrated in studying the distributional association be- +tween heart rate reserve and key demographics while adjusting for physical activity. Our +findings provide important insights about age and gender differences in distribution of +heart rate. Beyond the considered epidemiological application, the proposed regression +model could be used in other epidemiological studies to more flexibly model distributional +aspect of high frequency and high intensity data. Additionally, it can be used for estima- +tion of treatment effects in primary or secondary endpoints quantified via distributions. +There are multiple research directions that remain to be explored based on this cur- +rent work. In developing our method we have implicitly assumed that there are enough +measurements available per subject to accurately estimate quantile functions. Scenarios +with only a few sparse measurements pose a practical challenge and will need careful +handling. Other aspects of studies collecting distributional data such as distributional +measurements being multilevel (Goldsmith et al., 2015) or incorporating spatio-temporal +structure (Yang, 2020; Ghosal et al., 2022b) would be important to consider. Another +interesting direction of research could be to extend these models beyond the additive +paradigm, for example the single index model (Jiang et al., 2011) could be employed +to accommodated interaction and nonlinear effects of multiple scalar and distributional +predictors. Extending the proposed method to such more general and complex models +would be computationally challenging, nonetheless merits future attention because of +their potentially diverse applications. +22 + +Supplementary Material +Appendix A-E along with the Supplementary Tables and Supplementary Figures refer- +enced in this article are available online as Supplementary Material. +Software +Software implementation via R (R Core Team, 2018) and illustration of the proposed +framework is available upon request from the authors. +References +Antelmi, I., De Paula, R. S., Shinzato, A. R., Peres, C. A., Mansur, A. J., and Grupi, +C. J. (2004), “Influence of age, gender, body mass index, and functional capacity on +heart rate variability in a cohort of subjects without heart disease,” The American +journal of cardiology, 93, 381–385. +Augustin, N. H., Mattocks, C., Faraway, J. J., Greven, S., and Ness, A. R. (2017), +“Modelling a response as a function of high-frequency count data: The association +between physical activity and fat mass,” Statistical methods in medical research, 26, +2210–2226. +Carnicer, J. M. and Pena, J. M. (1993), “Shape preserving representations and optimality +of the Bernstein basis,” Advances in Computational Mathematics, 1, 173–196. +Chen, Y., Lin, Z., and M¨uller, H.-G. (2021), “Wasserstein regression,” Journal of the +American Statistical Association, 1–40. +Cui, E., Leroux, A., Smirnova, E., and Crainiceanu, C. M. (2022), “Fast univariate +inference for longitudinal functional models,” Journal of Computational and Graphical +Statistics, 31, 219–230. +Fan, Y., James, G. M., and Radchenko, P. (2015), “Functional additive regression,” The +Annals of Statistics, 43, 2296–2325. +23 + +Foy, A. J., Mandrola, J., Liu, G., and Naccarelli, G. V. (2018), “Relation of obesity +to new-onset atrial fibrillation and atrial flutter in adults,” The American journal of +cardiology, 121, 1072–1075. +Gellish, R. L., Goslin, B. R., Olson, R. E., McDONALD, A., Russi, G. D., and Moudgil, +V. K. (2007), “Longitudinal modeling of the relationship between age and maximal +heart rate.” Medicine and science in sports and exercise, 39, 822–829. +Ghodrati, L. and Panaretos, V. M. (2021), “Distribution-on-Distribution Regression via +Optimal Transport Maps,” arXiv preprint arXiv:2104.09418. +Ghosal, R., Ghosh, S., Urbanek, J., Schrack, J. A., and Zipunnikov, V. (2022a), “Shape- +constrained estimation in functional regression with Bernstein polynomials,” Compu- +tational Statistics & Data Analysis, 107614. +Ghosal, R. and Maity, A. (2022), “A Score Based Test for Functional Linear Concurrent +Regression,” Econometrics and Statistics, 21, 114–130. +Ghosal, R., Varma, V. R., Volfson, D., Hillel, I., Urbanek, J., Hausdorff, J. M., Watts, +A., and Zipunnikov, V. (2021), “Distributional data analysis via quantile functions +and its application to modelling digital biomarkers of gait in Alzheimer’s Disease,” +Biostatistics. +Ghosal, R., Varma, V. R., Volfson, D., Urbanek, J., Hausdorff, J. M., Watts, A., and +Zipunnikov, V. (2022b), “Scalar on time-by-distribution regression and its application +for modelling associations between daily-living physical activity and cognitive functions +in Alzheimer’s Disease,” Scientific reports, 12, 1–16. +Goldfarb, D. and Idnani, A. (1982), “Dual and primal-dual methods for solving strictly +convex quadratic programs,” in Numerical analysis, Springer, pp. 226–239. +— (1983), “A numerically stable dual method for solving strictly convex quadratic pro- +grams,” Mathematical programming, 27, 1–33. +Goldsmith, J., Zipunnikov, V., and Schrack, J. (2015), “Generalized multilevel function- +on-scalar regression and principal component analysis,” Biometrics, 71, 344–353. +24 + +Hron, K., Menafoglio, A., Templ, M., Hruzova, K., and Filzmoser, P. (2016), “Simplicial +principal component analysis for density functions in Bayes spaces,” Computational +Statistics & Data Analysis, 94, 330–350. +Huang, J. Z., Wu, C. O., and Zhou, L. (2002), “Varying-coefficient models and basis +function approximations for the analysis of repeated measurements,” Biometrika, 89, +111–128. +— (2004), “Polynomial spline estimation and inference for varying coefficient models with +longitudinal data,” Statistica Sinica, 14, 763–788. +Irpino, A. and Verde, R. (2013), “A metric based approach for the least square regression +of multivariate modal symbolic data,” in Statistical Models for Data Analysis, Springer, +pp. 161–169. +Jiang, C.-R., Wang, J.-L., et al. (2011), “Functional single index models for longitudinal +data,” The Annals of Statistics, 39, 362–388. +Kostis, J. B., Moreyra, A., Amendo, M., Di Pietro, J., Cosgrove, N., and Kuo, P. (1982), +“The effect of age on heart rate in subjects free of heart disease. Studies by ambulatory +electrocardiography and maximal exercise stress test.” Circulation, 65, 141–145. +Leary, A. C., Struthers, A. D., Donnan, P. T., MacDonald, T. M., and Murphy, M. B. +(2002), “The morning surge in blood pressure and heart rate is dependent on levels of +physical activity after waking,” Journal of hypertension, 20, 865–870. +Lorentz, G. G. (2013), Bernstein polynomials, American Mathematical Soc. +Matabuena, M. and Petersen, A. (2021), “Distributional data analysis with accelerometer +data in a NHANES database with nonparametric survey regression models,” arXiv. +Matabuena, M., Petersen, A., Vidal, J. C., and Gude, F. (2021), “Glucodensities: a +new representation of glucose profiles using distributional data analysis,” Statistical +Methods in Medical Research, 30, 1445–1464. +Meyer, M. J., Coull, B. A., Versace, F., Cinciripini, P., and Morris, J. S. (2015), “Bayesian +function-on-function regression for multilevel functional data,” Biometrics, 71, 563– +574. +25 + +Parzen, E. (2004), “Quantile probability and statistical data modeling,” Statistical Sci- +ence, 19, 652–662. +Pegoraro, M. and Beraha, M. (2022), “Projected Statistical Methods for Distributional +Data on the Real Line with the Wasserstein Metric.” J. Mach. Learn. Res., 23, 37–1. +Petersen, A. and M¨uller, H.-G. (2016), “Functional data analysis for density functions by +transformation to a Hilbert space,” The Annals of Statistics, 44, 183–218. +Petersen, A., Zhang, C., and Kokoszka, P. (2021), “Modeling Probability Density Func- +tions as Data Objects,” Econometrics and Statistics. +Powley, B. W. (2013), “Quantile function methods for decision analysis,” Ph.D. thesis, +Stanford University. +Prabhavathi, K., Selvi, K. T., Poornima, K., and Sarvanan, A. (2014), “Role of biological +sex in normal cardiac function and in its disease outcome–a review,” Journal of clinical +and diagnostic research: JCDR, 8, BE01. +R Core Team (2018), R: A Language and Environment for Statistical Computing, R +Foundation for Statistical Computing, Vienna, Austria. +Ramsay, J. and Silverman, B. (2005), Functional Data Analysis, New York: Springer- +Verlag. +Ramsay, J. O. et al. (1988), “Monotone regression splines in action,” Statistical science, +3, 425–441. +Sergazinov, R., Leroux, A., Cui, E., Crainiceanu, C., Aurora, R. N., Punjabi, N. M., and +Gaynanova, I. (2022), “A case study of glucose levels during sleep using fast function +on scalar regression inference,” arXiv preprint arXiv:2205.08439. +Talsk´a, R., Hron, K., and Grygar, T. M. (2021), “Compositional Scalar-on-Function +Regression with Application to Sediment Particle Size Distributions,” Mathematical +Geosciences, 1–29. +Tanaka, H., Monahan, K. D., and Seals, D. R. (2001), “Age-predicted maximal heart +rate revisited,” Journal of the american college of cardiology, 37, 153–156. +26 + +Tang, B., Zhao, Y., Venkataraman, A., Tsapkini, K., Lindquist, M., Pekar, J. J., and +Caffo, B. S. (2020), “Differences in functional connectivity distribution after transcra- +nial direct-current stimulation: a connectivity density point of view,” bioRxiv. +Vanbrabant, L. and Rosseel, Y. (2019), Restricted Statistical Estimation and Inference +for LinearModels, 0.2-250. +Verde, R. and Irpino, A. (2010), “Ordinary least squares for histogram data based on +wasserstein distance,” in Proceedings of COMPSTAT’2010, Springer, pp. 581–588. +Wang, J. and Ghosh, S. K. (2012), “Shape restricted nonparametric regression with +Bernstein polynomials,” Computational Statistics & Data Analysis, 56, 2729–2741. +Yang, H. (2020), “Random distributional response model based on spline method,” Jour- +nal of Statistical Planning and Inference, 207, 27–44. +Yang, H., Baladandayuthapani, V., Rao, A. U., and Morris, J. S. (2020), “Quantile +function on scalar regression analysis for distributional data,” Journal of the American +Statistical Association, 115, 90–106. +Yao, F., M¨uller, H.-G., and Wang, J.-L. (2005), “Functional linear regression analysis for +longitudinal data,” The Annals of Statistics, 2873–2903. +27 + +Supplementary Material for Distributional +outcome regression and its application to +modelling continuously monitored heart +rate and physical activity +Rahul Ghosal1,∗, Sujit Ghosh2, Jennifer A. Schrack3, Vadim Zipunnikov4 +1 Department of Epidemiology and Biostatistics, University of South Carolina +2Department of Statistics, North Carolina State University +3 Department of Epidemiology, Johns Hopkins Bloomberg +School of Public Health +4 Department of Biostatistics, Johns Hopkins Bloomberg +School of Public Health +January 30, 2023 +1 +arXiv:2301.11399v1 [stat.ME] 26 Jan 2023 + +1 +Appendix A: Proof of Theorem 1 +The predicted outcome quantile function is the conditional expectation of the outcome +quantile function based on the distribution-on-scalar and distribution regression (DOSDR) +model (2) and is given by, +E(QY (p) | z1, z2, . . . , zq, qx(p)) = β0(p) + +q +� +j=1 +zjβj(p) + h(qx(p)). +(1) +We will show conditions (1)-(3) are sufficient conditions to ensure E(QY (p) | z1, z2, . . . , zq, qx(p)) +is non-decreasing. Let us assume 0 ≤ zj ≤ 1, ∀j = 1, 2, . . . , J, without loss of generality. +It is enough to show T1(p) = β0(p) + �q +j=1 zjβj(p) and T2(p) = h(qx(p)) both are non +decreasing. The second part is immediate as both qx(·) and h(·) (by condition (3)) are +non decreasing. To complete the proof we only need to show T1(p) is non decreasing. +T ′ +1(p) = β′ +0(p)+�q +j=1 zjβ′ +j(p). Enough to show T ′ +1(p) ≥ 0 for all (z1, z2, . . . , zq) ∈ [0, 1]q. +Note that this is a linear function in (z1, z2, . . . , zq) ∈ [0, 1]q. By the well-known Bauer’s +principle the minimum is attained at the boundary points B = {(z1, z2, . . . , zq) : zj ∈ +{0, 1}}. Hence, the sufficient conditions are β′ +0(p) ≥ 0 and β′ +0(p) + �r +k=1 β′ +jk(p) ≥ 0 for +any sub-sample {j1, j2, . . . , jr} ⊂ {1, 2, . . . , q}, which follows from condition (1) and (2). +2 +Appendix B: Example of DOSDR +Example 2: Two scalar covariates (q = 2) and a distributional predictor +We illustrate the estimation for DOSDR where there are two scalar covariates z1, z2 +(q = 1) and a single distribution predictor QX(p). The DOSDR model (2) is given by +QiY (p) = β0(p)+zi1β1(p)+zi2β2(p)+h(QiX(p))+ϵi(p). The sufficient conditions (1)-(3) of +Theorem 1 in this case reduce to : A) The distributional intercept β0(p) is non-decreasing. +B) β0(p) + β1(p), β0(p) + β2(p), β0(p) + β1(p) + β2(p) is non-decreasing. C) h(·) is non- +decreasing. Note that condition B) illustrates that as the number of scalar covariates +increase we have more and more combinatorial combinations of the coefficint functions +restricted to be non-decreasing. Similar to Example 1, Conditions (A)-(C) again become +linear restrictions on the basis coefficients of the form Dψ ≥ 0, where the constraint +2 + +matrix is given by D = +� +� +� +� +� +� +� +� +� +� +� +AN +0 +0 +0 +AN +AN +0 +0 +AN +0 +AN +0 +AN +AN +AN +0 +0 +0 +0 +AN−1 +� +� +� +� +� +� +� +� +� +� +� +. +As the number of restrictions increase the parameter space becomes smaller and smaller, +which can result in a faster convergence of the optimization algorithm. +3 +Appendix C: Estimation of Asymptotic Covariance +Matrix +The DOSDR model (8) in the paper was reformulated as +QiY += Tiψ + ϵi, , +where Ti = [B0 Wi1 Wi2, . . . , Wiq Si]. Under suitable regularity conditions (Huang et al., +2004), √n( ˆψur − ψ0) can be shown to be asymptotically distributed as N(0, ∆) (also +holds true for finite sample sizes if ϵ(p) is Gaussian). In reality, ∆ is unknown and we +want to estimate ∆ by an estimator ˆ∆. +We derive a sandwich covariance estimator +ˆ∆ corresponding to the above model. Based on the ordinary least square optimization +criterion for model (11) (of the paper), the unrestricted estimator is given by ˆψur = +(TTT)−1TTQY , where QT +Y = (Q1Y , Q2Y , . . . , QnY )T and T = [TT +1 , TT +2 , . . . , TT +n]T. Hence, +V ar( ˆψur) = (TTT)−1TTΣT(TTT)−1. Here Σ = V ar(ϵ), which is typically unknown. We +apply an FPCA based estimation approach (Ghosal and Maity, 2022) to estimate Σ. +Let us assume (Huang et al., 2004) the error process ϵ(p) can be decomposed as +ϵ(p) = V (p) + wp, where V (p) is a smooth mean zero stochastic process with covariance +kernel G(p1, p2) and wp is a white noise with variance σ2. The covariance function of the +error process is then given by Σ(p1, p2) = cov{ϵ(p1), ϵ(p2)} = G(p1, p2)+σ2I(p1 = p2). For +data observed on dense and regular grid P, the covariance matrix of the residual vector +ϵi is Σm×m, the covariance kernel Σ(p1, p2) evaluated on the grid P = {p1, p2, . . . , pm}. +We can estimate Σ(·, ·) nonparametrically using functional principal component analysis +(FPCA) if the original residuals ϵij were available. Given ϵi(pj)s, FPCA (Yao et al., 2005) +3 + +can be used to get ˆφk(·), ˆλks and ˆσ2 to form an estimator of Σ(p1, p2) as +ˆΣ(p1, p2) = +K +� +k=1 +ˆλk ˆφk(p1)ˆφk(p2) + ˆσ2I(p1 = p2), +where K is large enough such that percent of variance explained (PVE) by the selected +eigencomponents exceeds some pre-specified value such as 99%. +In practice, we don’t have the original residuals ϵij. Hence we fit the unconstrained +DOSDR model (11) and and obtain the residuals eij = QiY (pj) − ˆ +QiY (pj). Then treating +eij as our original residuals, we can obtain ˆΣ(p1, p2) and ˆΣm×m using the FPCA approach +outlined above. Then +ˆ +V ar(ϵ) = ˆΣ = diag{ˆΣm×m, ˆΣm×m, . . . , ˆΣm×m}. Ghosal and Maity +(2022) discusses consistency of ˆΣ under standard regularity conditions. Hence an consis- +tent estimator of the covariance matrix is given by +ˆ +V ar( ˆψur) = (TTT)−1TT ˆΣT(TTT)−1. +In particular, ˆ∆n = ˆ∆/n = ˆ +cov( ˆψur) = (TTT)−1TT ˆΣT(TTT)−1. +4 + +4 +Appendix D: Algorithm 1 for Joint Confidence Band +Algorithm 1 Joint confidence band of β0 +1(p) +1. Fit the unconstrained model and obtain the unconstrained estimator +ˆψur = +argmin +ψ∈RKn +�n +i=1 ||QiY − Tiψ||2 +2. +2. Fit +the +constrained +model +and +obtain +the +constrained +estimator +ˆψr += +argmin +ψ∈ΘR +�n +i=1 ||QiY −Tiψ||2 +2. Obtain the constrained estimator of β0 +1(p) as ˆβ1r(p) = +ρKn(p) +′ ˆβ1r. +3. Let ˆ∆n be an estimate of the asymptotic covariance matrix of the unconstrained +estimator given by ˆ∆n = ˆ∆/n = ˆ +cov( ˆψur) +4. For b = 1 to B +- generate Zb ∼ NKn( ˆψur, ˆ∆n). +- compute the projection of Zb as ˆψr,b = argmin +ψ∈ΘR +||ψ − Zb||2 +ˆΩ. +- End For +5. For each generated sample ˆψr,b calculate estimate of β0 +1(p) as ˆβ1r,b(p) = ρKn(p) +′ ˆβ1r,b +(b = 1, . . . , B). Compute V ar(ˆβ1r(p)) based on these samples. +6. For b = 1 to B +- calculate ub = max +p∈P +|ˆβ1r,b(p)−ˆβ1r(p)| +√ +V ar(ˆβ1r(p)) . +- End For +7. Calculate q1−α the (1 − α) empirical quantile of {ub}B +b=1. +8. 100(1−α)% joint confidence band for β0 +1(p) is given by ˆβ1r(p)±q1−α +� +V ar(ˆβ1r(p)). +5 +Appendix E: Bootstrap Test for Global Distribu- +tional Effects +A practical question of interest in the DOSDR model is to directly test for the global +distributional effect of the scalar covariates Zj or test for the distributional effect of the +distributional predictor QX(p). In this section, we illustrate an nonparametric bootstrap +test based on our proposed estimation method which also easily lends itself to the required +5 + +shape constraints of the regression problem. In particular, we obtain the residual sum of +squares of the null and the full model and come up with the F-type test statistic defined +as +TD = RSSN − RSSF +RSSF +. +(2) +Here RSSN, RSSF are the residual sum of squares under the null and the full model +respectively. For example, let us consider the case of testing +H0 : βr(p) = 0 for all p ∈ [0, 1] +versus +H1 : βr(p) ̸= 0 for some p ∈ [0, 1]. +Let r = q without loss of generality. The residual sum of of squares for the full model +is given by RSSF = �n +i=1 ||QiY − B0 ˆβ0 − �q +j=1 Wij ˆβj − Si ˆθ||2 +2, where the estimates are +obtained from the optimization criterion (9) in the paper, with the constraint DFψ ≥ 0 +(denoting the constraint matrix for the full model as DF). Similarly, we have RSSN = +�n +i=1 ||QiY − B0 ˆβ0 − �q−1 +j=1 Wij ˆβj − Si ˆθ||2 +2, where the estimates are again obtained from +(9) with the constraint DNψ ≥ 0. Note that, in this case the constraint matrix is denoted +by DN and this is essentially a submatrix of DF as the conditions for monotinicity in (1)- +(3) (Theorem 1) for the reduced model is a subset of the original constrains for the full +model. The null distribution of the test statistic TD is nonstandard, hence we use residual +bootstrap to approximate the null distribution. The complete bootstrap procedure for +testing the distributional effect of a scalar predictor is presented in algorithm (2) below. +Similar strategy could be employed for testing the distributional effect of a distributional +predictor or multiple scalar predictors. +6 + +Algorithm 2 Bootstrap algorithm for testing the distributional effect of a scalar predictor +1. Fit the full DOSDR model in the paper using the optimization criterion +ˆψF = argmin +ψ +n +� +i=1 +||QiY − B0β0 − +q +� +j=1 +Wijβj − Siθ||2 +2 +s.t +DFψ ≥ 0. +and calculate the residuals ei(pl) = QiY (pl) − ˆQiY (pl), for i = 1, 2, . . . , n and l = +1, 2, . . . , m. +2. Fit the reduced model corresponding to H0 (the null) and estimate the parameters +using the minimization criteria, +ˆψN = argmin +ψ +n +� +i=1 +||QiY − B0β0 − +q−1 +� +j=1 +Wijβj − Siθ||2 +2 +s.t +DNψ ≥ 0. +Denote the estimates of the distributional effects as ˆβN +j (p) for j = 0, 1, . . . , q − 1 +and ˆhN(x). +3. Compute test statistic TD (2) based on these null and full model fits, denote this +as Tobs. +4. Resample B sets of bootstrap residuals {e∗ +b,i(p)}n +i=1 from residuals {ei(p)}n +i=1 ob- +tained in step 1. +5. for b = 1 to B +6. Generate distributional response under the reduced DOSDR model as +Q∗ +b,iY (p) = ˆβN +0 (p) + +q−1 +� +j=1 +zij ˆβN +j (p) + ˆhN(QiX(p)) + e∗ +b,i(p). +7. Given the bootstrap data set {QiX(p), Q∗ +b,iY (p), z1, z2, . . . , zq}n +i=1 fit the null and the +full model to compute the test statistic T ∗ +b . +8. end for +9. Calculate the p-value of the test as ˆp = +�B +b=1 I(T ∗ +b ≥Tobs) +B +. +7 + +6 +Supplementary Tables +Table S1: Average Wasserstein distance (standard error) between true and predicted +quantile functions in the test set over 100 Monte-Carlo replications, Scenario A1. +Sample Size +L=200 +L=400 +n= 200 +0.2587 (0.0154) +0.1882 (0.0138) +n= 300 +0.2568 (0.0132) +0.1858 (0.0105) +n= 400 +0.2554 (0.0141) +0.1865 (0.0120) +Table S2: Coverage of the projection-based 95% joint confidence interval for β1(p), for +various choices of the order of the Bernstein polynomial (BP) basis, scenario A1, based +on 100 M.C replications with L = 200. Average width of the joint confidence interval is +given in the parenthesis. The average choices of N from cross-validation for this scenario +are highlighted in bold. +BP order (N) +Sample size (n=200) +Sample size (n=300) +Sample size (n=400) +2 +0.92 (0.29) +0.9 (0.24) +0.9 (0.20) +3 +0.92 (0.31) +0.94 (0.25) +0.96 (0.22) +4 +0.93 (0.33) +0.93 (0.26) +0.93 (0.23) +Table S3: Descriptive statistics of age and BMI for the complete, male and female samples +in the BLSA analysis. +Characteristic +Complete (n=890) +Male (n=432) +Female (n=458) +P value +Mean +SD +Mean +SD +Mean +SD +Age +66.66 +13.35 +68.03 +13.41 +65.37 +13.17 +0.003 +BMI (kg/m2) +27.40 +4.96 +27.52 +4.23 +27.28 +5.57 +0.45 +Table S4: Results from multiple linear regression model of mean heart rate on age, sex +(Male), BMI and mean activity count. Reported are the estimated fixed effects along +with their standard error and P-values. +Dependent variable : Mean heart rate +Value +Std.Error +P-value +Intercept +82.47 +3.458 +< 2 × 10−16∗∗∗ +age +−0.18 +0.026 +< 1.2 × 10−11∗∗∗ +sex +−4.19 +0.659 +< 3.2 × 10−10∗∗∗ +BMI +0.18 +0.067 +0.0091∗∗ +Mean activity +2.44 +0.697 +0.0005∗∗∗ +Observations +890 +Adjusted R2 +0.142 +Note: +∗p<0.05; ∗∗p<0.01; ∗∗∗p<0.001 +8 + +7 +Supplementary Figures +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.2 +0.4 +0.6 +0.8 +1.0 +d +Power +n=200 +n=300 +n=400 +Figure S1: +Displayed are the estimated power curves for simulation scenario A2. +The parameter d controls the departure from the null and the power curves for n ∈ +{200, 300, 400} are shown by solid, dashed and dotted lines. The dashed horizontal line +at the bottom corresponds to the nominal level of α = 0.05. +9 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +2 +3 +4 +5 +6 +7 +DODSR +p +γ(p) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +2 +3 +4 +5 +6 +7 +PAVA +p +γ(p) +Figure S2: Displayed are estimates of additive effect γ(p) = β0(p) + h(qx(p)) (solid) +at at qx(p) = +1 +n +�n +i=1 QiX(p) and its estimate ˆγ(p) averaged over 100 M.C replications +(dashed) along with point-wise 95% confidence interval (dotted) for scenario B, n = 400. +Left: Estimates from the proposed DOSDR method. Right: Isotonic regression method +with PAVA. +10 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +50 +100 +150 +200 +250 +Heartrate +p +Heartrate QF (raw) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +2 +4 +6 +8 +Activity +p +Activity QF (log) +Figure S3: Subject-specific quantile functions of heart rate and log-transformed activity +counts during 8 a.m.- 8 p.m. period. Color profiles show four randomly chosen partici- +pants. +11 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +50 +100 +150 +200 +250 +HR +p +Q(p) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +50 +100 +150 +200 +250 +DODSR Predicted HR +p +Q(p) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +50 +100 +150 +200 +250 +HR +p +Q(p) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +50 +100 +150 +200 +250 +PAVA Predicted HR +p +Q(p) +Figure S4: Top: LOOCV predictions of quantile functions of heart rate from DOSDR +method based on age, sex, BMI and PA distribution. Bottom: LOOCV predictions of +quantile functions of heart rate from PAVA method (Ghodrati and Panaretos, 2021) based +on PA distribution. +References +Ghodrati, L. and V. M. Panaretos (2021). Distribution-on-distribution regression via +optimal transport maps. arXiv preprint arXiv:2104.09418. +Ghosal, R. and A. Maity (2022). A score based test for functional linear concurrent +regression. Econometrics and Statistics 21, 114–130. +12 + +Huang, J. Z., C. O. Wu, and L. Zhou (2004). Polynomial spline estimation and inference +for varying coefficient models with longitudinal data. Statistica Sinica 14, 763–788. +Yao, F., H.-G. M¨uller, and J.-L. Wang (2005). Functional linear regression analysis for +longitudinal data. The Annals of Statistics, 2873–2903. +13 + diff --git a/5dFIT4oBgHgl3EQf7yth/content/tmp_files/load_file.txt b/5dFIT4oBgHgl3EQf7yth/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c1436ca52eada63d20e0db2c4e728261e555366 --- /dev/null +++ b/5dFIT4oBgHgl3EQf7yth/content/tmp_files/load_file.txt @@ -0,0 +1,1205 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf,len=1204 +page_content='Distributional outcome regression and its application to modelling continuously monitored heart rate and physical activity Rahul Ghosal1, Sujit K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosh2, Jennifer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Schrack3, Vadim Zipunnikov4 1 Department of Epidemiology and Biostatistics, University of South Carolina 2Department of Statistics, North Carolina State University 3 Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health 4 Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health January 30, 2023 Abstract We propose a distributional outcome regression (DOR) with scalar and distribu- tional predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Distributional observations are represented via quantile functions and the dependence on predictors is modelled via functional regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' DOR expands existing literature with three key contributions: handling both scalar and distributional predictors, ensuring jointly monotone regression structure with- out enforcing monotonicity on individual functional regression coefficients, pro- viding a statistical inference for estimated functional coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Bernstein poly- nomial bases are employed to construct a jointly monotone regression structure without over-restricting individual functional regression coefficients to be mono- tone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Asymptotic projection-based joint confidence bands and a statistical test of global significance are developed to quantify uncertainty for estimated functional regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Simulation studies illustrate a good performance of DOR model in accurately estimating the distributional effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The method is applied to continuously monitored heart rate and physical activity data of 890 participants of Baltimore Longitudinal Study of Aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Daily heart rate reserve, quantified via a subject-specific distribution of minute-level heart rate, is modelled additively as a function of age, gender, and BMI with an adjustment for the daily distribution of minute-level physical activity counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Findings provide novel scientific insights in epidemiology of heart rate reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Keywords: Distributional Data Analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Distribution-on-distribution regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Quan- tile function-on-scalar Regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' BLSA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Physical Activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Heart Rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='11399v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='ME] 26 Jan 2023 1 Introduction Distributional data analysis is an emerging area of research with diverse applications in digital medicine and health (Augustin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Matabuena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Matabuena and Petersen, 2021), radiomics (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2020), neuroimaging (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2020) among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' With the advent of modern medical devices and wear- ables, many studies collect subject-specific high frequency or high density observations including heart rate, physical activity (steps, activity counts), continuously monitored blood glucose, functional and structured brain images, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The central idea of distributional data analysis is to capture the distributional aspect in this data and model it within regression frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Thus, distributional data analysis inherently deals with data objects which are distributions typically represented via histograms, densities, quan- tile functions or other distributional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021) provide an in-depth overview of recent developments in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Similar to functional regression models, depending on whether the outcome or the predictor is distributional, there are various types of distributional regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Petersen and M¨uller (2016) and Hron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2016) developed functional compositional methods to analyze samples of densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For scalar outcome and distributional predictors represented via densities, a common idea has been to transform densities by mapping them to a proper Hilbert space L2 and then use existing functional regression approaches for modelling scalar outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Petersen and M¨uller (2016) used a log-quantile density transformation, whereas Talsk´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021) used a centered log-ratio transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Other approaches for modelling scalar outcomes and distributional predictors include scalar-on-quantile function regression (Ghosal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2021), kernel-based approaches using quantile functions (Matabuena and Petersen, 2021) and many others (see Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021) and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In parallel, there was also a substantial work on developing models with distributional outcome and scalar predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2020) developed a quantile function-on- scalar (QFOSR) regression model, where subject-specific quantile functions of data were modelled via scalar predictors of interest using a function-on-scalar regression approach (Ramsay and Silverman, 2005), which make use of data-driven basis functions called quantlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' One limitation of the approach is a no guarantee of underlying monotonicity of the predicted quantile functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' To address this, Yang (2020) extended this approach 2 using I-splines (Ramsay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 1988) or Beta CDFs which enforce monotonicity at the estimation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' One important limitation of this approach is enforcement of jointly monotone (non-decreasing) regression structure via enforcement of monotonicity on each individual functional regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' As we demonstrate in our application, this assumption could be too restrictive in real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Distribution-on-distribution regression models when both outcome and predictors are distributions have been studied by Verde and Irpino (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Irpino and Verde (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghodrati and Panaretos (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Pegoraro and Beraha (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' These models aim to understand the association between distributions within a pre-specified, often linear, regression structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Verde and Irpino (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Irpino and Verde (2013) used an ordinary least square approach based on the squared L2 Wasserstein distance between distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Outcome quantile function QiY (p) was modelled as a non-negative linear combination of other quantile functions QiXj(p)s using a multiple linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' This model although useful and adequate for some applications, may not be flexible enough as it assumes a linear association between the distribution valued response and predictors, which are additionally assumed to be constant across all quantile levels p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021) used a geometric approach taking distributional valued outcome and predictor to a tangent space, where regular tools of function-on-function regression (Ramsay and Silverman, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2005) were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Pegoraro and Beraha (2022) used an approximation of the Wasserstein space using monotone B-spline and developed methods for PCA and regression for distributional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Recently, Ghodrati and Panaretos (2021) developed a shape-constrained approach linking Frechet mean of the outcome distribution to the predictor distribution via an optimal transport map that was estimated by means of isotonic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Many of above-mentioned methods mainly focused on dealing with constraints en- forced by a specific functional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Developing inferential tools is somewhat under-developed are of distributional data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021) derived the asymp- totic convergence rates for the estimated regression operator in their proposed method for Wasserstein regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2020) developed joint credible bands for distribu- tional effects, but monotonocity of the quantile function was not imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yang (2020) developed a global statistical test for estimated functional coefficients in the distribu- tional outcome regression, however, no confidence bands was proposed to identify and 3 test local quantile effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In this paper, we propose a distributional outcome regression that expands exist- ing literature in three main directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' First, our model includes both scalar and dis- tributional predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Second, it ensures jointly monotone (non-decreasing) additive regression structure without enforcing monotonicity of individual functional regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Thirdly, it provides a toolbox of statistical inference tools for estimated functional coefficients including asymptotic projection-based joint confidence bands and a statistical test of global significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We capture distributional aspect in outcome and predictors via quantile functions and construct a jointly monotone regression model via a specific shape-restricted functional regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The distributional effects of the scalar covariates are captured via functional coefficient βj(p)’s varying over quantile lev- els and the effect of the distributional predictor is captured via a monotone function h(·), similar to an optimal transport approach in Ghodrati and Panaretos (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In the special case, when there is no distributional predictor, the model resembles a quantile function-on-scalar regression model, but with much more flexible constraints compared to Yang (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In the absence of scalar predictors the model reduces to a distribution on distribution regression model, where the monotone function representing the optimal transport map is estimated by a non-parametric functional regression model under shape constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We use Bernstein polynomial (BP) basis functions to model the distribu- tional effects βj(p)s and the monotone map h(·), which are known to enjoy attractive and optimal shape-preserving properties (Lorentz, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Carnicer and Pena, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ad- ditionally, BP is instrumental in constructing and enforcing a jointly monotone regression structure without over-restricting individual functional regression coefficients to be mono- tone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Finally, inferential tools are developed including joint asymptotic confidence bands for distributional functional effects and p-values for testing the distributional effects of predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' As a motivating application, we study continuously monitored heart rate and physical activity collected in Baltimore Longitudinal Study of Aging (BLSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We aim to study the association between the distribution of heart rate as a distributional outcome and age, sex and body mass index (BMI) while also adjusting for a key confounder, the distribution of minute-level physical activity aggregated over 8am-8pm time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Figure 1 displays daily profiles of heart rate and physical activity between 8am-8pm for a BLSA participant 4 along with the corresponding subject-specific quantile functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 8 10 12 14 16 18 20 40 60 80 100 120 Time of Day Heartrate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 40 60 80 100 120 p Heartrate QF 8 10 12 14 16 18 20 0 200 600 1000 Time of Day Activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0 200 600 1000 p Activity QF Figure 1: Diurnal profile of heart rate and physical activity between 8 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='- 8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and the corresponding subject specific quantile functions for a randomly chosen subject in the BLSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We present our distributional modeling framework and illustrate the proposed estimation method in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In Section 3, we perform numerical simulations to evaluate the performance of the proposed method and provide comparisons with existing methods for distributional regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In Section 4, we demonstrate application of the proposed method in modelling continuously monitored heart rate reserve in BLSA study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Section 5 concludes with a brief discussion of our proposed method and some possible extensions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='1 Modelling Framework and Distributional Representations We consider the scenario, where there are repeated subject-specific measurements of a distributional response Y along with several scalar covariates zj, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , q and we 5 also have a distributional predictor X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Let us denote the subject-specific response and covariates as Yik, Xil, zij (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , n1i, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , n2i), for subject i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Here n1i, n2i denotes the number of repeated observations of the distributional response and predictor respectively for subject i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Assume Yik (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , n1i) ∼ FiY (y), a subject- specific cumulative distribution function (cdf), where FiY (y) = P(Yik ≤ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Then, the subject-specific quantile function is defined as QiY (p) = inf{y : FiY (y) ≥ p}, p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The quantile function completely characterizes the distribution of the individual obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Given Yik s, the empirical quantile function can be calculated based on linear interpolation of order statistics (Parzen, 2004) and serves as an estimate of the latent subject specific quantile function QiY (p) (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yang, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In particular, for a sample (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , Xn), letX(1) ≤ X(2) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , ≤ X(n) be the corresponding order statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The empirical quantile function, for p ∈ [ 1 n+1, n n+1], is then given by, ˆQ(p) = (1 − w)X([(n+1)p]) + wX([(n+1)p]+1), (1) where and w is a weight satisfying (n + 1)p = [(n + 1)p] + w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Based on this formulation and observations Yik, Xil, we can obtain the subject specific quantile functions ˆQiY (p) and ˆQiX(p) which are estimators of the true quantile functions QiY (p),QiX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The em- pirical quantile functions are consistent (Parzen, 2004) and are suitable for distributional representaion for several attractive mathematical properties (Powley, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2021) without requiring any smoothing parameter selection as in density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 Distribution-on-scalar and Distribution Regression We assume that the scalar covariates (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq) ∈ [0, 1]q without any loss of gener- ality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', achievable by linear transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We posit the following distributional regression model, associating the distributional response QiY (p) to the scalar covariates zij, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , q, and a distributional predictor QiX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We will refer to this as a distribution-on-scalar and distribution regression (DOSDR) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' QiY (p) = β0(p) + q � j=1 zijβj(p) + h(QiX(p)) + ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2) 6 Here β0(p) is a distributional intercept and βj(p) s are the distributional effects of the scalar covariates Zj at quantile level p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The unknown nonparametric function h(·) captures the additive effect of the distributional predictor QiX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The residual er- ror process ϵi(p) is assumed to be a mean zero stochastic process with an unknown covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We make the following flexible and interpretable assumptions on the coefficient functions βj(·), j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , q and on h(·) which ensures the pre- dicted value of the response quantile function QiY (p) conditionally on the predictors, E(QiY (p) | zi1, zi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , ziq, qix(p)) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Theorem 1 Let the following conditions hold in the model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The distributional intercept β0(p) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Any additive combination of β0(p) with distributional slopes βj(p) is non-decreasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', β0(p) + �r k=1 βjk(p) is non-decreasing for any sub-sample {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , jr} ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' h(·) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Then E(QY (p) | z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq, qx(p)) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that E(QY (p) | z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq, qx(p)) is the predicted quantile function under the squared Wasserstein loss function, which is same as the squared error loss for the quan- tile functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The proof is illustrated in Appendix A of the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Assumptions (1) and (2) are much weaker and flexible than the monotonicity conditions of the QFOSR model in Yang (2020), where each of the function coefficients βj(p)s is re- quired to be monotone, whereas, we only impose monotonicity on the sum of functional coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' This is not just a technical aspect but this flexibility is important from a prac- tical perspective, as it allows for capturing possible non-monotone association between the distributional response and individual scalar predictors zj’s while still maintaining the required monotonicity of the predicted response quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Condition (3) matches with the monotonicity assumption of the distributional regression model in Ghodrati and Panaretos (2021) and in the absence of any scalar predictors, essentially captures the op- timal transport map between the two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that this optimal transport map is constructed after adjusting for scalars of interest - thus, it provides a model general in- ferential framework compared to that in Ghodrati and Panaretos (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Thus, the above 7 DOSDR model extends the previous inferential framework for distributional response on scalar and contains both the QFOSR model and the distribution-on-distribution regres- sion model as its submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' More succinctly, in absence of distributional predictor we have, QiY (p) = β0(p) + q � j=1 zijβj(p) + ϵi(p), (3) which is a quantile-function-on-scalar regression (QFOSR) model ensuring monotonon- icity under conditions (1),(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Similarly, in absence of any scalar covariates, we have a distribution-on-distribution regression model QiY (p) = β0(p) + h(QiX(p)) + ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (4) Model (4) is a bit more general than the one considered in Ghodrati and Panaretos (2021), including a transnational effect β0(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' As a technical note, in models (2) and (4), function h(·) is identifiable only up to an additive constant, and in particular, the estimable quantity is the additive effect β0(p) + h(qx(p)) for a fixed QX(p) = qx(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='3 Estimation in DOSDR We follow a shape constrained estimation approach (Ghosal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2022a) for estimating the distributional effects βj(p) and the nonparamatric function h() which naturally in- corporates the constraints (1)-(3) of Theorem 1 in the estimation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The univariate coefficient functions βj(p) (j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , p) are modelled in terms of univariate expansions of Bernstein basis polynomials as βj(p) = N � k=0 βjkbk(p, N), where bk(p, N) = �N k � pk(1 − p)N−k, for 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (5) The number of basis polynomials depends on the degree of the polynomial basis N (which is assumed to be same for all βj(·) for computational tractability in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The Bernstein polynomials bk(p, N) ≥ 0 and �N k=0 bk(p, N) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Wang and Ghosh (2012) and Ghosal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022a) illustrate that various shape constraints e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', monotonicity, convexity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' can be reduced to linear constraints on the basis coefficients of the form ANβN j ≥ 0, where βN j = (βj0, βj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , βjN)T and AN is the constraint matrix chosen in 8 a way to guarantee a desired shape restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In particular, in our context of DOSDR, we need to choose constraint matrices AN in such a way which jointly ensure conditions (1),(2) in Theorem 1 and thus guarantee a non-decreasing predicted value of the response quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The nonparametric function h(·) is modelled similarly using univariate Bernstein polynomial expansion as h(x) = N � k=0 θkbk(x, N), where bk(x, N) = �N k � xk(1 − x)N−k, for 0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (6) Since the domain of h(·) modelled via Bernstein basis is [0, 1], the quantile functions of the distributional predictor QX(p) are transformed to a [0, 1] scale using linear transformation of the observed predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We make the assumption here that the distributional predic- tors are bounded, which is reasonable in the applications we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Henceforth, we assume QX(p) ∈ [0, 1] without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Further, note that, b0(x, N) = 1, and since β0(p) already contains this constant term in the DOSDR model (2), including the constant basis while modelling h(·) will lead to model singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Hence we drop the constant basis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' the first term) while modelling h(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In particular, h(QiX(p)) is modelled as h(QiX(p)) = �N k=1 θkbk(QiX(p), N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that this is equivalent to imposing the constraint h(0) = θ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The non-decreasing condition in (3) of Theorem 1 can again be specified as a linear constraint on the basis coefficients of the form Rθ ≥ 0, where θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , θN)T, and R is the constraint matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The DOSDR model (2) can be reformulated in terms of basis expansions as QiY (p) = N � k=0 β0kbk(p, N) + q � j=1 zij N � k=0 βjkbk(p, N) + N � k=1 θkbk(QiX(p), N) + ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (7) = bN(p)Tβ0 + q � j=1 ZT ij(p)βj + bN(QiX(p))Tθ + ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Here βj = (βj0, βj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , βjN)T, bN(p)T = (b0(p, N), b1(p, N), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , bN(p, N)), bN(QiX(p))T = (b1(QiX(p), N), b2(QiX(p), N), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , bN(QiX(p), N)) and ZT ij(p) = zij ∗ bN(p)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Suppose that we have the qunatile functions QiY (p), QiX(p) evaluated on a grid P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , pm} ⊂ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' De- note the stacked value of the quantiles for ith subject as QiY = (QiY (p1), QiY (p2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , QiY (pm))T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 9 The DOSDR model in terms of Bernstein basis expansion (7) can be reformulated as QiY = B0β0 + q � j=1 Wijβj + Siθ + ϵi, (8) where B0 = (bN(p1), bN(p2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , bN(pm))T,Wij = (Zij(p1), Zij(p2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , Zij(pm))T and Si = (bN(QiX(p1)), bN(QiX(p2)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , bN(QiX(pm)))T and ϵi are the stacked residuals ϵi(p)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The parameters in the above model are the basis coefficients ψ = (βT 0 , βT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , βT q , θT)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For estimation of the parameters, we use the following least square criterion, which reduces to a shape constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Namely, we obtain the estimates ˆψ by minimizing residual sum of squares as ˆψ = argmin ψ n � i=1 ||QiY − B0β0 − q � j=1 Wijβj − Siθ||2 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='t Dψ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (9) The universal constraint matrix D on the basis coefficients is chosen to ensure the con- ditions (1),(2),(3) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Later in this section, we illustrate examples how the constraint matrix is formed in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The above optimization problem (9) can be iden- tified as a quadratic programming problem (Goldfarb and Idnani, 1982, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R package restriktor (Vanbrabant and Rosseel, 2019) can be used for performing the above opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Example 1: Single scalar covariate (q = 1) and a distributional predictor We consider the case where there is a single scalar covariate z1 (q = 1) and a dis- tribution predictor QX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In this case, the DOSDR model (2) is given by QiY (p) = β0(p) + zi1β1(p) + h(QiX(p)) + ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The sufficient conditions (1)-(3) for non-decreasing quantile functions in this case reduces to: A) The distributional intercept β0(p) is non- decreasing B) β0(p) + β1(p) is non-decreasing C) h(·) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that the above conditions do no enforce β1(p) to be non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Once the coefficient func- tions are modelled in terms of Bernstein basis expansions as in (4) and (5), conditions (A)-(C) can be be enforced via the following linear restrictions on the basis coefficients i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', ANβ0 ≥ 0, [AN AN](βT 0 , βT 1 )T ≥ 0, AN−1θ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Here AN is a constraint matrix which imposes monotonicity on functions fN(x) modelled with Bernstein polynomials as fN(x) = �N k=0 βkbk(x, N), where bk(x, N) = �N k � xk(1 − x)N−k, for 0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The 10 derivative is given by f ′ N(x) = N �N−1 k=0 (βk+1 − βk)bk(x, N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Hence if βk+1 ≥ βk for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , N −1, fN(x) is non decreasing, which is achieved with the constraint matrix AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The combined linear restrictions on the parameter ψ = (βT 0 , βT 1 , θT)T is given by Dψ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The matrices AN, D are given by AN ≡ � � � � � � � � −1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 0 0 −1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 0 −1 1 � � � � � � � � , D = � � � � � AN 0 0 AN AN 0 0 0 AN−1 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (10) Similar example with two scalar covariates (q = 2) and a distributional predictor is given in Appendix B of the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Our estimation ensures that the shape restrictions are enforced everywhere and hence the predicted quantile functions are nondecreasing in the whole domain p ∈ [0, 1] as opposed to fixed quantile levels or design points in Ghodrati and Panaretos (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The order of the Bernstein polynomial basis N controls the smoothness of the coefficient functions βj(·) and h(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We follow a truncated basis approach (Ramsay and Silverman, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2015), by restricting the number of BP basis to ensure the resulting coefficient functions are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The optimal order of the basis functions is chosen via V -fold cross-validation method (Wang and Ghosh, 2012) using cross-validated residual sum of squares defined as, CV SSE = �V v=1 �nv i=1 ||QiY,v − ˆQ−v iY,v||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Here ˆQ−v iY is the fitted quantile values of observation i within the v th fold obtained from the constrained optimization criterion (9) and trained on the rest (V − 1) folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 Uncertainty Quantification and Joint Confidence Bands To construct confidence intervals, we use the result that the constrained estimator ˆψ in (9) is the projection of the corresponding unconstrained estimator (Ghosal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2022a) onto the restricted space: ˆψr = argmin ψ∈ΘR ||ψ − ˆψur||2 ˆΩ, for a non-singular matrix ˆΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The restricted parameter space is given by ΘR = {ψ ∈ RKn : Dψ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The DOSDR model (8) can be reformulated as QiY = Tiψ + ϵi, where Ti = [B0 Wi1 Wi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , Wiq Si] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The 11 unrestricted and restricted estimators are given by, ˆψur = argmin ψ∈RKn n � i=1 ||QiY − Tiψ||2 2 (11) ˆψr = argmin ψ∈ΘR n � i=1 ||QiY − Tiψ||2 2 Let us denote QT Y = (Q1Y , Q2Y , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , QnY )T and T = [TT 1 , TT 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , TT n]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Then we can write, 1 n||QY − Tψ||2 2 = 1 n||QY − T ˆψur||2 2 + 1 n||T ˆψur − Tψ||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Hence ˆψr = argmin ψ∈ΘR ||ψ− ˆψur||2 ˆΩ, where ˆΩ = 1 n �n i=1 TT i Ti and Ω = E( ˆΩ) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Thus, we can use the projection of the large sample distribution of √n( ˆψur − ψ0) to approximate the distribution of √n( ˆψr − ψ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Now √n( ˆψur − ψ0) is asymptotically distributed as N(0, ∆) under suitable regularity conditions (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2004, 2002) for general choice of basis fucntions (holds true for finite sample sizes if ϵ(p) is Gaussian), where ∆ can be estimated by a consistent estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In particular, we use a sandwich covariance estimator corresponding to model QiY = Tiψ + ϵi, for estimating ∆ following a functional principal component analysis (FPCA) approach (Ghosal and Maity, 2022) for estimation of the covariance matrix of the residuals ϵi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Details of this estimation procedure is included in Appendix C of the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Let us consider the scenario with a single scalar covariate and distributional pre- dictor for simplicity of illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The Bernstein polynomial approximation of β1(p) be given by β1N(p) = �N k=0 βkb1k(p, N) = ρKn(p) ′β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Algorithm 1 in Appendix D is used to obtain an asymptotic 100(1 − α)% joint confidence band for the true coefficient function β0 1(p), corresponding to a scalar predictor of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Here β0 1(p) denotes the true distributional coefficient β1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The algorithm relies on two steps i) Use the asymp- totic distribution of √n( ˆψr − ψ0) to generate samples from the asymptotic distribution of ˆβ1r(p) (these can be used to get point-wise confidence intervals) ii) Use the gener- ated samples and the supremum test statistic (Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2022) to obtain joint confidence band for β0 1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Similar strategy can also be employed for ob- taining an asymptotic joint confidence band for the additive effect β0(p) + h(qx(p)), for a fixed value of QX(p) = qx(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Based on the joint confidence band, it is possible to directly test for the global distributional effects β(p) (or h(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The p-value for the test 12 H0 : β(p) = 0 for all p ∈ [0, 1] versus H1 : β(p) ̸= 0 for at least one p ∈ [0, 1], could be obtained based on the 100(1 − α)% joint confidence band for β(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In particular, follow- ing Sergazinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022), the p-value for the test can be defined as the smallest level α for which at least one of the 100(1 − α)% confidence intervals around β(p) (p ∈ P) does not contain zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Alternatively, a nonparametric bootstrap procedure for testing the global effects of scalar and distributional predictors is illustrated in Appendix E of the Supplementary Material which could be useful for finite sample sizes and non Gaussian error process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 3 Simulation Studies In this Section, we investigate the performance of the proposed estimation and testing method for DOSDR via simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' To this end, we consider the following data gener- ating scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='1 Data Generating Scenarios Scenario A1: DOSDR, Both distributional and scalar predictor We consider the DOSDR model given by, QiY (p) = β0(p) + zi1β1(p) + h(QiX(p)) + ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (12) The distributional effects are taken to be β0(p) = 2+3p, β1(p) = sin( π 2p) and h(x) = ( x 10)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The scalar predictor zi1 is generated independently from a U(0, 1) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The dis- tributional predictor QiX(p) is generated as QiX(p) = ciQN(p, 10, 1), where QN(p, 10, 1) denotes the pth quantile of a normal distribution N(10, 1) and ci ∼ U(1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The resid- ual error process ϵ(p) is independently sampled from N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='1) for each p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Since we do not directly observe these quantile functions QiX(p), QiY (p) in practice we assume we have the subject-specific observations Xi = {xi1 = QiX(ui1), xi2 = QiX(ui2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , xiLi1 = QiX(uiLi1)} and Yi = {yi1 = QiY (vi1), yi2 = QiY (vi2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , yiLi2 = QiY (viLi2)}, where ui, vj s are independently generated from U(0, 1) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For simplicity, we assume that Li1 = Li2 = L many subject specific observations are available for both the distributional outcome and the predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Based on the observations Xi, Yi the subject specific quantile 13 functions QiX(p) and QiY (p) are estimated based on empirical quantiles as illustrated in equation (1) on a grid of p values ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We consider number of individual measure- ments L = 200, 400 and training sample size n = 200, 300, 400 for this data generating scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The grid P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , pm} ⊂ [0, 1] is taken to be a equi-spaced grid of length m = 100 in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='995].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' A separate sample of size nt = 100 is used as a test set for each of the above cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Scenario A2: DOSDR, Testing the effect of scalar predictor We consider the data generating scheme (12) in scenario A1 above and test for the distri- butional effect of the scalar predictor z1 using the proposed joint-confidence band based test in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' To this end we let β1(p) = d × sin( π 2p), where the parameter d controls the departure from the null hypothesis H0 : β1(p) = 0 for all p ∈ [0, 1] versus H1 : β1(p) ̸= 0 for some p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The number of subject-specific measurements L is set to 200 and sample sizes n ∈ {200, 300, 400} are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Scenario B: DOSDR, Only distributional predictor We consider the following distribution on distribution regression model QiY (p) = h(QiX(p)) + ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (13) The distributional outcome QiY (p), the distributional predictor QiX(p) and the error process ϵi(p) are generated similarly as in Scenario A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The number of subject-specific measurements L is set to 200 and sample sizes n ∈ {200, 300, 400} are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' This scenario is used to compare the performance of the proposed DOSDR method with that of the isotonic regression approach illustrated in Ghodrati and Panaretos (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We consider 100 Monte-Carlo (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C) replications from simulation scenarios A1 and B to assess the performance of the proposed estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For scenario A2, 200 replicated datasets are used to assess type I error and power of the proposed testing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 Simulation Results Performance under scenario A1: We evaluate the performance of our proposed method in terms of integrated mean squared error (MSE), integrated squared Bias (Bias2) and integrated variance (Var).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For the distributional effect β1(p), these are defined as MSE = 1 M �M j=1 � 1 0 {ˆβj 1(p) − β1(p)}2dp, Bias2 = � 1 0 {ˆ¯β1(p) − β1(p)}2dp, V ar = 1 M �M j=1 � 1 0 {ˆβj 1(p) − ˆ¯β1(p)}2dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Here ˆβj 1(p) is the estimate of β1(p) from the jth replicated dataset and ˆ¯β1(p) = 1 M �M j=1 ˆβj 1(p) is the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C average estimate based on the M replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Table 1 reports the squared Bias, Variance and MSE of the estimates of β1(p) for all cases considered in scenario A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' MSE as well as squared Bias and Variance are found to decrease and be negligible as sample size n or number of measurements L increase, illustrating satisfactory accuracy of the proposed estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Table 1: Integrated squared bias, variance and mean square error of estimated β1(p) over 100 Monte-Carlo replications, Scenario A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Sample Size L=200 L=400 β1(p) Bias2 Var MSE Bias2 Var MSE n= 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0035 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0019 n= 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='9 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0026 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='7 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0016 n= 400 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0018 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='5 × 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0010 Since, h(x) is not directly estimable in the DOSDR model (12), we consider estimation of the estimable additive effect γ(p) = β0(p) + h(qx(p)) at qx(p) = 1 n �n i=1 QiX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The performance of the estimates in terms of squared Bias, variance and MSE are reported in Table 2, which again illustrates satisfactory performance of the proposed method in capturing the distributional effect of the distributional predictor QX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Table 2: Integrated squared bias, variance and mean square error of the estimated additive effect γ(p) = β0(p)+h(qx(p)) at qx(p) = 1 n �n i=1 QiX(p) over 100 Monte-Carlo replications, Scenario A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Sample Size L=200 L=400 β0(p) + h(qx(p)) Bias2 Var MSE Bias2 Var MSE n= 200 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='9 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='023 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='023 n= 300 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='3 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='017 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='017 n= 400 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='013 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='013 The estimated M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C mean for the distributional effect β1(p) and γ(p) along with their 15 respective 95% point-wise confidence intervals are displayed in Figure 2, for the case n = 400, L = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C mean estimates are superimposed on the true curves and along with the narrow confidence intervals, they illustrate low variability and high accuracy of the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 p β1(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 4 6 8 10 p γ(p) Figure 2: Left: True distributional effect β1(p) (solid) and estimated ˆβ1(p) averaged over 100 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C replications (dashed) along with point-wise 95% confidence interval (dotted), scenario A1, n = 400, L = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Right: Additive effect γ(p) = β0(p) + h(qx(p)) (solid) at qx(p) = 1 n �n i=1 QiX(p) and its estimate ˆγ(p) averaged over 100 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C replications (dashed) along with point-wise 95% confidence interval (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' As a measure of out-of-sample prediction performance, we report the average Wasser- stein distance between the true quantile functions and the predicted ones in the test set defined as WD = 1 nt �nt i=1[ � 1 0 {Qtest i (p)− ˆQi test(p)}2dp] 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Supplementary Table S1 reports the summary of the average Wasserstein distance across the 100 Monte-Carlo replica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The low values of the average WD metric and their M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C standard error indicate a satisfactory prediction performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The prediction accuracy appears to be improving with an increase in the number of measurements L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The perfor- mance of the proposed projection based joint confidence intervals for β1(p) is investigated in Supplementary Table S2 which reports the coverage and width of the joint confidence bands for β1(p) for various choices of N and for the case L = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' It is observed that the nominal coverage of 95% lies within the two standard error limit of the estimated coverage in the all the cases, particularly for choices of N picked by our proposed cross 16 validation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Performance under scenario A2: We assess the performance of the proposed testing method in terms of estimated type I error and power calculated from the Monte-Carlo replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We set the order of the Bernstein polynomial basis N = 3 based on our results from previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The estimated power curve is displayed as a function of the parameter d in Supplementary Figure S1, using a nominal level of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' At d = 0, the null hypothesis holds and the power corresponds to the type I error of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The nominal level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='05 lies within its two standard error limit for all the sample sizes, illustrating that the test maintains proper size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For d > 0, we see the power quickly increase to 1, showing that the proposed test is able to capture small departures from the null hypothesis successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Performance under scenario B: We again consider estimation of the estimable additive effect we consider estimation of the estimable additive effect γ(p) = β0(p) + h(qx(p)) at qx(p) = 1 n �n i=1 QiX(p), which can be estimated by both the proposed DOSDR (2) method and the isotonic regression method (Ghodrati and Panaretos, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that true β0(p) = 0, but we include a distributional intercept in our DOSDR model, nonetheless, as this information is not available to practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For the isotonic regression method we directly fit the model (13) without any intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The performance of the estimates are compared in terms of squared Bias, variance and MSE and are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We observe a similar performance of the proposed method with the PAVA based isotnic regression method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Table 3: Integrated squared bias, variance and mean square error of the estimated additive effect γ(p) = β0(p)+h(qx(p)) at qx(p) = 1 n �n i=1 QiX(p) over 100 Monte-Carlo replications, Scenario B, from the DOSDR method and the isotonic regression method with PAVA (Ghodrati and Panaretos, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Sample Size DOSDR PAVA β0(p) + h(qx(p)) Bias2 Var MSE Bias2 Var MSE n= 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='022 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='022 n= 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='016 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='016 n= 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='013 3 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='012 The estimated M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C mean for the distributional effect γ(p) along with their respective 17 95% point-wise confidence intervals are displayed in Supplementary Figure S2, for the case n = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Again, both the method are observed to perform a good job in capturing γ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The proposed DOSDR method enables conditional estimation of γ(p) = β0(p) + h(qx(p)) on the entire domain p ∈ [0, 1], where as for the isotonic regression method, interpolation is required from grid level estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The PAVA based isotonic regression method failed to converge in 5% of the cases for sample size n = 200, where as, this issue was not faced by our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In terms of model flexibility, the isotonic regression method do not directly accommodate scalar predictors, or a distributional intercept, and keeping these points in mind our proposed method certainly provide a uniform and flexible approach for modelling distributional outcome, in the presence of both distributional and scalar predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 4 Modelling Distribution of Heart Rate in Baltimore Longitudinal Study of Aging In this section, we apply our proposed framework to continuously monitored heart rate and physical activity data collected in Baltimore Longitudinal Study of Aging (BLSA), the longest-running scientific study of aging in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Specifically, the distri- bution of minute-level heart rate is modelled via age, sex and BMI and the distribution of minute-level activity counts capturing daily composition of physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We set our study period to be 8 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' - 8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and calculate distributional representation of minute-level heart rate and (log-transformed) activity counts of BLSA participants via subject-specific quantile functions QiY (p) (heart rate) and QiX(p) (represented via log- transformed AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For each participant, we consider only their first BLSA visit while obtaining the subject-specific quantile functions QiY (p), QiX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Our final sample con- stitutes of n = 890 BLSA participants, who had heart rate, physical activity and other covariates used for the analysis available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Supplementary Table S3 presents the descrip- tive statistics of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Supplementary Figure S3 shows the subject-specific quantile functions of heart rate and physical activity (log-transformed, during 8 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' time period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' As a starting point, we study the dependence of mean heart rate on mean activity count and age, sex 18 (Male=1, Female=0) and BMI via the multiple regression model, µH,i = θ0 + θ1agei + θ2sexi + θ3BMIi + θ4µA,i + ϵi, where µH,i, µA,i are the subject specific means of heart rate and activity counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Supple- mentary Table S4 reports the results of the model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Mean heart rate is found to be negatively associated with age and mean activity, and positively associated with BMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The above results although useful, does not paint the whole picture about how the dis- tribution of hear rate depends on these biological factors and the distribution of physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Therefore, we use the proposed DOR model QiY (p) = β0(p) + ageiβage(p) + BMIiβBMI(p) + sexiβsex(p) + h(QiX(p)) + ϵi(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (14) The scalar covariates age, BMI as well as activity counts are transformed to be [0, 1] scale using monotone linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The distributional effects of age, sex (Male=1, Female=0) and BMI on heart rate are captured by βage(p), βsex(p), βBMI(p), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The monotone nonparametric function h(·) is used to link the distribution of heart rate and the distribution of activity counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We use the proposed estimation method for estimation of the distributional effects βj(p)s and h(·) (h(0) = 0 is imposed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The common degree of the Bernstein polynomial basis used to model all the distributional coefficient was chosen via five-fold cross-validation method that resulted in N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The estimated distributional effects along with their asymptotic 95% joint confidence bands using the proposed projection based method are displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The p-values from the joint confidence band based global test for the intercept and the effect of age, BMI, sex, and distribution of activity counts are found to be 1 × 10−6, 1 × 10−6, 5 × 10−5, 3 × 10−4 and 1 × 10−6, respectively, resulting in the significance of all the predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The estimated distributional intercept ˆβ0(p) is monotone and represents the baseline distribution of heart rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The estimated distributional effect of age is found to be significant for all p, in particular, ˆβage(p) is negative and appears to be decreasing and then stabilizing in p ∈ [0, 1] illustrating moderate-high levels of heart rate decrease at an accelerated rate with age compared to sedentary levels of activity (Antelmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The maximal levels of heart rate (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8) are found to be decreasing with age (βage(p) < 0) (Kostis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Gellish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The distributional 19 effect of BMI ˆβBMI(p) is found to be positive and increasing in p (especially at higher quantiles), indicating that a higher maximal heart rate is associated with a higher BMI after adjusting for age, sex and the daily distribution of activity counts (Foy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The estimated effect of sex (Male) ˆβsex(p) illustrates that females have higher heart rate (Antelmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Prabhavathi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2014) compared to males across all quantile levels after adjusting for age, BMI and PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The lower heart rate in males compared to females can be attributed to size of the heart, which is typically smaller in females than males (Prabhavathi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2014) and thus need to beat faster to provide the same output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The estimated monotone regression map between PA and heart rate distribution ˆh(x) (estimated under constraint h(0) = 0) is found to be highly nonlinear and convex, illustrating a non-linear dependence of heart rate on physical activity, especially at higher values of PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The convex nature of the map points out an accelerated increase in the heart rate quantiles with an increase in the corresponding quantile levels of PA (Leary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The estimated distributional effects especially for age and gender in our analysis, illustrate that the distributional effects have no reason to be non-decreasing, as enforced in the qunatile function-on-scalar regression model in Yang (2020), which might lead to wrong conclusions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The proposed DOR method is more flexible in this regard and enforces the monotonicity of the quantile functions without requiring the distributional effects to be monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We also compare the predictive performance of the proposed DOSDR model with that of the distribution-on-distribution regression model by Ghodrati and Panaretos (2021) based on isotonic regression (DODR-ISO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Supplementary Figure S4 displays the leave-one-out-cross-validated (LOOCV) predicted quantile functions of heart rate from both the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We define the measure LOOCV R-Squared as R2 loocv = 1 − �N i=1 � 1 0 {Qi(p)− ˆ Qi loocv(p)}2dp �N i=1 � 1 0 {Qi(p)− ¯Q}2dp , where ¯Q = 1 N �N i=1 � 1 0 Qi(p)dp to compare the out-of-sample prediction accuracy of the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The R2 loocv value for the DOSDR and the DODR- ISO model are calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='60 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='49 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' This illustrates the proposed DOSDR method is able to predict the heart rate quantile functions more accurately with the use of additional information from the biological scalar factors age, sex and BMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 60 80 100 120 intercept for HR p beta0_intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 −30 −10 10 30 age effect on HR p beta_age 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 −20 0 10 30 bmi effect on HR p beta_bmi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 −10 −6 −4 −2 0 sex effect on HR p beta_sex(M) 0 2 4 6 8 0 50 100 150 h function h(x) Figure 3: Estimated distributional effects (solid) along with their joint 95% confidence bands (dotted) for age, BMI (both scaled to [0, 1]) and sex (Male) on heart rate along with the estimated link function h(·) (solid) (under the constraint h(0) = 0) between the distribution of heart rate and physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 5 Discussion In this article, we have developed a flexible distributional outcome regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The dis- tributional functional effects are modelled via Bernstein polynomial basis with appro- priate shape constraints to ensure monotonicity of the predicted quantile functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' A novel construction of BP-based regression structure results in imposing much less restric- tive compared to existing methods for modelling monotone quantile function outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 21 Thus, the proposed framework enables more flexible dependencies between distributional outcome and scalar and distributional predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Inferential tools are developed that include projection-based asymptotic joint confidence bands and a global test of statisti- cal significance for estimated functional regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Numerical analysis using simulations illustrate an accurate performance of the estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The proposed test is also shown to maintain the nominal test size and have a satisfactory power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' An additional nonparametric bootstrap test provided in the supplementary material could be particularly useful in finite sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Application of DOR is demonstrated in studying the distributional association be- tween heart rate reserve and key demographics while adjusting for physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Our findings provide important insights about age and gender differences in distribution of heart rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Beyond the considered epidemiological application, the proposed regression model could be used in other epidemiological studies to more flexibly model distributional aspect of high frequency and high intensity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Additionally, it can be used for estima- tion of treatment effects in primary or secondary endpoints quantified via distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' There are multiple research directions that remain to be explored based on this cur- rent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In developing our method we have implicitly assumed that there are enough measurements available per subject to accurately estimate quantile functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Scenarios with only a few sparse measurements pose a practical challenge and will need careful handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Other aspects of studies collecting distributional data such as distributional measurements being multilevel (Goldsmith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2015) or incorporating spatio-temporal structure (Yang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2022b) would be important to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Another interesting direction of research could be to extend these models beyond the additive paradigm, for example the single index model (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2011) could be employed to accommodated interaction and nonlinear effects of multiple scalar and distributional predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Extending the proposed method to such more general and complex models would be computationally challenging, nonetheless merits future attention because of their potentially diverse applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 22 Supplementary Material Appendix A-E along with the Supplementary Tables and Supplementary Figures refer- enced in this article are available online as Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Software Software implementation via R (R Core Team, 2018) and illustration of the proposed framework is available upon request from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' References Antelmi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', De Paula, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Shinzato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Peres, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Mansur, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Grupi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2004), “Influence of age, gender, body mass index, and functional capacity on heart rate variability in a cohort of subjects without heart disease,” The American journal of cardiology, 93, 381–385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Augustin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Mattocks, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Faraway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Greven, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Ness, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2017), “Modelling a response as a function of high-frequency count data: The association between physical activity and fat mass,” Statistical methods in medical research, 26, 2210–2226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Carnicer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Pena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (1993), “Shape preserving representations and optimality of the Bernstein basis,” Advances in Computational Mathematics, 1, 173–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and M¨uller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), “Wasserstein regression,” Journal of the American Statistical Association, 1–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Cui, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Leroux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Smirnova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Crainiceanu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022), “Fast univariate inference for longitudinal functional models,” Journal of Computational and Graphical Statistics, 31, 219–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', James, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Radchenko, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2015), “Functional additive regression,” The Annals of Statistics, 43, 2296–2325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 23 Foy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Mandrola, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Naccarelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2018), “Relation of obesity to new-onset atrial fibrillation and atrial flutter in adults,” The American journal of cardiology, 121, 1072–1075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Gellish, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Goslin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Olson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', McDONALD, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Russi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Moudgil, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2007), “Longitudinal modeling of the relationship between age and maximal heart rate.” Medicine and science in sports and exercise, 39, 822–829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghodrati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Panaretos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), “Distribution-on-Distribution Regression via Optimal Transport Maps,” arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='09418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Urbanek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Schrack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Zipunnikov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022a), “Shape- constrained estimation in functional regression with Bernstein polynomials,” Compu- tational Statistics & Data Analysis, 107614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Maity, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022), “A Score Based Test for Functional Linear Concurrent Regression,” Econometrics and Statistics, 21, 114–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Varma, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Volfson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Hillel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Urbanek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Hausdorff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Watts, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Zipunnikov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), “Distributional data analysis via quantile functions and its application to modelling digital biomarkers of gait in Alzheimer’s Disease,” Biostatistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Varma, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Volfson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Urbanek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Hausdorff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Watts, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Zipunnikov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022b), “Scalar on time-by-distribution regression and its application for modelling associations between daily-living physical activity and cognitive functions in Alzheimer’s Disease,” Scientific reports, 12, 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Goldfarb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Idnani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (1982), “Dual and primal-dual methods for solving strictly convex quadratic programs,” in Numerical analysis, Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 226–239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' — (1983), “A numerically stable dual method for solving strictly convex quadratic pro- grams,” Mathematical programming, 27, 1–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Goldsmith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Zipunnikov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Schrack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2015), “Generalized multilevel function- on-scalar regression and principal component analysis,” Biometrics, 71, 344–353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 24 Hron, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Menafoglio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Templ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Hruzova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Filzmoser, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2016), “Simplicial principal component analysis for density functions in Bayes spaces,” Computational Statistics & Data Analysis, 94, 330–350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2002), “Varying-coefficient models and basis function approximations for the analysis of repeated measurements,” Biometrika, 89, 111–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' — (2004), “Polynomial spline estimation and inference for varying coefficient models with longitudinal data,” Statistica Sinica, 14, 763–788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Irpino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Verde, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2013), “A metric based approach for the least square regression of multivariate modal symbolic data,” in Statistical Models for Data Analysis, Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 161–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2011), “Functional single index models for longitudinal data,” The Annals of Statistics, 39, 362–388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Kostis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Moreyra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Amendo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Di Pietro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Cosgrove, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Kuo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (1982), “The effect of age on heart rate in subjects free of heart disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Studies by ambulatory electrocardiography and maximal exercise stress test.” Circulation, 65, 141–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Leary, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Struthers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Donnan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', MacDonald, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Murphy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2002), “The morning surge in blood pressure and heart rate is dependent on levels of physical activity after waking,” Journal of hypertension, 20, 865–870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Lorentz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2013), Bernstein polynomials, American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Matabuena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Petersen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), “Distributional data analysis with accelerometer data in a NHANES database with nonparametric survey regression models,” arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Matabuena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Petersen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Gude, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), “Glucodensities: a new representation of glucose profiles using distributional data analysis,” Statistical Methods in Medical Research, 30, 1445–1464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Coull, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Versace, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Cinciripini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Morris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2015), “Bayesian function-on-function regression for multilevel functional data,” Biometrics, 71, 563– 574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 25 Parzen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2004), “Quantile probability and statistical data modeling,” Statistical Sci- ence, 19, 652–662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Pegoraro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Beraha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022), “Projected Statistical Methods for Distributional Data on the Real Line with the Wasserstein Metric.” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 23, 37–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Petersen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and M¨uller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2016), “Functional data analysis for density functions by transformation to a Hilbert space,” The Annals of Statistics, 44, 183–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Petersen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Kokoszka, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), “Modeling Probability Density Func- tions as Data Objects,” Econometrics and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Powley, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2013), “Quantile function methods for decision analysis,” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' thesis, Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Prabhavathi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Selvi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Poornima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Sarvanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2014), “Role of biological sex in normal cardiac function and in its disease outcome–a review,” Journal of clinical and diagnostic research: JCDR, 8, BE01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R Core Team (2018), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ramsay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Silverman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2005), Functional Data Analysis, New York: Springer- Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ramsay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (1988), “Monotone regression splines in action,” Statistical science, 3, 425–441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Sergazinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Leroux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Cui, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Crainiceanu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Aurora, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Punjabi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Gaynanova, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2022), “A case study of glucose levels during sleep using fast function on scalar regression inference,” arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='08439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Talsk´a, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Hron, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Grygar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2021), “Compositional Scalar-on-Function Regression with Application to Sediment Particle Size Distributions,” Mathematical Geosciences, 1–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Tanaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Monahan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Seals, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2001), “Age-predicted maximal heart rate revisited,” Journal of the american college of cardiology, 37, 153–156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 26 Tang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Venkataraman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Tsapkini, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Lindquist, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Pekar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Caffo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2020), “Differences in functional connectivity distribution after transcra- nial direct-current stimulation: a connectivity density point of view,” bioRxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Vanbrabant, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Rosseel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2019), Restricted Statistical Estimation and Inference for LinearModels, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2-250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Verde, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Irpino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2010), “Ordinary least squares for histogram data based on wasserstein distance,” in Proceedings of COMPSTAT’2010, Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 581–588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2012), “Shape restricted nonparametric regression with Bernstein polynomials,” Computational Statistics & Data Analysis, 56, 2729–2741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2020), “Random distributional response model based on spline method,” Jour- nal of Statistical Planning and Inference, 207, 27–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Baladandayuthapani, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', Rao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Morris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2020), “Quantile function on scalar regression analysis for distributional data,” Journal of the American Statistical Association, 115, 90–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', M¨uller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', and Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2005), “Functional linear regression analysis for longitudinal data,” The Annals of Statistics, 2873–2903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 27 Supplementary Material for Distributional outcome regression and its application to modelling continuously monitored heart rate and physical activity Rahul Ghosal1,∗, Sujit Ghosh2, Jennifer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Schrack3, Vadim Zipunnikov4 1 Department of Epidemiology and Biostatistics, University of South Carolina 2Department of Statistics, North Carolina State University 3 Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health 4 Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health January 30, 2023 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='11399v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='ME] 26 Jan 2023 1 Appendix A: Proof of Theorem 1 The predicted outcome quantile function is the conditional expectation of the outcome quantile function based on the distribution-on-scalar and distribution regression (DOSDR) model (2) and is given by, E(QY (p) | z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq, qx(p)) = β0(p) + q � j=1 zjβj(p) + h(qx(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (1) We will show conditions (1)-(3) are sufficient conditions to ensure E(QY (p) | z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq, qx(p)) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Let us assume 0 ≤ zj ≤ 1, ∀j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , J, without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' It is enough to show T1(p) = β0(p) + �q j=1 zjβj(p) and T2(p) = h(qx(p)) both are non decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The second part is immediate as both qx(·) and h(·) (by condition (3)) are non decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' To complete the proof we only need to show T1(p) is non decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' T ′ 1(p) = β′ 0(p)+�q j=1 zjβ′ j(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Enough to show T ′ 1(p) ≥ 0 for all (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq) ∈ [0, 1]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that this is a linear function in (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq) ∈ [0, 1]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' By the well-known Bauer’s principle the minimum is attained at the boundary points B = {(z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq) : zj ∈ {0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Hence, the sufficient conditions are β′ 0(p) ≥ 0 and β′ 0(p) + �r k=1 β′ jk(p) ≥ 0 for any sub-sample {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , jr} ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , q}, which follows from condition (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2 Appendix B: Example of DOSDR Example 2: Two scalar covariates (q = 2) and a distributional predictor We illustrate the estimation for DOSDR where there are two scalar covariates z1, z2 (q = 1) and a single distribution predictor QX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The DOSDR model (2) is given by QiY (p) = β0(p)+zi1β1(p)+zi2β2(p)+h(QiX(p))+ϵi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The sufficient conditions (1)-(3) of Theorem 1 in this case reduce to : A) The distributional intercept β0(p) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' B) β0(p) + β1(p), β0(p) + β2(p), β0(p) + β1(p) + β2(p) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' C) h(·) is non- decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that condition B) illustrates that as the number of scalar covariates increase we have more and more combinatorial combinations of the coefficint functions restricted to be non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Similar to Example 1, Conditions (A)-(C) again become linear restrictions on the basis coefficients of the form Dψ ≥ 0, where the constraint 2 matrix is given by D = � � � � � � � � � � � AN 0 0 0 AN AN 0 0 AN 0 AN 0 AN AN AN 0 0 0 0 AN−1 � � � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' As the number of restrictions increase the parameter space becomes smaller and smaller, which can result in a faster convergence of the optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 3 Appendix C: Estimation of Asymptotic Covariance Matrix The DOSDR model (8) in the paper was reformulated as QiY = Tiψ + ϵi, , where Ti = [B0 Wi1 Wi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , Wiq Si].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Under suitable regularity conditions (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2004), √n( ˆψur − ψ0) can be shown to be asymptotically distributed as N(0, ∆) (also holds true for finite sample sizes if ϵ(p) is Gaussian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In reality, ∆ is unknown and we want to estimate ∆ by an estimator ˆ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We derive a sandwich covariance estimator ˆ∆ corresponding to the above model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Based on the ordinary least square optimization criterion for model (11) (of the paper), the unrestricted estimator is given by ˆψur = (TTT)−1TTQY , where QT Y = (Q1Y , Q2Y , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , QnY )T and T = [TT 1 , TT 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , TT n]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Hence, V ar( ˆψur) = (TTT)−1TTΣT(TTT)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Here Σ = V ar(ϵ), which is typically unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We apply an FPCA based estimation approach (Ghosal and Maity, 2022) to estimate Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Let us assume (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2004) the error process ϵ(p) can be decomposed as ϵ(p) = V (p) + wp, where V (p) is a smooth mean zero stochastic process with covariance kernel G(p1, p2) and wp is a white noise with variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The covariance function of the error process is then given by Σ(p1, p2) = cov{ϵ(p1), ϵ(p2)} = G(p1, p2)+σ2I(p1 = p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For data observed on dense and regular grid P, the covariance matrix of the residual vector ϵi is Σm×m, the covariance kernel Σ(p1, p2) evaluated on the grid P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , pm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' We can estimate Σ(·, ·) nonparametrically using functional principal component analysis (FPCA) if the original residuals ϵij were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Given ϵi(pj)s, FPCA (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', 2005) 3 can be used to get ˆφk(·), ˆλks and ˆσ2 to form an estimator of Σ(p1, p2) as ˆΣ(p1, p2) = K � k=1 ˆλk ˆφk(p1)ˆφk(p2) + ˆσ2I(p1 = p2), where K is large enough such that percent of variance explained (PVE) by the selected eigencomponents exceeds some pre-specified value such as 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In practice, we don’t have the original residuals ϵij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Hence we fit the unconstrained DOSDR model (11) and and obtain the residuals eij = QiY (pj) − ˆ QiY (pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Then treating eij as our original residuals, we can obtain ˆΣ(p1, p2) and ˆΣm×m using the FPCA approach outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Then ˆ V ar(ϵ) = ˆΣ = diag{ˆΣm×m, ˆΣm×m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , ˆΣm×m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal and Maity (2022) discusses consistency of ˆΣ under standard regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Hence an consis- tent estimator of the covariance matrix is given by ˆ V ar( ˆψur) = (TTT)−1TT ˆΣT(TTT)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In particular, ˆ∆n = ˆ∆/n = ˆ cov( ˆψur) = (TTT)−1TT ˆΣT(TTT)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 4 4 Appendix D: Algorithm 1 for Joint Confidence Band Algorithm 1 Joint confidence band of β0 1(p) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Fit the unconstrained model and obtain the unconstrained estimator ˆψur = argmin ψ∈RKn �n i=1 ||QiY − Tiψ||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Fit the constrained model and obtain the constrained estimator ˆψr = argmin ψ∈ΘR �n i=1 ||QiY −Tiψ||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Obtain the constrained estimator of β0 1(p) as ˆβ1r(p) = ρKn(p) ′ ˆβ1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Let ˆ∆n be an estimate of the asymptotic covariance matrix of the unconstrained estimator given by ˆ∆n = ˆ∆/n = ˆ cov( ˆψur) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For b = 1 to B generate Zb ∼ NKn( ˆψur, ˆ∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' compute the projection of Zb as ˆψr,b = argmin ψ∈ΘR ||ψ − Zb||2 ˆΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' End For 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For each generated sample ˆψr,b calculate estimate of β0 1(p) as ˆβ1r,b(p) = ρKn(p) ′ ˆβ1r,b (b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Compute V ar(ˆβ1r(p)) based on these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For b = 1 to B calculate ub = max p∈P |ˆβ1r,b(p)−ˆβ1r(p)| √ V ar(ˆβ1r(p)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' End For 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Calculate q1−α the (1 − α) empirical quantile of {ub}B b=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 100(1−α)% joint confidence band for β0 1(p) is given by ˆβ1r(p)±q1−α � V ar(ˆβ1r(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 5 Appendix E: Bootstrap Test for Global Distribu- tional Effects A practical question of interest in the DOSDR model is to directly test for the global distributional effect of the scalar covariates Zj or test for the distributional effect of the distributional predictor QX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In this section, we illustrate an nonparametric bootstrap test based on our proposed estimation method which also easily lends itself to the required 5 shape constraints of the regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' In particular, we obtain the residual sum of squares of the null and the full model and come up with the F-type test statistic defined as TD = RSSN − RSSF RSSF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' (2) Here RSSN, RSSF are the residual sum of squares under the null and the full model respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' For example, let us consider the case of testing H0 : βr(p) = 0 for all p ∈ [0, 1] versus H1 : βr(p) ̸= 0 for some p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Let r = q without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The residual sum of of squares for the full model is given by RSSF = �n i=1 ||QiY − B0 ˆβ0 − �q j=1 Wij ˆβj − Si ˆθ||2 2, where the estimates are obtained from the optimization criterion (9) in the paper, with the constraint DFψ ≥ 0 (denoting the constraint matrix for the full model as DF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Similarly, we have RSSN = �n i=1 ||QiY − B0 ˆβ0 − �q−1 j=1 Wij ˆβj − Si ˆθ||2 2, where the estimates are again obtained from (9) with the constraint DNψ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Note that, in this case the constraint matrix is denoted by DN and this is essentially a submatrix of DF as the conditions for monotinicity in (1)- (3) (Theorem 1) for the reduced model is a subset of the original constrains for the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The null distribution of the test statistic TD is nonstandard, hence we use residual bootstrap to approximate the null distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The complete bootstrap procedure for testing the distributional effect of a scalar predictor is presented in algorithm (2) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Similar strategy could be employed for testing the distributional effect of a distributional predictor or multiple scalar predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 6 Algorithm 2 Bootstrap algorithm for testing the distributional effect of a scalar predictor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Fit the full DOSDR model in the paper using the optimization criterion ˆψF = argmin ψ n � i=1 ||QiY − B0β0 − q � j=1 Wijβj − Siθ||2 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='t DFψ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and calculate the residuals ei(pl) = QiY (pl) − ˆQiY (pl), for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , n and l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Fit the reduced model corresponding to H0 (the null) and estimate the parameters using the minimization criteria, ˆψN = argmin ψ n � i=1 ||QiY − B0β0 − q−1 � j=1 Wijβj − Siθ||2 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='t DNψ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Denote the estimates of the distributional effects as ˆβN j (p) for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , q − 1 and ˆhN(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Compute test statistic TD (2) based on these null and full model fits, denote this as Tobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Resample B sets of bootstrap residuals {e∗ b,i(p)}n i=1 from residuals {ei(p)}n i=1 ob- tained in step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' for b = 1 to B 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Generate distributional response under the reduced DOSDR model as Q∗ b,iY (p) = ˆβN 0 (p) + q−1 � j=1 zij ˆβN j (p) + ˆhN(QiX(p)) + e∗ b,i(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Given the bootstrap data set {QiX(p), Q∗ b,iY (p), z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' , zq}n i=1 fit the null and the full model to compute the test statistic T ∗ b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' end for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Calculate the p-value of the test as ˆp = �B b=1 I(T ∗ b ≥Tobs) B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 7 6 Supplementary Tables Table S1: Average Wasserstein distance (standard error) between true and predicted quantile functions in the test set over 100 Monte-Carlo replications, Scenario A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Sample Size L=200 L=400 n= 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2587 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0154) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='1882 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0138) n= 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2568 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0132) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='1858 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0105) n= 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2554 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0141) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='1865 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0120) Table S2: Coverage of the projection-based 95% joint confidence interval for β1(p), for various choices of the order of the Bernstein polynomial (BP) basis, scenario A1, based on 100 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C replications with L = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Average width of the joint confidence interval is given in the parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The average choices of N from cross-validation for this scenario are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' BP order (N) Sample size (n=200) Sample size (n=300) Sample size (n=400) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='92 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='29) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='20) 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='92 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='31) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='96 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='22) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='93 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='33) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='93 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='93 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='23) Table S3: Descriptive statistics of age and BMI for the complete, male and female samples in the BLSA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Characteristic Complete (n=890) Male (n=432) Female (n=458) P value Mean SD Mean SD Mean SD Age 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='66 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='35 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='03 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='41 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='37 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='003 BMI (kg/m2) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='96 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='23 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='45 Table S4: Results from multiple linear regression model of mean heart rate on age, sex (Male), BMI and mean activity count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Reported are the estimated fixed effects along with their standard error and P-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Dependent variable : Mean heart rate Value Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='Error P-value Intercept 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='458 < 2 × 10−16∗∗∗ age −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='026 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 × 10−11∗∗∗ sex −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='659 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 × 10−10∗∗∗ BMI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0091∗∗ Mean activity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0005∗∗∗ Observations 890 Adjusted R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='142 Note: ∗p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' ∗∗p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' ∗∗∗p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='001 8 7 Supplementary Figures 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 d Power n=200 n=300 n=400 Figure S1: Displayed are the estimated power curves for simulation scenario A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The parameter d controls the departure from the null and the power curves for n ∈ {200, 300, 400} are shown by solid, dashed and dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The dashed horizontal line at the bottom corresponds to the nominal level of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 2 3 4 5 6 7 DODSR p γ(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 2 3 4 5 6 7 PAVA p γ(p) Figure S2: Displayed are estimates of additive effect γ(p) = β0(p) + h(qx(p)) (solid) at at qx(p) = 1 n �n i=1 QiX(p) and its estimate ˆγ(p) averaged over 100 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='C replications (dashed) along with point-wise 95% confidence interval (dotted) for scenario B, n = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Left: Estimates from the proposed DOSDR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Right: Isotonic regression method with PAVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0 50 100 150 200 250 Heartrate p Heartrate QF (raw) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0 2 4 6 8 Activity p Activity QF (log) Figure S3: Subject-specific quantile functions of heart rate and log-transformed activity counts during 8 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='- 8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Color profiles show four randomly chosen partici- pants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 11 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0 50 100 150 200 250 DODSR Predicted HR p Q(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0 50 100 150 200 250 HR p Q(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='0 0 50 100 150 200 250 PAVA Predicted HR p Q(p) Figure S4: Top: LOOCV predictions of quantile functions of heart rate from DOSDR method based on age, sex, BMI and PA distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Bottom: LOOCV predictions of quantile functions of heart rate from PAVA method (Ghodrati and Panaretos, 2021) based on PA distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' References Ghodrati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Panaretos (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Distribution-on-distribution regression via optimal transport maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='09418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Ghosal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Maity (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' A score based test for functional linear concurrent regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Econometrics and Statistics 21, 114–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 12 Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Wu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Zhou (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Polynomial spline estimation and inference for varying coefficient models with longitudinal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Statistica Sinica 14, 763–788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Yao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' M¨uller, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Wang (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' Functional linear regression analysis for longitudinal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' The Annals of Statistics, 2873–2903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFIT4oBgHgl3EQf7yth/content/2301.11399v1.pdf'} diff --git a/8NE5T4oBgHgl3EQfQg5R/content/tmp_files/2301.05513v1.pdf.txt b/8NE5T4oBgHgl3EQfQg5R/content/tmp_files/2301.05513v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd245ac0f8c53cf5a3db2c6d4f0585569653cbf5 --- /dev/null +++ b/8NE5T4oBgHgl3EQfQg5R/content/tmp_files/2301.05513v1.pdf.txt @@ -0,0 +1,1532 @@ +1 + +Exploring the substrate-driven morphological changes +in Nd0.6Sr0.4MnO3 thin films + +R S Mrinaleni 1, 2, E P Amaladass1, 2*, S Amirthapandian 1, 2, A. T. Sathyanarayana 1, 2, +Jegadeesan P 1, 2, Ganesan K 1, 2, R M Sarguna 1, 2, P. N. Rao 3, Pooja Gupta 3, 4, T +Geetha Kumary1, 2, and S. K. Rai 3, 4, Awadhesh Mani1, 2 + +1Material Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, 603102, +India +2Homi Bhabha National Institute, Indira Gandhi Centre for Atomic Research, Kalpakkam +603102, India +3Synchrotrons Utilisation Section, Raja Ramanna Centre for Advanced Technology, PO +RRCAT, Indore, Madhya Pradesh 452013, India +4Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai, +Maharashtra 400094, India +*Corresponding author: edward@igcar.gov.in + +ABSTRACT +Manganite thin films are promising candidates for studying the strongly correlated electron +systems. Understanding the growth-and morphology-driven changes in the physical properties +of manganite thin films is vital for their applications in oxitronics. This work reports the +morphological, +structural, +and +electrical +transport +properties +of +nanostructured +Nd0.6Sr0.4MnO3 (NSMO) thin films fabricated using the pulsed laser deposition technique. +Scanning electron microscopy (SEM) imaging of the thin films revealed two prominent surface +morphologies: a granular and a unique crossed-nano-rod-type morphology. From X-ray +diffraction (XRD) and atomic force microscopy (AFM) analysis, we found that the observed +nanostructures resulted from altered growth modes occurring on the terraced substrate surface. +Furthermore, investigations on the electrical-transport properties of thin films revealed that the +films with crossed-nano-rod type morphology showed a sharp resistive transition near the +metal-to-insulator transition (MIT). An enhanced temperature coefficient of resistance (TCR) +of up to one order of magnitude was also observed compared to the films with granular +morphology. Such enhancement in TCR % by tuning the morphology makes these thin films +promising candidates for developing oxide-based temperature sensors and detectors. + + +2 + +INTRODUCTION +Nd0.6Sr0.4MnO3 (NSMO) belongs to the class of magnetic oxides RE1-xAxMnO3 (where RE= +La3+, Nd3+, Pr3+, Sm3+, and A = Ca2+, Sr2+, Ba2+, etc.) with perovskite (ABO3) structure which +exhibits a variety of magnetic phases by tuning the dopant concentration x (x = 0 to 0.9)1–3. +Manganites are known for their exotic properties such as the Colossal magnetoresistive (CMR) +phenomenon4, Metal-insulator-transition (MIT) accompanied by a magnetic transition from +paramagnetic (PM) to ferromagnetic (FM) state5, half-metallicity6, and tuneable in-plane and +out of plane magnetic anisotropy7. These properties are exploited for potential spintronics +applications such as spin injection devices8, Magnetic tunnel junctions9–11, and magnetic +storage devices (MRAMs)12. In recent times, the perovskite-manganite systems are the ideal +oxide candidates for developing superlattices, self-assembled nano-arrays13, nano-ribbons14, +nano-wires, vertically aligned nanocomposite (VAN) thin films15–19, etc. which offer enhanced +Low-field magnetoresistance (LFMR), switchable magnetic anisotropy and for studying other +interesting interface effects such as magnetic exchange bias20. Focus on growth dynamics is +required to tune exclusive nano-architectures in the thin film as it offers additional handles to +tailor its physical properties such as a high CMR %, high Curie & MIT temperature, high- +temperature coefficient of resistance (TCR %), and enhanced magnetoresistive (MR) +phenomenon. The manganite system is highly sensitive to external perturbations due to the +strong connection between the spin-charge and lattice degrees of freedom21,22. This poses a +major challenge in obtaining epitaxial/patterned thin films for useful applications. +The pulsed laser deposition (PLD) technique has been extensively used to fabricate oxide- +based manganite thin films. This is because it offers good stoichiometric transfer of the target +material onto the substrate in addition to deposition in an oxygen background. Various studies +have been carried out to obtain epitaxial thin films by tuning the deposition parameters such as +the oxygen partial pressure, substrate temperature, laser energy density, and repetition rate, +affecting its growth and physical properties23,24. Additionally, the growth of the thin film is +influenced by the substrate. The strain offered by the substrate affects the surface morphology +and microstructure of the manganite thin film. Different methodologies such as i) varying the +substrates for different lattice matching25–27 (ii) choice of substrates with different +crystallographic orientations with corresponding chemical terminations14 iii) varying the +thickness of the thin films28, and iv) high-temperature annealing17 are adopted to tune the strain +and morphology of the thin films. Therefore, thin films with unique morphology and long- +range ordered nanostructures can be obtained by fine-tuning the growth parameters. Compared + +3 + +to the previous works on VAN and other nanostructures of the popular manganite system La- +Sr-Mn-O, we have observed a granular nanostructure and another distinct nanostructure with +crossed-nano-rods in our thin films. We have synthesized NSMO thin films using the PLD +technique on single-crystal SrTiO3 (100) oriented substrates (STO). The effects of PLD +parameters and annealing conditions on the surface morphology were investigated. Using +SEM, AFM, and XRD techniques, the growth mechanism leading to a specific type of nano- +structuring in the NSMO thin films is studied. Additionally, the morphology-driven changes in +the temperature dependence of resistivity are investigated, and we observed a signature trend +in the MIT corresponding to the particular morphology. +EXPERIMENTAL METHODS +The NSMO thin films were fabricated using the PLD technique using a commercial NSMO +pellet as the target. Before deposition, SrTiO3 (STO) (1 0 0) single crystals substrate was +cleaned by boiling in de-ionized(DI) water for 3 minutes, followed by ultra-sonication in DI +water, acetone, and iso-propyl alcohol followed by rinsing in DI water. With the water leaching +procedure, the SrO terminations present in the substrate surface can be effectively dissolved +and removed with DI at elevated temperatures > 60 oC followed by ultra-sonication. A KrF +Excimer laser source (λ = 248 nm) operated with a laser energy density of 1.75 J/cm2 at 3Hz +was used to ablate the target. The films were deposited in an oxygen partial of 0.36 mbar with +substrate temperature fixed at 750 oC. After deposition, the films were in situ annealed at 750 +oC for 2h, and the PLD chamber was maintained with O2 background pressure of 0 to 1 bar. +Further, the films were ex-situ annealed in a tube furnace at 950 oC in an oxygen atmosphere +with a flow rate of ~ 20 sccm for 2h. +The surface morphology of the thin films was examined using a Scanning electron microscope +(SEM) from Carl Zeiss, crossbeam 340, and the images were collected in inlens-duo mode at +3-5 kV. Atomic force microscopy (AFM) was used for 2D and 3D visualization of the surface +of substrates and the films. XRD studies have been carried out at Engineering Applications +Beamline, BL-02, Indus-2 synchrotron source, India using beam energy of 15 keV for the +structural characterization of the films29. The Grazing incidence (GI) and ω-2θ scans were +performed, and data were collected using the Dectris detector (MYTHEN2 X 1K) in reflection +geometry. In the GI-scan, the incident angle is kept fixed at ω = 0.5o, and the detector moves +along the given 2θ range. The monochromatic high-resolution mode of the beamline was used, + +4 + +keeping the beam energy at 15 keV (λ = 0.826 Å). The peaks were indexed with reference to +the ICDD data30 (ICDD number - 01-085-6743) + +RESULTS AND DISCUSSION: +1. Morphology studies of the nanostructured thin films: +The NSMO thin films prepared under the above conditions possessed two prominent +surface morphology – granular and rod-type. Two representative films with granular +nanostructure and crossed-rod nanostructure were chosen to study the physical properties. +These two systems will be referred to as NS-G and NS-R, where NS stands for NSMO thin +film, and ‘G’/’R’ stands for the type of morphology. The thickness of NS-G and NS-R thin +films is determined to be ~ 100 nm by cross-sectional SEM. +Figure 1(a) shows the SEM image of NS-G thin films with granular morphology. The +film is uniformly covered with multifaceted grains. Figure 1(c) (i) shows the average grain size + + +0 +20 +40 +60 +80 +100 +0 +100 +200 +300 +400 +Frequency (count) +Grain size (nm) + Frequency count + Lognormal fitting +Avg grain size += 38.9 nm 0.1 nm +0 +200 +400 +600 +800 1000 +0 +100 +200 +300 +400 +500 +Frequency (count) +Length of rod (nm) + Frequency count + Lognormal fitting +Avg length of rod += 188.7 nm 1.7 nm +0 +20 +40 +60 +80 +100 +120 +0 +100 +200 +300 +Frequency (count) +Width of rod (nm) + Frequency count + Lognormal fitting +Avg width of rod += 39.6 nm 0.2 nm +c) (i) +(ii) +(iii) +Figure 1: Scanning electron microscopy images of NSMO thin films on STO. a) NS-G - +granular morphology. b) NS-R – self-aligned-crossed-Nano-rod-morphology. c) The +histograms illustrate the grain size calculation for NS-G and NS-R thin film. (i) Average grain +size estimated for NS-G. (ii) Average rod length estimated for NS-R. (iii) Average rod width +estimated for NS-R. + +100 nm100 nm5 + +estimated to be 38.9 nm. Figure 1(b) shows the SEM image of NS-R thin films with unique +surface morphology. The thin film surface is uniformly covered with nano-rods crossed at right +angles embedded in a matrix of NSMO containing square/rectangular pits. In NS-R, the +average rod length is estimated to be 188 nm with an average width of 39.6 nm, as shown in +the Figure 1(c), (ii) and (iii). Further, AFM measurements have been carried out on the NS-G +and NS-R thin films. The 2D and 3D AFM scan in Figure 2 show columnar/island-type features +in the NS-G thin film and crossed-rod features in the NS-R thin film. +2. Structural analysis of the thin film: +The bulk NSMO compound has an orthorhombic crystal structure belonging to the Pbnm space +group. In the pseudo-cubic (pc) representation, the unit cell parameter is given by apc ≈ c / 2 ≈ +3.849 Å. The substrate STO has a cubic crystal structure with a lattice constant aSTO = 3.905 +Å. NSMO grown on the STO substrate experiences a tensile strain due to the lattice mismatch + + + + + + +Fig. 2: a), b), 2D and c), d) 3D AFM scans of grain-type NSMO thin film (left) and rod- +type sample NSMO thin film on STO substrate. +a) +b) +c) +d) +Figure 2: a), b), 2D, and c), d) 3D AFM scans of grain-type NSMO thin film (left) and rod-type +sample NSMO thin film on STO substrate. + +nm +20 +0 +2.0 +0 +1.8 +0.2 +1.6 +0.4 +1.4 +0.6 +1.2 +0.8 +1.0 +1.0 +0.8 +1.2 +0.6 +1.4 +0.4 +1.6 +0.2 +1.8 +0 +2.048.3 1F:Height +2 +8 +1.6 +61 +4- +21 +9 +nm +2 +41 +0 +0.2 +0.4 +0.6 +8'0 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +um21. 1F:Height +2 +51 +1.8... +1.6 +51 +1.4 +1.2 +1.0 +51 +nm +8'0 +0.6 +51 +0.4 +2 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +umnm +8 +0 +0 +2.0 +0.2 +1.8 +0.4 +1.6 +0.6 +1.4 +0.8 +1.2 +1.0 +1.0 +0.8 +1.2 +0.6 +1.4 +0.4 +1.6 +0.2 +1.8 +0 +2.06 + +of 1.4 %. The GI-XRD and high-resolution XRD (HR-XRD) reflections of the films are shown +in Figure 3(a) and (b). The presence of multiple reflections in the GI-XRD scan of NS-G in +Figure 3(a) reveals that the granular thin film is polycrystalline. In NS-R, the reflections of +NSMO are absent, as seen in Figure 3(b). This may be due to its out-of-plane orientation with +respect to the substrate. At the high 2θ angle ≈ 39.1o , the (3 1 0) STO plane gets aligned, +resulting in high STO (3 1 0) reflection along with the NSMO (2 4 0) peak. This shows that the +films are well-oriented, mirroring the substrate. Though NS-G is oriented, the crystallographic +difference between NS-G and NS-R is attributed to the type of nano-structuring in the films. + +Figure 3: GI-XRD scans of NSMO thin films a) NS-G b) NS-R indexed using ICDD data (* - +STO peaks) + +3. Effect of ex-situ annealing on morphology: +To gain insight into the type of growth across these films, we compare the +morphological changes in the in-situ annealed and ex-situ annealed samples in Figure 4. In +granular thin films, no significant changes have been observed after in-situ and ex-situ +annealing, apart from a minor increase in grain size, as seen in Figure 4(a). Whereas the sample +with rod-type morphology obtained after ex-situ annealing in Figure 4(d) exhibits facetted +droplets embedded in a matrix with rectangular holes and rod features in the in-situ annealed + +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +1E+02 +1E+03 +1E+04 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +1E+02 +1E+03 +1E+04 +* (310) + (240),(332) + (040), (224) + (024), (132) +* (200) + (220), (004) + (202), (022) + (020), (112) + (002), (110) +Intensity (arb. units) +2(deg) + NS-G +(a) +(b) +* (100) + (240),(332) +* (310) +* (200) +2(deg) + NS-R + +7 + +case, Figure 4(c). It is evident that once the initial growth mode is set, the ex-situ annealing +aids in increasing grain size, relieving the strain in thin films in addition to decreasing oxygen +defects in NSMO thin films23. We inspect the HR-XRD scans of the NS-G and NS-R thin films +in the in-situ and ex-situ annealed cases to verify this claim. +Figure 5(a) and (b) show the HR-XRD scan performed over a range of 2θ (10o – 40o) +for the films NS-G and NS-R after in-situ and ex-situ annealing. It is observed that the (0 0 4) +NSMO peak is absent in the in-situ annealed NS-G thin film, whereas upon ex-situ annealing, +NS-G shows improved texturing with the (0 0 4) NSMO peak close to the (0 0 2) substrate +peak. In the case of NS-R, along with the substrate’s (002) reflection, corresponding (0 0 l) +pseudo-cubic reflections from NSMO are present with significant intensity even in the in-situ +annealed condition. Further, as we compare HR-XRD scans of NS-G and NS-R after ex-situ +annealing, the NS-R thin film has increased relative intensity compared with the NS-G. +a) +b) + +Pristine thin film after in-situ annealing Further upon ex-situ annealing + +a) + + + + + +b) + +Fig. 3: Effect ex-situ annealing on NSMO thin films with a) granular b)rod-type +surface morphology +c) +d) +Figure 4: Illustration of the effect of ex-situ annealing on NSMO thin films. a) SEM image of +in-situ annealed granular thin film b) SEM image of the granular thin film after ex-situ +annealing c) SEM image of the thin film after in-situ annealing showing rods and squared +blocks in the encircled regions. d) SEM image of the same thin film after ex-situ annealing +showing crossed-rod type morphology. + +100nm100 nm100 nm100nm8 + + +Figure 5: High resolution-XRD scan of NSMO thin films around the STO-(200) reflection +inset: fine scan of NSMO (004) of NS-R sample showing double peaks – P1 and P2 (* - STO +peaks) +Therefore, NS-R is highly oriented and more crystalline, which can be attributed to its +epitaxial nature of growth. Additionally, the HRXRD scan of NS-R thin films after ex-situ +annealing shows a doublet feature at its (004) reflection. A high-resolution fine scan was +performed on the NS-R thin film to confirm the double peaks. Referring to the literature, we +found that a similar doublet feature has been reported due to strain relaxation in PSMO thin +films on STO substrate31. By fitting the peaks using the pseudo-Voigt function, as shown in +figure S1 of supplementary information, the peaks were de-convoluted to evaluate the out-of- +plane lattice parameter (tabulated in table T1 – supplementary information). The first peak was +at 2θ=24.88o with a c-lattice constant of 7.66 Å, and the second peak was at 2θ=24.97o with a +c-lattice constant of 7.63 Å. The reduction in the c-lattice constant of the second peak shows +that there is compression of the lattice along the c-axis because of the tensile strain experienced +by the thin film due to the substrate. Such a splitting in the peak was absent in films of thickness +< 80 nm, indicating that this double peak is due to partial strain relaxation in the thicker film +initiated by ex-situ annealing. +Thus, from the detailed XRD studies and discussions in the previous section, it is inferred +that difference in initial-growth mode, and subsequent ex-situ annealing has prominently tuned +the resulting surface morphology of the NSMO thin films. The granular thin film NS-G has +multiple orientations similar to a polycrystalline system, whereas NS-R shows improved +crystallinity and orientation mirroring the substrate. The parameters affecting the initial growth +are discussed in the upcoming section. + +22 +24 +26 +28 +10 +-1 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +10 +8 +22 +24 +26 +28 +10 +-1 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +10 +8 +b) +Intensity (arb. units) +2(deg) +Ex-situ annealed + In-situ annealed +* (200) +(004) +NS- R +P1 +a) +NS- G +2(deg) +Intensity (arb. units) + Ex-situ annealed + In-situ annealed +* (200) +(004) +24.8 +25.0 +25.2 +1000 +10000 +P2 +2(deg) + +9 + +4. Effect of PLD parameters in tuning the morphology: +PLD Parameters like laser energy density, oxygen partial pressure, and substrate temperature +highly influence the type of growth. Changes in these parameters lead to variations in the +energy of the ad-atoms deposited on the substrate. To understand the role of O2 partial pressure +and laser energy density during the deposition, we have prepared NSMO thin films by varying +these parameters. Post deposition, the films were in-situ annealed at 750 oC for 2h in an oxygen +background pressure of 1 bar. Ex-situ annealing was carried out subsequently. +Figure 6 presents the morphology of films deposited under different laser energy +densities varied from 1 to 1.75 J/cm2. During deposition, the oxygen partial pressure and +substrate temperature were maintained at 0.36 mbar and 750 oC. Figure 7 presents the +morphology of films obtained at different oxygen partial pressure of 0.3mbar, 0.4 mbar, and +0.5 mbar while the laser energy density and substrate temperature were maintained at 1 J/cm2 +and 750 oC during deposition, respectively. +We found that changes in oxygen partial pressure and laser energy density did not +influence the surface morphology, as both type of morphologies have been observed in +different deposition runs with the same parameters. Further, as we have obtained granular and +rod-type films for the same substrate temperature of 750 oC, the role of substrate temperature +is also ruled out. Thus, irrespective of changes in the parameters mentioned above, thin films +of either granular or crossed-rod nanostructure were obtained. Therefore we suspect the + +a) +b) +c) +d) +e) +f) +1 J/cm2 +1 J/cm2 +1.5 J/cm2 +1.5 J/cm2 +1.75 J/cm2 +1.75 J/cm2 +Figure 6: The SEM images of NSMO thin films with granular morphology (a), (b), and (c) and +rod morphology from (d), (e), and (f) obtained at corresponding laser energy density- 1 J/cm2, +1.5 J/cm2, and 1.75 J/cm2. + +200 nm200nm200nm200nm200nm200nm10 + +substrate and the strain it offers to plays a vital role in altering the growth mode of the thin +film. +5. Effect of miscut angle in tuning the morphology: +The commercial STO substrates used here are one-sided polished, and their surface was +found to have a miscut. In commercially purchased wafers, the occurrence of a miscut in the +range of 0.05o-0.3o is well known and unavoidable due to mechanical cutting and polishing of +single crystal STO wafers14,32. In Figure 7(a), the as-received STO substrate, after cleaning, +shows clear terrace features in the AFM scan, confirming the presence of miscut on the +substrate surface. In a given wafer, the miscut can be in-plane or out-of-plane or both (some +works refer to this as miscut directions φ and θ instead of in-plane and out-of-plane, +respectively). The miscut angle and direction can alter the growth mode as the lattice strain is +anisotropic along the substrate surface and step edges6, thus resulting in different surface +morphology by forming anisotropic structural domains33. Several works are available in +literature 33–35 on the growth of manganite thin film on STO substrate with miscut. These +reports claim that the value of the miscut angle and appropriate adjustments in growth +conditions can control the number of structural domains in the thin film. As we have already +ruled out the possibility of growth conditions influencing the resulting morphology, we tried +to evaluate the value of miscut present in our STO substrates to see if it has affected the +resulting morphology. +a) +b) +c) +d) +e) +f) + +0.3 mbar +0.4 mbar +0.5 mbar +0.5 mbar +0.4 mbar +0.3 mbar +Figure 7: The SEM images of NSMO thin films prepared at oxygen partial pressure of 0.3 mbar, +0.4 mbar, and 0.5 mbar. a), b), and c) are granular NSMO thin, and films with rod morphology +are shown in d), e), and f) at corresponding oxygen partial pressure. + +200 nm200nm200nm200 nm200 nm200 nm11 + +To determine the value of miscut present in the substrates, we have followed the XRD- +protocol from literature36. This was carried out in a BRUKER D8, Lab source XRD setup. +According to the protocol, a low incident angle (~0.2o) rocking-scan was initially performed to +ensure that the sample was aligned with the X-ray. This was done to optimize the angle of the +sample holder, and the offset in the 2θ value (~0.4o) was noted as ζ. Following that, a rocking +scan was performed around the (200) peak of STO (46.483o), and phi & chi scans were done +to orient the wafer. Further, the rocking scan around the (200) peak of STO was repeated, fixing +the X-ray tube position. Finally, a detector scan was performed around the (200) peak of STO +and this time the offset in 2θ was noted as ζ’. The difference δζ, between ζ and ζ’, gives the +estimate of miscut. Next, the sample was rotated by 90o, and the scans mentioned above are +repeated in the same order. The difference between the offsets obtained this time was denoted +as δξ. Finally, the out-of-plane miscut angle was evaluated using equation (1). After +determining miscut on various STO wafers, we found that the value out of plane miscut angle +varies from 0.13o up to 0.48o. +𝜃𝑜𝑢𝑡−𝑜𝑓−𝑝𝑙𝑎𝑛𝑒 = 𝑎𝑟𝑐𝑡𝑎𝑛 √tan2(𝛿𝜁) + tan2(𝛿𝜉) (1) +Table 1 : This table illustrates the morphology of NSMO thin films obtained on STO substrates +with different values of miscut. + +Sample +Miscut +angle +Granular +Morphology +Sample +Miscut +angle +Rod type +Morphology +NS-G +0.48 o + +NS-R +0.31 o + +G1 +0.31 o + +R1 +0.19 o + +G2 +0.30 o + +R2 +0.25 o + +Table T1 : This table illustrates the morphology of NSMO thin films obtained on STO +substrates with different values of miscut. + +100nm100nm100nm100 nm100nm100nm12 + + +We see from table T1 that, both granular and rod-type morphology was observed on substrates +with miscut angle varying from 0.13o up to 0.48o. Sample G1 with granular morphology and +NS-R with rod-type morphology, possess the same miscut angle of ~ 0.3o. This is very +interesting, as the value of the miscut angle has not influenced the altered growth modes present +in our samples. Therefore to comprehend the resulting morphology, we have further +investigated the type of growth occurring on the terraced surface. +6. Thin film growth on the terraced surface: +A miscut on the substrate is useful for epitaxial thin films37 as the steps and terrace +edges act as nucleation centres and result in a step flow growth mode38. But the actual processes +governing the step-flow growth are more complex. The basic parameters driving this type of +growth are the coefficient of diffusion and the height of the Ehrlich-Schwoebel (ES) barrier39. +The diffusion of the adatoms on the surface and their incorporation into the crystal structure +govern the formation of different morphologies at the surface. Additionally, the ES barrier at +the terrace/step edges introduces an asymmetry in the potential energy at the edge. An adatom, +reaching the terrace, either nucleates or descends into the step depending on the ES barrier +height. Similarly, an adatom reaching below the step experiences an inverse step barrier which +prevents the particles from attaching to the step from below. +If the barrier height is appropriate, ad-atoms can properly attach themselves to the step +edges resulting in a step flow growth. However, the existence of the barrier makes the growth +on the stepped surfaces highly unstable resulting in modified surface features such as step +meandering, nano-columns/wire formation, spirals/mound formations, and faceted pits. In a +recent work by Magdalena et al.40, a simulation using the Cellular Automaton model in (2+1)D +gave rise to different patterns of surface morphology on vicinal surfaces. According to the +simulation, different processes occurred depending on the values assigned to the barrier height +at step edges. The adatom could either attach to the step to build the crystal by jumping/ +descending at the step edge or scatter away from the barrier resulting in the formation of +islands. For a fixed adatom flux, diffusion of adatoms takes place on the vicinal surface, and +probabilities are assigned for each of the processes mentioned above. Depending on the +probability value, various surface patterns were simulated for three cases. In case (i), for a high +ES barrier, the three-dimensional surface formation resulted in square/rectangular islands +following the cubic lattice symmetry at the middle of the terraces. In case (ii), with a reduced + +13 + +ES barrier height, more atoms were trapped at the top of the step, and a new pattern of +nanocolumns emerged consisting of cubic formations with deep narrow cubic pits. Finally, in +case (iii), when the height of the barrier was adjusted such that the probability of the adatoms +descending the step is equal/of the same order as the probability of the adatoms jumping up to +the step from below, it resulted with nano-wire or a columnar growth. Further, the presence of +additional local sinks that alters the potential barrier also resulted in nano-columns/islands at +random positions. +Thus, we can understand that our resulting granular morphology on the miscut STO +substrate is precisely similar to the surface morphology resulting from the case (iii). In the STO +susbtrate, the presence of disoriented terraces and improperly removed SRO terminations may +have altered the ES barrier resulting in local sinks at the substrate surface, thus resulting in the +island/columnar growth. Finally, the surface morphology of the NS-R thin film resembles the + +As received substrate +after cleaning +After treatment for +TiO2 termination + a) + +b) + + c) + + + + + + + + + +d) +Figure 8: AFM scan of the STO substrate a) as-received commercial substrate after +cleaning b) the same substrate after TiO2 termination obtained after heat treatment method +with a step height of ~ 0.4 Å (one-unit cell height of STO). c), d) NSMO thin films grown on +the corresponding substrates + +10-3nm +400 +um +um +0 +1.0 +1.0 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +010-3nm +600 +0 +0 +0.2 +0.2 +0.4 +0.4 +0.6 +0.6 +0.8 +0.8 +1.0 +1.0 +um +wn100 nm100 nm14 + +morphology they obtained in case (ii). This fact can be verified from a close inspection of the +surface of NS-R at high magnification in Figure 9(a). The surface morphology clearly shows +layer-by-layer growth with squared pits. +Further, in attempting to reduce the local sinks, NSMO thin films were synthesized on +pure TiO2 terminated substrates. The substrates are treated with DI water and then annealed at +high temperatures according to the protocol for TiO2-termination41. The treatment produced +clear step and terrace characteristics in the substrate, as observed in the AFM scan shown in +Figure 8(b). NSMO thin films were deposited on these substrates and, subsequently, ex-situ +annealed. The SEM imaging revealed that they exhibited similar rod-type morphology where +the rods are self-aligned and crossed at right angles embedded in a matrix of NSMO with +rectangular features, shown in figure 8(d). This procedure was repeated on several TiO2 - +terminated STO substrates, and we could reproduce the same morphology. This is because the +complete removal of SrO assures the absence of local sinks and suppresses the island/columnar +growth. However, rods in the thin film are believed to arise from droplets deposited due to high +laser energy density (1.75 J/cm2). This is verified in the SEM images of in-situ annealed +NSMO thin film shown in Figure 4(b), where the droplets are elongated into rods upon ex-situ +annealing. +Figure 9: SEM images of NSMO thin films under high magnification. a) NSMO thin film grown +on as-received, cleaned STO substrate deposited b) NSMO thin film grown with laser fluence +on TiO2 terminated STO substrate +Lastly, to obtain smoother films, we have synthesized NSMO thin films on fully TiO2 +terminated STO substrate at low laser energy density (1 J/cm2), reducing droplets' density. As +expected, we obtained thin films with reduced density of rods with the same type of +morphology. The SEM image of the film is shown in Figure 9(b), free of nano-rods. The films +have rectangular faceted pits, and layer-by-layer growth is evident through the holes. +a) +b) + + +100 nm100 nm15 + +Therefore the ES barrier plays a significant role in vicinal surfaces and can result in the +spontaneous ordering of adatoms resulting in unique surface nanostructures. Thus we +emphasize that when films are grown on a commercial substrate, the resulting morphology can +be either granular or rod-type depending on the potential energy landscape that depends upon +a wide range of parameters, including the size, shape of terraces, and type of terminations +present at the substrate. +7. Electrical-transport measurements: +The nanostructure plays a vital role in the transport behaviour of a manganite thin film +system30. To understand the transport behaviour of the nanostructured NSMO thin films, the +resistivity measurements are carried out using the standard 4-probe geometry 42 and plotted as +a function of temperature in Figure 10. It is observed that the granular film NS-G has higher +resistivity as compared to NS-R. Both thin films, NS-G and NS-R, exhibit the insulator-to- +metal transition (MIT), and the transition temperature TMIT is found to be 147 K for sample N- +G and 135 K for NS-R. The transition into the metallic regime is sharper in the case of NS-R +compared to NS-G thin film. The electrical transport behaviour has been analysed using +different theoretical models and fitted in the corresponding temperature regimes. The best fit +in each region is chosen based on the reduced 2 value. +𝜌 (𝑇) = 𝜌𝑅 𝑇𝑒𝑥𝑝 (𝐸𝑎 𝜅𝐵𝑇 +⁄ +) …(2) + 𝜌 (𝑇) = 𝜌0 𝑒𝑥𝑝 (𝑇0 𝑇 +⁄ ) +1 4 +⁄ + …(3) + 𝐸ℎ𝑜𝑝𝑝𝑖𝑛𝑔 = +𝜅𝐵 𝑇𝑜 +1 4 +⁄ 𝑇 +3 4 +⁄ +4 + …(4) +The high-temperature insulating phase is studied using the small polaron hopping +(SPH) model and the variable range hopping (VRH) mechanism given by equations (2) and +(3), and hopping energy is calculated from equation (4) 42,43. The VRH model better fits the +high-temperature region (≈ 195 K to 300 K) for both films. The hopping energy in the case of +NS-G is 128 meV and 125 meV for NS-R, in agreement with the order of value reported for +manganite thin films (~100 meV) 43,44. The resistivity in the metallic region below TMIT is +generally fitted with an empirical equation (5). At low temperatures, in addition to the +temperature-independent scattering effects from defects and grain boundaries (GBs) (ρo), etc., + +16 + +scattering effects due to electron-electron (ρ2), electron-magnon (ρ4.5) and electron-phonon (ρP) +dominate along with the strong correlation effects (ρ0.5) 45. +A low-temperature resistive upturn is observed below 50 K in both films. In figure S3 +of supplementary information, the resistive upturn in the low-temperature region from 4 K up +to 60 K is fitted using equation (6), which considers all the scattering mechanisms mentioned +above. An enhanced resistive upturn is observed at low temperatures in NS-G. This is due to +the enhanced GB-scattering effect and the contribution from other scattering mechanisms at +low-temperature. The contributions from different scattering mechanisms are analysed, and the +values are tabulated in supplementary information Table T2. +The intermediate temperature regime from 90 K to 134 K in the ferromagnetic-metallic +state is fitted using the equation (7). The addition of the polaronic term to the resistivity gives +a better fitting in this region as theoretical models claim the formation of polaron near the +MIT46. +𝜌 (𝑇) = 𝜌𝑜 + 𝜌𝑚𝑇𝑚 (5) + +0 +50 +100 +150 +200 +250 +300 +0 +5 +10 +15 +20 +25 +30 +0 +100 +200 +300 +0.0 +1.5 +3.0 +4.5 +0 +100 +200 +300 +0.0 +0.2 +0.4 +0.6 +46 +.0 + + + +T + + +Temperature (K) + NS-G + NS-R + Linear fit from + 110K to 125K- NS-G + Linear fit from + 110K to 125K- NS-R +33 +.1 + + + +T + +(b) + Resistivity - NS -G + VRH fit + FM-metallic fit + Low-temp uptrun +TIMT  147 K +(cm) +Temperature (K) +(a) +(c) + Resistivity - NS-R + VRH fit + FM-metallic fit + Low-temp uptrun +TIMT  135 K +(cm) +Temperature (K) + Figure 10: a), b) Resistivity vs. temperature curve of NSMO thin films – NS-G and NS-R +showing insulator to metal transition with decreasing temperature and fitted according to +theoretical models in different temperature regimes c) Normalized resistivity plot of NS-G and +NS-R thin film. Inset: Plot of variation of TCR with respect to temperature. + +17 + +An interesting feature is observed in the resistivity plots of the NS-G and NS-R thin +films apart from the low-temperature resistive upturn. In Figure 10(c), the resistivity of both +the thin films (NS-G and NS-R) has been normalized with their resistivity at 300 K, and a linear +fitting in the metallic region below TMIT (110 K to 125 K) is carried out to determine the slope. +The resistivity slope of samples with rod morphology differs from samples with granular +morphology up to an order of magnitude. The increase in slope value below the transition +temperature indicates the sharpness of the resistive transition for the samples with rod +morphology. This characteristic increase in slope up to an order is evident in all our thin films +with rod-type morphology (see supplementary figure S2). To characterize the sensitivity of +resistance with respect to changes in temperature, the temperature-coefficient of resistance +(TCR) has been evaluated using equation (7). It was found that NS-R has a higher value of +TCR %, ~ 12 %, compared to NS-G with TCR %, ~ 7 %. Additionally, the samples with rod +morphology were found to have enhanced TCR% (supplementary information figure S2). To +comprehend this result, we discuss the effect of GBs on the conduction mechanism. +The manganite system undergoes a disorder-induced phase transition from PM to FM +state with decreasing temperature21. Due to phase co-existence during the transition, the +conduction channel is presumed to have filamentary FM paths in the PM matrix 47. Conduction +takes place through the percolation of current across the well-connected FM regions. In +addition to the FM filamentary path, the GBs also play a significant role in the conduction +mechanism. We refer to Verutruyen et al.’s 48 work which explores the effect of a single GB +in the La-Ca-Mn-O (LCMO) system. They showed that the resistivity falls sharply at the +transition temperature when measured on a single grain of LCMO (free of GBs). However, +when measured across a single GB, the resistivity initially decreased, followed by a broad +resistive feature near the transition temperature. Thus, in a granular system, though the +conduction takes place through the percolation paths of well-connected FM regions, the GBs +cause increased resistivity due to increased spin-dependent scattering across the GB47. The +above explanation is consistent with our results, where the thin film with granular morphology +(NS-G) shows a broad resistive transition below the transition temperature with reduced TCR +𝜌 (𝑇) = 𝜌𝑜 + 𝜌2𝑇2 + 𝜌4.5𝑇4.5 + 𝜌𝑃𝑇5 + 𝜌0.5𝑇0.5 (6) +𝜌 (𝑇) = 𝜌𝑜 + 𝜌2𝑇2 + 𝜌4.5𝑇4.5 + 𝜌𝑃𝑇5 + 𝜌0.5𝑇0.5 + 𝜌7.5𝑇7.5 (7) +𝑇𝐶𝑅 % = +1 +𝜌 ( +𝑑𝜌 +𝑑𝑇) x 100 (8) + +18 + +%. If the connectivity is enhanced between the grains, a sharper decrease in the resistivity can +occur in the metallic regime. Remarkably, we observe that all of our thin films with rod- +morphology show sharp resistive transition near MIT irrespective of the thickness of the film. +Thus, this nanostructure aids improved conduction in the FM metallic phase, leading to the +sharp resistive transition with enhanced TCR % comparable to that of a highly-crystalline +system. Attempts to enhance the TCR % have been carried out by doping with elements such +as Ag, as high TCR % is required for applications in sensors and infrared detectors49,50. These +elements precipitate as nanocomposite in the manganite system and improve the conductivity, +leading to a sharper resistive transition. However, in our study, we have substantiated that the +enhancement of TCR % is possible with proper tuning of the nanostructured morphology of +thin films. +CONCLUSION: +In conclusion, the PLD-grown NSMO thin films were observed to have two prominent +surface morphologies – granular and crossed-nano rods. The metal-to-insulator transition +(MIT) temperature, TMIT, was found to be 147 K for a granular NSMO (NS-G) thin film and +135 K for a thin film with crossed-rod morphology (NS-R). The nature of the resistive +transition is broad in the former, whereas the latter exhibits a sharp MIT feature. The +temperature coefficient of resistance (TCR) was evaluated, and NS-R thin film has a higher +value of TCR %, ~ 12 %, compared to NS-G with TCR % ~ 7 %. Additionally, we have +observed that all the films with rod-type morphology exhibit a significant enhancement in +TCR% up to one order of magnitude compared to the granular thin film. Thus, we have +demonstrated that TCR % can be enhanced with proper tuning of the nanostructures in thin +films, which is relevant for technological applications. The reason for such nano-structuring is +explored in great detail. It was found that parameters like laser energy density, O2 partial +pressure, and the substrate miscut angle had minimal effect. At the same time, the difference +in the potential landscape of the Ehrlich-Schwoebel (ES) barrier is believed to play a vital role +in the growth dynamics of the films. Films grown with reduced laser energy density (1 J/cm2) +on the TiO2 terminated substrates exhibited highly reproducible layer-by-layer growth. This +substantiates the presence of reduced local sinks and ES barrier height, resulting in epitaxial +growth of NSMO thin films. Therefore, a fine-tuning of a wide range of parameters, including +strain and surface terminations, is required to obtain a fine control of the ES barrier that +influences the growth process of thin films. This paves the way for investigation into the role +of the ES barrier in manganite thin film growth. Using RHEED and in-situ STM techniques, a + +19 + +few groups have already attempted to experimentally determine the value of the ES barrier on +SrTiO3 substrates for the growth of La-Ca-Mn-O manganite system51. It would be interesting +to explore the relationship between the value of the ES-barrier and the type of morphology +experimentally in the future. + +Author contributions +The division of work is as follows: NSMO thin film samples were prepared by R.S.M. SEM +imaging was carried out by S.A, J.P. AFM measurements were carried out by K.G. XRD +measurements were carried out by R.M.S, P.N.R, PG, S.K.R. Magneto-transport measurements +were carried out by R.S.M and E.P.A. Analysis were done by R.S.M, E.P.A, and S.A. Writing +was carried out by R.S.M, and all authors discussed the results and commented on the +manuscript. E.P.A., T.G.K and A.M. supervised this research work. + +Conflict of interest: +The authors declare no conflict of interest. +Acknowledgments +One of the authors (R S Mrinaleni) would like to acknowledge the Department of Atomic +Energy, India for the provision of experimental facilities. We thank UGC-DAE CSR, +Kalpakkam node, for providing access to magnetic and magnetotransport measurement +systems. The authors are grateful to RRCAT, Indore, for beam line facilities. +Funding statement: +One of the authors (R S Mrinaleni) would like to acknowledge the funding support from the +Department of Atomic Energy, India. + +References: +1. +E. Dagotto. Nanoscale phase seperation and CMR. +2. +Tokura, Y. Critical features of colossal magnetoresistive manganites. Reports Prog. +Phys. 69, 797–851 (2006). +3. +Ebata, K. et al. Chemical potential shift induced by double-exchange and polaronic +effects in Nd1-x Srx Mn O3. Phys. Rev. B - Condens. Matter Mater. Phys. 77, (2008). + +20 + +4. +Haghiri-Gosnet, A. M. & Renard, J. P. CMR manganites: Physics, thin films and +devices. J. Phys. D. Appl. Phys. 36, (2003). +5. +Tokura, Y. & Tomioka, Y. Colossal magnetoresistive manganites. J. Magn. Magn. +Mater. 200, 1–23 (1999). +6. +Perna, P. et al. Tailoring magnetic anisotropy in epitaxial half metallic +La0.7Sr0.3MnO3 thin films. J. Appl. Phys. 110, 013919 (2011). +7. +Song, C. et al. Emergent perpendicular magnetic anisotropy at the interface of an oxide +heterostructure. Phys. Rev. B 104, (2021). +8. +Li, X., Lindfors-Vrejoiu, I., Ziese, M., Gloter, A. & van Aken, P. A. Impact of +interfacial coupling of oxygen octahedra on ferromagnetic order in +La0.7Sr0.3MnO3/SrTiO3 heterostructures. Sci. Rep. 7, 40068 (2017). +9. +Liu, Q. et al. Perpendicular Manganite Magnetic Tunnel Junctions Induced by +Interfacial Coupling. ACS Appl. Mater. Interfaces 14, 13883–13890 (2022). +10. +Liu, Q. et al. Perpendicular Manganite Magnetic Tunnel Junctions Induced by +Interfacial Coupling. ACS Appl. Mater. Interfaces 14, 13883–13890 (2022). +11. +Chi, X. et al. Enhanced Tunneling Magnetoresistance Effect via Ferroelectric Control +of Interface Electronic/Magnetic Reconstructions. ACS Appl. Mater. Interfaces 13, +56638–56644 (2021). +12. +Gajek, M. et al. Tunnel junctions with multiferroic barriers. Nat. Mater. 6, 296–302 +(2007). +13. +Kim, D. H., Ning, S. & Ross, C. A. Self-assembled multiferroic perovskite–spinel +nanocomposite thin films: epitaxial growth, templating and integration on silicon. J. +Mater. Chem. C 7, 9128–9148 (2019). +14. +Sánchez, F., Ocal, C. & Fontcuberta, J. Tailored surfaces of perovskite oxide +substrates for conducted growth of thin films. Chemical Society Reviews vol. 43 2272– +2285 (2014). +15. +Ning, X., Wang, Z. & Zhang, Z. Large, Temperature-Tunable Low-Field +Magnetoresistance in La0.7Sr0.3MnO3:NiO Nanocomposite Films Modulated by +Microstructures. Adv. Funct. Mater. 24, 5393–5401 (2014). + +21 + +16. +Zhang, W., Ramesh, R., MacManus-Driscoll, J. L. & Wang, H. Multifunctional, self- +assembled oxide nanocomposite thin films and devices. MRS Bull. 40, 736–745 +(2015). +17. +Chen, A., Bi, Z., Jia, Q., MacManus-Driscoll, J. L. & Wang, H. Microstructure, +vertical strain control and tunable functionalities in self-assembled, vertically aligned +nanocomposite thin films. Acta Mater. 61, 2783–2792 (2013). +18. +Zhang, C. et al. Large Low-Field Magnetoresistance (LFMR) Effect in Free-Standing +La0.7Sr0.3MnO3 Films. ACS Appl. Mater. Interfaces 13, 28442–28450 (2021). +19. +Huang, J. et al. Exchange Bias in a La0.67Sr0.33MnO3/NiO Heterointerface +Integrated on a Flexible Mica Substrate. ACS Appl. Mater. Interfaces 12, 39920–39925 +(2020). +20. +Qin, Q. et al. Interfacial antiferromagnetic coupling between SrRu O3 and L a0.7 S +r0.3Mn O3 with orthogonal easy axis. Phys. Rev. Mater. 2, 104405 (2018). +21. +Dagotto, E., Hotta, T. & Moreo, A. Colossal magnetoresistant materials: the key role +of phase separation. Phys. Rep. 344, 1–153 (2001). +22. +Krivoruchko, V. N. The Griffiths phase and the metal-insulator transition in substituted +manganites (Review Article). Low Temperature Physics vol. 40 586–599 (2014). +23. +Bhat, S. G. & Kumar, P. S. A. Tuning the Curie temperature of epitaxial +Nd0.6Sr0.4MnO3 thin films. J. Magn. Magn. Mater. 448, 378–386 (2018). +24. +Kumari, S. et al. Effects of Oxygen Modification on the Structural and Magnetic +Properties of Highly Epitaxial La0.7Sr0.3MnO3 (LSMO) thin films. Sci. Rep. 10, +(2020). +25. +Wang, H. S., Li, Q., Liu, K. & Chien, C. L. Low-field magnetoresistance anisotropy in +ultrathin Pr0.67Sr0.33MnO3 films grown on different substrates. Appl. Phys. Lett. 74, +2212–2214 (1999). +26. +Huang, J., Wang, H., Sun, X., Zhang, X. & Wang, H. Multifunctional La 0.67 Sr 0.33 +MnO 3 (LSMO) Thin Films Integrated on Mica Substrates toward Flexible Spintronics +and Electronics. ACS Appl. Mater. Interfaces 10, 42698–42705 (2018). +27. +Boileau, A. et al. Textured Manganite Films Anywhere. ACS Appl. Mater. Interfaces + +22 + +11, 37302–37312 (2019). +28. +Greculeasa, S. G. et al. Influence of Thickness on the Magnetic and Magnetotransport +Properties of Epitaxial La0.7Sr0.3MnO3 Films Deposited on STO (0 0 1). +Nanomaterials vol. 11 (2021). +29. +Gupta, P. et al. BL-02: A versatile X-ray scattering and diffraction beamline for +engineering applications at Indus-2 synchrotron source. J. Synchrotron Radiat. 28, +1193–1201 (2021). +30. +Arun, B., Suneesh, M. V. & Vasundhara, M. Comparative Study of Magnetic Ordering +and Electrical Transport in Bulk and Nano-Grained Nd0.67Sr0.33MnO3 Manganites. +J. Magn. Magn. Mater. 418, 265–272 (2016). +31. +Zhang, B. et al. Effects of strain relaxation in Pr0.67Sr0.33MnO3 films probed by +polarization dependent X-ray absorption near edge structure. Sci. Rep. 6, 19886 +(2016). +32. +Pai, Y. Y., Tylan-Tyler, A., Irvin, P. & Levy, J. Physics of SrTiO3-based +heterostructures and nanostructures: A review. Reports on Progress in Physics vol. 81 +036503 (2018). +33. +Paudel, B. et al. Anisotropic domains and antiferrodistortive-transition controlled +magnetization in epitaxial manganite films on vicinal SrTiO3 substrates. Appl. Phys. +Lett. 117, (2020). +34. +Boschker, J. E. et al. In-plane structural order of domain engineered La0.7Sr 0.3MnO3 +thin films. Philos. Mag. 93, 1549–1562 (2013). +35. +Konstantinović, Z., Sandiumenge, F., Santiso, J., Balcells, L. & Martínez, B. Self- +assembled pit arrays as templates for the integration of Au nanocrystals in oxide +surfaces. Nanoscale 5, 1001–1008 (2013). +36. +Wang, J. et al. Quick determination of included angles distribution for miscut +substrate. Meas. J. Int. Meas. Confed. 89, 300–304 (2016). +37. +Scheel, H. J. Control of Epitaxial Growth Modes for High‐Performance Devices. +Cryst. growth Technol. 621–644 (2003). +38. +Chae, R. H., Rao, R. A., Gan, Q. & Eom, C. B. Initial Stage Nucleation and Growth of + +23 + +Epitaxial SrRuO3 Thin Films on (0 0 1) SrTiO3 Substrates. J. Electroceramics 4, 345– +349 (2000). +39. +Schwoebel, R. L. & Shipsey, E. J. Step Motion on Crystal Surfaces. J. Appl. Phys. 37, +3682–3686 (1966). +40. +Załuska-Kotur, Magdalena, Hristina Popova, and V. T. Step Bunches, Nanowires and +Other Vicinal “Creatures”—Ehrlich–Schwoebel Effect by Cellular Automata. Crystals +11, 1135 (2021). +41. +Connell, J. G., Isaac, B. J., Ekanayake, G. B., Strachan, D. R. & Seo, S. S. A. +Preparation of atomically flat SrTiO3 surfaces using a deionized-water leaching and +thermal annealing procedure. Appl. Phys. Lett. 101, (2012). +42. +Miccoli, I., Edler, F., Pfnür, H. & Tegenkamp, C. The 100th anniversary of the four- +point probe technique: the role of probe geometries in isotropic and anisotropic +systems. J. Phys. Condens. Matter 27, 223201 (2015). +43. +Gopalarao, T. R., Ravi, S. & Pamu, D. Electrical transport and magnetic properties of +epitaxial Nd0.7Sr0.3MnO3 thin films on (001)-oriented LaAlO3 substrate. J. Magn. +Magn. Mater. 409, 148–154 (2016). +44. +Gopalarao, T. R. & Ravi, S. Study of Electrical Transport and Magnetic Properties of +Nd0.7Sr0.3MnO3/Nd0.8Na0.2MnO3 Bilayer Thin Films. J. Supercond. Nov. Magn. +31, 1149–1154 (2018). +45. +Arun, B., Suneesh, M. V & Vasundhara, M. Comparative Study of Magnetic Ordering +and Electrical Transport in Bulk and Nano-Grained Nd0.67Sr0.33MnO3 Manganites. +J. Magn. Magn. Mater. 418, 265–272 (2016). +46. +Sudakshina, B., Supin, K. K. & Vasundhara, M. Effects of Nd-deficiency in +Nd0.67Ba0.33MnO3 manganites on structural, magnetic and electrical transport +properties. J. Magn. Magn. Mater. 542, 168595 (2022). +47. +de Andrés, A., García-Hernández, M. & Martínez, J. L. Conduction channels and +magnetoresistance in polycrystalline manganites. Phys. Rev. B 60, 7328–7334 (1999). +48. +Vertruyen, B. et al. Magnetotransport properties of a single grain boundary in a bulk +La-Ca-Mn-O material. J. Appl. Phys. 90, 5692–5697 (2001). + +24 + +49. +Li, J. et al. Improvement of electrical and magnetic properties in +La0.67Ca0.33Mn0.97Co0.03O3 ceramic by Ag doping. Ceram. Int. (2022) +doi:10.1016/j.ceramint.2022.08.255. +50. +Jin, F. et al. La0.7Ca0.3MnO3-δ:Ag nanocomposite thin films with large temperature +coefficient of resistance (TCR). J. Mater. (2022) doi:10.1016/j.jmat.2022.01.010. +51. +Gianfrancesco, A. G., Tselev, A., Baddorf, A. P., Kalinin, S. V & Vasudevan, R. K. +The Ehrlich–Schwoebel barrier on an oxide surface: a combined Monte-Carlo and in +situ scanning tunneling microscopy approach. Nanotechnology 26, 455705 (2015). + +SUPPLEMENTARY INFORMATION + +I- +Deconvolution of NSMO (004) reflection: + +Figure S1: The double peak in the HR-XRD scan of NS-R thin film is confirmed by a HR-fine +scan. The individual peak positions are noted as the centre of the fitted peaks P1 and P2. +Table ST1: The table illustrates the values of c-lattice parameters evaluated from the (004) NSMO +reflection. +24.4 +24.6 +24.8 +25.0 +25.2 +25.4 +25.6 +25.8 +0.0 +5.0x10 +4 +1.0x10 +5 +1.5x10 +5 +2.0x10 +5 +Intensity (arb. units) +2(deg) + NS-R - fine scan + Fit Peak 1 + Fit Peak 2 + Cumulative Fit Peak +Sample +2θ for (004) reflection +(o deg) +Calculated c-lattice +parameter (Å) +NS-G +25.05o +7.61 +NS-R +24.88 o – P1 +24.97 o – P2 +7.66 +7.63 + +25 + +II- +Transport studies on NSMO thin films +Three samples with granular morphology G-A, G-B, G-C, and rod morphology R-A, R-B, R- +C, were selected and their resistivity was measured using 4-probe technique. The normalized- +resistivity plot for the selected NSMO thin films are shown Figure. 4. The value of resistivity +is different across the NSMO thin films, since they are deposited under slightly different PLD +conditions but all of them exhibited MIT. Observing the nature of MIT transition in these +selected samples, G-A, G-B, G-C with granular morphology have a broad resistive transition +below their MIT temperature. The samples R-A, R-B, R-C with rod-morphology show a sharp +resistive transition in the FM-metallic state below their MIT temperature. The value of slope is +evaluated from the linear fit in the metallic region and it shows that samples with rod-type +morphology have increased slope up to one order as compared to the granular samples. +Temperature coefficient of resistance (TCR) is evaluated for these films and it is found that +samples G-A, G-B, G-C have peak-TCR % of 5 %, 4 %, and 8 % at 105 K, 77 K, and 121 K, +respectively. An enhanced TCRpeak % is obtained for samples with rod-morphology. The +samples R-A, R-B, R-C have peak-TCR % of 21 %, 14.5 %, and 18 % at 98 K, 80 K, and 100 +K, respectively. + +26 + + +Figure S2: Plots of normalized resistivity vs. temperature of NSMO films with granular and +rod-type morphology. (a),(c),(e): Samples with granular morphology G-A, G-B, G-C. +(b),(d),(f): Samples with rod morphology R-A, R-B, R-C. A linear fit in the FM-metallic region +give the rate of change of resistivity with respect to temperature. + +III- Low-temperature studies on NSMO thin films – NS-G +and NS-R + +To study the low-temperature transport across the thin films with different morphology, the plot of low- +temperature resistivity of the granular thin film NS-G and rod-type thin film NS-R is shown in figure +S5. An enhanced low-temperature resistive upturn is observed in NS-G from figure. S5. Using the low- +temperature transport equation the resistivity data is fit and the fitting parameters are summarized in +table ST2. The first term, ρo which represents the contribution from grain-boundary (GB) scattering is +0 +50 +100 +150 +200 +250 +300 +0 +2 +4 +6 +8 +0 +50 +100 +150 +200 +250 +300 +0 +30 +60 +90 +0 +50 +100 +150 +200 +250 +300 +0 +2 +4 +6 +8 +10 +12 +14 +16 +0 +50 +100 +150 +200 +250 +300 +0 +30 +60 +90 +120 +0 +50 +100 +150 +200 +250 +300 +0 +2 +4 +6 +8 +10 +12 +14 +0 +50 +100 +150 +200 +250 +300 +0 +5 +10 +15 +20 +25 +15 +.0 + + + +T + + Sample: G - A + Linear fit in + FM-metallic region + +Temperature (K) +TCRpeak %  5 % + +TCRpeak %  21 % +74 +.5 + + + +T + + Sample: R - A + Linear fit in + FM-metallic region + +Temperature (K) +(b) +TCRpeak %  4 % +0.28 + + + +T + + +Temperature (K) + Sample: G - B + Linear fit in + FM-metallic region +(c) +TCRpeak %  14 % +76 +.6 + + + +T + + +Temperature (K) + Sample: R - B + Linear fit in + FM-metallic region +(d) +TCRpeak %  8 % +0.35 + + + +T + + Sample: G - C + Linear fit in + FM-metallic region + +Temperature (K) +(e) +TCRpeak %  18 % +20 +.1 + + + +T + + +Temperature (K) + Sample: R - C + Linear fit in + FM-metallic region +(a) +(f) + +27 + +found to be higher by more than one-order in NS-G as compared to NS-R. This is expected as NS-G +has a granular morphology and increased contribution from GB scattering affects the transport +mechanism even at low-temperatures. Additionally, ρo’s value is higher by orders of magnitude as +compared to the other coefficients. This shows that GB scattering effects dominate the transport +mechanism compared to other contributions to the electronic transport. + + +Figure S3: Low-temperature resistive up-turn is observed in the NSMO thin films NS-G and +NS-R. The temperature regime from 4 K up to 60K is fit using the low-temperature transport +equation. + + + +0 +10 +20 +30 +40 +50 +0.07 +0.08 +0.09 +0.10 +0 +10 +20 +30 +40 +50 +0.0020 +0.0025 +0.0030 +0.0035 +0.0040 + Resistivity - NS-G + Low-temp uptrun +TIMT  147 K +(cm) +Temperature (K) +TIMT  135 K + Resistivity - NS-R + Low-temp uptrun +(cm) +Temperature (K) +Sample +𝝆𝒐 +𝝆𝟐 +𝝆𝟒.𝟓 +𝝆𝑷 +𝝆𝟎.𝟓 +R2 (%) +NS-G +0.09272 +1.16E-5 +-1.00E-9 +1.33E-10 +-0.0038 +99.99 +NS-R +0.00305 +3.53E-7 +-2.90E-11 +4.90E-12 +-8.38E-5 +99.99 +Table ST2: The table illustrates the values of coefficients of low-temperature transport after +fitting. + diff --git a/9tE0T4oBgHgl3EQffwDm/content/tmp_files/2301.02410v1.pdf.txt b/9tE0T4oBgHgl3EQffwDm/content/tmp_files/2301.02410v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf054d6ef5be09e9c4b512445a617899a6f207e3 --- /dev/null +++ b/9tE0T4oBgHgl3EQffwDm/content/tmp_files/2301.02410v1.pdf.txt @@ -0,0 +1,1798 @@ +Codepod: A Namespace-Aware, Hierarchical Jupyter +for Interactive Development at Scale +Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian +Dept. of Computer Science, Iowa State University +Ames, Iowa, USA +{hebi,qxiao,jtian}@iastate.edu,forrest.bao@gmail.com +ABSTRACT +Jupyter is a browser-based interactive development environment +that has been popular recently. Jupyter models programs in code +blocks, and makes it easy to develop code blocks interactively by +running the code blocks and attaching rich media output. How- +ever, Jupyter provides no support for module systems and names- +paces. Code blocks are linear and live in the global namespace; +therefore, it is hard to develop large projects that require modular- +ization in Jupyter. As a result, large-code projects are still devel- +oped in traditional text files, and Jupyter is only used as a surface +presentation. We present Codepod, a namespace-aware Jupyter +that is suitable for interactive development at scale. Instead of +linear code blocks, Codepod models code blocks as hierarchical +code pods, and provides a simple yet powerful module system for +namespace-aware incremental evaluation. Codepod is open source +at https://github.com/codepod-io/codepod. +ACM Reference Format: +Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian. 2023. Codepod: A Namespace- +Aware, Hierarchical Jupyter for Interactive Development at Scale. In Pro- +ceedings of (Conference’23). ACM, New York, NY, USA, 10 pages. https: +//doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +Traditional software development is typically closely tied with file +systems. Developers write code into a set of files in the file-system +hierarchy. For example, developers write functions in files using +a text editor and invoke a compiler or an interpreter to run or +evaluate the code in the files. Modern Integrated Development +Environments (IDEs) provide a file system browser and integrate +debuggers to help run and debug over the files. +Jupyter notebook [6] is a browser-based interactive development +environment that has been widely adopted by many different com- +munities, both in science and industry. Jupyter notebooks support +literate programming that combines code, text, and execution re- +sults with rich media visualizations. Juyter models the code as a +sequence of "code cells". This provides a clean separation between +code blocks, whereas text editors do not partition code in the same +text file but instead relying on developers and editor plugins to do +so. Code cells can be interactively (re)-run and display results in rich +media such as data visualization right beside the cell, providing de- +velopers an interactive Read-Eval-Print-Loop (REPL) development +experience. Jupyter has been popular recently in software devel- +opment [10–12, 15], proving such interactive cycle is beneficial to +software development. +Conference’23, Jan, 2023, Ames, IA, USA +2023. ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +However, Jupyter falls short for module systems and namespaces. +Code blocks in Jupyter notebooks are linear and live in the global +namespace, making it non-scalable for large software projects of +hundreds of function definitions with potential naming conflicts. +As a result, large code projects are still developed in traditional +text files, and Jupyter is primarily used as a surface presentation +of the projects, consisting of only a fraction of the entire codebase. +Our case study in Section 4.3 found that Jupyter notebook shares +less than 5% of the code of real-world open-source projects. All +functions defined in the Jupyter notebook are only accessed in the +same notebook. There are calls from Jupyter to the code in the text +files, but no calls from text files to Jupyter code. +The Jupyter-file hybrid development model has several disadvan- +tages. Changes in files are not in sync with the Jupyter runtime. This +effectively breaks the REPL interactive development functionality. +The hybrid model still relies on text editors, external debuggers, +and IDEs and thus still suffers from the drawbacks of file-based +software development, which we will detail below. +Although computers store information into files, organizing code +into text files where information is linearly presented is counter- +productive. Complex software requires proper abstraction and seg- +mentation of code, typically by defining functions and hierarchical +modules. For simplicity, in this paper, we assume functions are the +building blocks of software projects and refer to the functions when +we talk about “code blocks”. File-based approaches force developers +to maintain the correspondence between code and files, which differ +significantly in granularity: code blocks are small in size, but large +in amount, while files are typically long but few. The unbalance +in granularity poses dilemmas to developers: including too many +code blocks into one file makes the hierarchy hard to maintain, +while including few code blocks into one file creates many small +files and deep directories that are also hard to work with. Besides, +programming languages typically design module systems around +file systems, e.g., a file is a module. It becomes tedious to reference +and import from different modules scattered over multiple files +and levels of directories. This is the case in the real world. Among +highly regarded open source projects, each project contains tens to +hundreds of files, possibly with levels of different directories. For +a file containing tens of functions, about half of the functions are +internal to the file and are not called in other files. +To overcome the above disadvantages of both Jupyter and text- +file-based development, we propose Codepod, a namespace-aware +Jupyter for interactive software development at scale. Codepod +models a program as hierarchical code blocks and represents it +accordingly. Developers write each function as a code pod and +place it at an appropriate hierarchy. In Codepod, the code blocks are +organized into a tree of code pods and decks. A code pod resembles +arXiv:2301.02410v1 [cs.SE] 6 Jan 2023 + +Conference’23, Jan, 2023, Ames, IA, USA +Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian +a cell in Jupyter. The partition of pods is maintained by grouping +them into decks. A deck can also contain child decks. All code of +the entire project can be developed without needing files. +In addition, Codepod features a simple yet powerful module sys- +tem that abstracts over the native module system of programming +languages to provide a consistent and straightforward evaluation +model for different languages. Codepod’s module system consists +of five namespace rules, inspired by the hierarchical nature of code +blocks and the access pattern among them. (1) namespace separa- +tion by default: in Codepod, each deck is a namespace, and the root +deck is the global namespace. Pods in different decks are defined in +separate namespaces and cannot see each other; (2) public pods: a +pod can be marked as "public" and is made available to its parent +deck; (3) utility pods: A pod or deck in Codepod can be marked +as a “utility pod/deck”. Such a pod is visible to the parent deck +node’s sub-tree; (4) testing pods: a testing pod or deck can access +its parent deck’s namespace; and (5) explicit path: a pod is always +accessible by specifying the full path within the tree, providing the +compatibility for arbitrary imports. The detailed rationale of the +rules is discussed in Section 2. +Last but not least, Codepod provides a namespace-aware incre- +mental evaluation model. In Codepod, every pod can be executed, +and the evaluation happens in the appropriate namespace. Similar +to Jupyter notebooks, the results are displayed right beside the +code pod for easy debugging and intuitive interactive development. +When a pod is changed, the pod can be re-evaluated, and the up- +dated definition is applied incrementally in that scope in the active +runtime, and the new definition is visible to all other pods using it +in the entire codebase without restarting the current runtime. +We have implemented a fully working Codepod as a Web ap- +plication and currently have implemented full namespace-aware +runtime support for four language kernels: Python, JavaScript, Julia, +and Scheme/Racket. New kernels can be easily developed based +on existing Jupyter notebook kernels. Codepod is open-sourced at +https://example.com +In summary, we make the following contributions in this work: +• we propose Codepod, a novel namespace-aware interactive +development environment +• we propose a simple yet powerful module system abstraction +for Codepod +• we provide a fully working Codepod implementation with +namespace-awareness and incremental runtime support for +four programming languages, and make it open source +• we conduct case studies of real-world open-source projects +to statistically show that our Codepod model will be useful +for real-world development. +2 +HIERARCHICAL PODS +In this section, we introduce the Codepod model and its namespace +rules. In the next section, we describe the incremental evaluation +runtime and algorithms. +2.1 +Codepod Interface +In Codepod, code blocks are organized into a tree. In the tree, non- +leaf nodes are decks, and leaf nodes are pods. Thus a deck may +contain a list of pods and a list of sub-decks. A pod is a text editor +containing the real code, and a deck is the container of the pods. +We will use “node” to refer to a node in the tree, which can be either +a deck or a pod. +An overview demo of the Codepod interface is shown in Figure +1, implementing a simplified Python regular expression compiler. +The code is organized into a tree, which starts from the leftmost +ROOT node, and grows to the right. The background level of grey +of the deck indicates the level of the deck in the tree. +In order to define the interactions between code pods in the tree, +Codepod provides simple yet powerful namespace rules abstracting +different languages’ native module systems and providing a consis- +tent module system for all languages. In the following sections, we +introduce the rules in detail. We will revisit this overview exam- +ple in Section 2.7 for the meaning of different kinds of pods after +introducing the namespace rules. +A typical workflow using Codepod starts from an empty tree +of a single ROOT deck. Developers can create pods as a child of +the deck and start to develop in the global namespace. To develop +hierarchical modules, developers can create a deck under the ROOT +deck and create pods under the new deck. Pods and decks can be +moved from one node to another node in different levels to group +the pods and re-order the code hierarchy. Pods and decks can be +folded so that only the pods of interest are displayed during the +development. A pod can be evaluated, and the possibly rich media +result will be displayed under the pod. +2.2 +NS Rule 1: Namespace Separation +In Codepod, the code blocks are organized into a tree of decks and +pods. Each deck can contain multiple pods and child decks. A pod +contains the actual code, and a deck declares a namespace. A pod +belongs to the namespace of its parent deck. The first rule is the +basic namespace separation: pods in the same namespace are visible +to each other, but pods in different namespaces are not. For example, +in Fig. 2, there are 5 decks, and thus 5 namespaces. In Deck-2, there +are two pods defining functions a and b. Functions a and b can call +each other without a problem because they are in the same deck +and thus the same namespace. In all other four decks, the reference +to either a or b will throw errors because they belong to different +namespaces. +2.3 +NS Rule 2: Public Interface to Parent +In order to build up the software, we have to establish connections +between the definitions of code pods in different namespaces. Soft- +ware programs are often highly hierarchical: lower-level functions +are composed together to build higher-level functions. This is a +natural fit to the Codepod model, where code blocks are ordered +hierarchically. Thus in this rule, we allow public interfaces to be +exposed from child decks to parent decks. More specifically, each +pod can be marked as “public”. Such public pods are visible in the +parent deck of the current deck. For example, in Fig. 3, there are +3 decks, thus 3 namespaces. The three namespaces are composed +hierarchically; Deck-A is the parent of Deck-B, which is the parent +of Deck-C. In Deck-C, there are four pods, defining four functions +c1, c2, c3, and c4. Those functions can see each other because they +are in the same namespace. The pods for c1, c2, and c4 are marked +public (indicated by highlight), while c3 is not. In its parent deck + +Codepod: A Namespace-Aware, Hierarchical Jupyter +for Interactive Development at Scale +Conference’23, Jan, 2023, Ames, IA, USA +Figure 1: Codepod overview example for Regular Expression code. +Figure 2: NS Rule 1: separate namespace by default +Figure 3: NS Rule 2: export to parent namespace. Yellow +highlights indicate pods to be exported/exposed to parent +decks. +(Deck-B), the call of c1, c2, c4 is allowed, meaning that they are +available in this parent namespace. However, the usage of c3 will +raise an error because it is not exposed. +The public functions are exported only to the parent deck but +not to the child decks. For example, function b1 is defined in Deck- +B, and the pod is marked public. This function b1 is visible to its +parent deck, Deck-A, but not to its child deck, Deck-C. +Lastly, the public interface is only exposed to one level up the +hierarchy. If the names are desired to be visible further up, the +names can be further exposed up to the root deck. For example, +although c4 is marked as public, it is only visible to its immediate +parent deck, Deck-B. Calling c4 in the for pod for a2 in Deck-A will +raise an error as c4 is not visible in Deck-A. In the middle deck, +Deck-B, the functions c1 and c2 are re-exported to the parent deck, +and thus c1 and c2 are available in the top deck, Deck-A. +In summary, this “up-rule” allows users to mark a pod public +and expose it to one-level deck up, and can be re-exported to upper +levels explicitly until the root pod. This namespace rule closely +resembles the hierarchical nature of software and is natural to use +this to build up complex functionalities from the ground up. +2.4 +NS Rule 3: Utility Pods +Although exposing pods from child decks to parent decks is natural +for building software, it cannot cover all use-cases. One particular +access pattern is utility functions that are supposed to be called +in many other pods at different levels. This is commonly used in +real-world software. For example, many software projects will have +a utils folder that implements utility functions such as string +manipulation, general parsing, logging functions. Such utility func- +tions are used by other functions at different hierarchy levels. In +the Codepod hierarchy, pods for such functions need to be children +for all other pods calling the utility functions; thus, the model will +no longer be a tree but a graph. However, modeling code blocks as +graphs is not as scalable as trees, and too many utility pods will + +CPiwTe3yqmmC +sre parse +Pattern +parse +class Pattern(): +1 def parse(): +def closegroup(): pass +2 +_parse_sub() +3 +def opengroup(): pass +3 +isstring() +sre_compile +4 +Tokenizer.match() +SubPattern +class SubPattern(): +compile +_parse_sub +2 +def closegroup(): pass +1 def compile(): +1 +def +_parse_sub(): +m +def append(): pass +re +2 +_code() +2 +_parse() +4 +def getwidth(): pass +3 +parse() +3 +Tokenizer.match() +compile +4 +isstring() +Tokenizer +def re.compile(): +_parse +1 class Tokenizer(): +compile() +_code +def _parse(): +2 +def get(): pass +1 +def _code(): +2 +_parse_sub() +3 +def match(): pass +match +2 +_compile() +3 +_escape() +def match(): +3 +compile_info() +4 +Pattern.opengroup() +_compile().search() +CPQW6M4jqdqG ++Test +5 +SubPattern.getwidth() +_compile_info +Tokenizer.get() +1 p = Pattern() +search +1 def _compile_info(): +2 p.opengroup("a") +def search(): +2 +SubPattern.getwidth() +ROOT +escape +3 print(p) +2 +_compile().match() +1 def _escape(): +_compile +1 _parse(p,"hello abc") +Pattern.checkgroup() +_compile +1 def _compile(): +1 _escape(p,"a") +def +_compileO: +2 +_simple() +2 +sre_compile.compile() +3 +SubPattern +CPbxCz8GETTx +AUtility +_simple +1 def _simple(): +isstring +2 +SubPattern.getwidth() +1 def isstring(obj): +CPGGBRbcaBGb ++Test +2 +return isinstance( +1 print("Testing ..") +3 +obj,(str,bytes)) +2 isstring("abc") +isnumber +1 isstring(1) +def isnumber(obj): +2 +return isinstance( +3 +obj,(int, float))Deck-2 +Deck-4 +a +# ERROR not defined +1 +def a(): +2 a() +2 +# ok +3 +return b() +Deck-1 +Deck-5 +b +1 # ERROR not defined +1 +def b(): +1 # ERROR not defined +2 a() +2 +# ok +2 a() +3 +return a() +Deck-3 +1 # ERROR not defined +2 a()Deck-B +Deck-C +Deck-A +b1 +def b1(): +c1 +1 +al +2 +# ok, cl exported +def cl(): +1 +def al(): +3 +return cl() +2 +return c1() + c3() +2 +# ok +3 +b1() +cl,c2 +c2 +4 +# ok +1 # dummy pod +def c2(): +5 +c1() +2 # re-export cl,c2 +2 +c3() +a2 +b3 +c3 +1 +def a2(): +1 +def b3(): +1 +def c3(): +2 +# ERROR not +defined +2 +# +ERROR +R not defined +2 +pass +3 +b4() +3 +c3() +c4 +4 +# ERROR not +defined +c4() +b4 +5 +1 +def c4(): +1 +def b4(): +2 +pass +c4()Conference’23, Jan, 2023, Ames, IA, USA +Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian +Figure 4: NS Rule 3: Utility pods/decks (indicated in green +icon Utility) +make it impossible to layout the pod hierarchy cleanly in a 2D space +without many intersections. +Thus, we design a “utility rule”: a deck/pod can be marked as +a utility deck/pod. Such a utility pod is meant to provide utility +functions to the parent deck’s sub-tree, and thus all the public +functions in the utility deck are visible in the parent deck’s whole +sub-tree. The utility pods are also namespace-aware: it is only +visible to the parent deck, but not the grand-parent deck and above. +Thus the utility decks can also be hierarchically ordered to build +utility functions at different abstraction levels. As a special case, a +utility deck under the root deck defines global utility functions that +can be accessed throughout the entire codebase. +For example, in Fig. 4, there are three regular decks and two +utility decks. The public functions utils_b1 and utils_b2 defined +in utility deck B are visible in its parent deck B’s sub-tree, including +decks B, C. Another utility deck, A, is defined in the upper level +and has a greater visible scope. +2.5 +NS Rule 4: Testing Pods +Figure 5: NS Rule 4: Testing pods/decks (indicated in green +icon Test) +Another essential pattern in interactive software development +is to test whether the functions work as expected by writing some +testing code and observing results. Such testing code must access +the functions being tested, thus having to be in the same namespace +or the parent namespace. However, either option has problems. +On the one hand, testing code might create variables, introduce +helper functions, and produce side effects. Thus they should be +in a separate namespace to avoid polluting the namespace of the +functions under testing. On the other hand, placing the testing deck +as the parent deck of the function under testing is not logically +natural because it does not provide upper-level functions. +Therefore, we allow a deck/pod to be marked as a testing deck/- +pod. A testing deck is placed as a child deck in the same namespace +of the functions being tested. Although the testing deck/pod is +a child-namespace, it can access the definitions visible within its +parent deck, thus is able to call and test the function of interest. +The testing pods are also namespace-aware: it can only access the +function definitions in its parent deck, but not the grand-parent +deck or siblings. Thus the testing decks can also be hierarchically +ordered to build testing functions at different abstraction levels. +For example, in Fig. 5, there are three regular decks and two +testing decks, and one testing pod inside the regular deck A. The +code pods in the testing deck are visible within the same testing +deck, allowing for a testing setup like defining variables x and y +and using them in other pods in the same testing deck. The pods in +the testing deck run in a separate namespace, thus will not pollute +other namespaces. A testing deck can access functions defined in its +parent deck and can thus call and test whether the function yields +expected results. A testing pod is similar to a testing deck, running +in a separate namespace, and has access to the function definitions +in the deck it belongs to. +2.6 +NS Rule 5: Explicit Access via Full Path +Figure 6: NS Rule 5: explicit imports by full path +Finally, the 5th rule is the “brute-force rule”: a pod can always +be accessible by specifying the full path within the tree. In other +words, all pods are accessible via an explicit full path. This provides +compatibility for arbitrary imports. This is considered the last resort +and is ideally not needed but can be helpful in some cases. The path +can be either a relative path connecting two pods or the absolute +path from the root deck. In order to specify the path, the decks have +to be named. In Codepod, an unnamed deck receives a UUID as the + +Deck-C +1 +def c1(): +2 +util_bl() +3 +Deck-B +1 +def c2(): +1 +def b(): +2 +util_b2() +2 +# OK +3 +3 +util_al() +Deck-A +4 +util_a2() +5 +# OK +UtilityDeck-B +AUtility +1 +def al(): +6 +util_bl() +2 +util_al() +7 +util_b2() +util_bl +3 +util_a2() +1 def util_b(): pass +L +def a() : +2 +util_b2 +# ERROR not defined +3 +util_bi() +1 def util_b2(): pass +UtilityDeck-A +AUtility +util_al +1 def util_al(): pass +util_a2 +1 def util a2(): passDeck-C +c1 +def c1(): +Deck-B +2 +pass +b1 +c2 +def b1(): +1 +def c2(): +Deck-A +2 +return c() +2 +pass +a +b2 +1 +def b2(): +1 +TestDeck-B ++Test +2 +def a() : +2 +1 +print("Testing b ..") +3 +2 +assert bl() == 1 +4 +assert b2() == 2 +5 +4 +assert cl() == 3 +1 +print("Testing a ++ Test +assert a() == 3 +TestDeck-A ++Test +1 # test context setup +2x=2 +3y=3 +1 # x and y are available +2 print("Testing a ..") +3 assert a(x) == yC +C +1 # explicit relative import +2 +import d from ../D +B +3 +def c() : +4 +()p +1 +2 +A +D +1 +d +2 +def d(): +2 +CPrFHVDyXTrC + # explicit absolute import +2 +import c from /A/B/C +3 +def f() : +4 +c()Codepod: A Namespace-Aware, Hierarchical Jupyter +for Interactive Development at Scale +Conference’23, Jan, 2023, Ames, IA, USA +name. Most of the time, developers do not need to specify names +to the decks, as the first four rules will make the modules system +usable without specifying names. Named decks are also helpful as +a document for naming important module hierarchies. +For example, in Fig. 6, there are 5 decks in the codebase. In Deck- +D, a function d is defined. In Deck-C, the function d is not accessible. +However, it is still able to be imported by a relative path ../D. As +another example, in the bottom deck, a full absolute path /A/B/C +is used to access the function c defined in Deck-C. +2.7 +Discussion +In summary, based on the hierarchical pod model, Codepod pro- +vides a simple yet powerful module system including five rules: +namespace separation, public pods, utility pods, testing pods, and +full-path explicit access. These rules are highly hierarchical, and +therefore are well suited for building hierarchical software projects +from the ground up. This module system abstracts over different +programming languages’ native module systems and provides a con- +sistent module system across languages. The following section will +discuss the runtime system and algorithms to support the Codepod +module system. +Let us revisit the Codepod example in Fig. 1, and see how these +namespace rules are useful in real-world applications. This ex- +ample implements a simplified Python regular expression com- +piler. The functions are ordered into the decks re, sre_compile +and sre_parse decks. sre_parse is the basic buiding block. It +contains a child deck that defines three internal classes, Pattern, +SubPattern, Tokenizer. These classes are only used in sre_parse +module and not exposed to the upper level. The sre_parse mod- +ule defines several helper functions including _parse_sub, _parse, +_escape, and they are used to build a parse function which is ex- +posed to parent module sre_compile. Similarly, module sre_compile +defines some internal helper functions that are composed to pro- +vide compile to the parent re module to build the top level API +compile, match and search. The general functions isstring and +isnumber are defined in a utility deck and are accessed through the +modules sre_compile and sre_parse. Finally, the testing pods at +different hierarchy make it easy to test and debug the functions at +different levels. +Ordering code blocks in Codepod is natural in building the dif- +ferent levels of abstractions, and the hierarchy of code is close to +the call graph, and is more cleanly maintained compared to file +editors. The Codepod implementation maintains a clear code hi- +erarchy and makes it easy to develop the project interactively. In +comparison, the file-based implementation using VSCode would +spread the functions into files, and within the file, the hierarchy of +the functions is not clearly maintained. Writing all the functions +into a Jupyter notebook is challenging due to the lack of namespace +support, and the code hierarchy cannot be maintained within a +single global namespace. +2.8 +Version Control +Implementing a version control system requires tremendous effort. +There already exists mature file-based version control systems such +as git and svn. We re-use the git version control system to apply +version control to Codepod. Specifically, the code pods are first +CodePod +Front-end +Runtime +Kernel +RunCode +Request Complete +EvalInNS(code, ns) +AddImport(from, to, name) +DeleteImport(from, name) +Jupyter Protocol +CodePod added protocol +Figure 7: Kernel Communication Protocols +exported to files. Then git version control is applied to the generated +files. For front-end rendering, we query the git version control +system for the diff between two versions, e.g., the current changes +or specific git commit changes. Then the diff results are parsed to +show pod-level diffs. +3 +INCREMENTAL RUNTIME +Codepod develops an intuitive, effective, consistent cross-language +scope-aware incremental runtime abstraction. In Codepod, the run- +time loads the code pods in the hierarchy. When a pod in some scope +is changed, the updated definition can be applied incrementally in +that scope in the active runtime without restarting the runtime, +and the new definition is visible for other pods depending on it. +3.1 +Runtime Kernel Communication +We build the Codepod Kernel communication based on the Jupyter +Message Queue protocol. The Jupyter kernel protocols and our +added protocols are shown in Fig. 7. In its simplest form, Jupyter +kernel messaging supports eval and complete. We add the fol- +lowing protocol to the messaging queue: EvalInNS, AddImport, +DeleteImport. The protocol EvalInNS instructs the langauge ker- +nel to evaluate code in a specific namespace. The AddImport proto- +col makes a function “name” defined in “from” namespace available +in the “to” namespace, and DeleteImport undo the change. The +algorithm for the language kernel is given in next section. +3.2 +Language Kernel Algorithm +The language kernel needs to support four functions to work with +Codepod: GetModule, EvalInNS, AddImport, DeleteImport. Most +languages do not natively support all these operations. We imple- +ment a thin wrapper of these functions as shown in Algorithm 1. +GetModule is a function that gets the module instance for a spe- +cific namespace. Since we need to refer to the same module given +the same name, we need to maintain a mapping from namespace +names to the module instance. In line 1, the nsmap object is created +as a global variable, initialized with an empty dictionary. The Get- +Module function will query this map, and if the module already +exists, return the module instance (lines 3-4). If the module is not +found, create a new module by calling the language’s createMod +API, record the module in the nsmap, and return the module (lines +6-8). + +Conference’23, Jan, 2023, Ames, IA, USA +Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian +Algorithm 1 Namespace-aware Runtime: Language Kernel +1: 𝑛𝑠𝑚𝑎𝑝 ← 𝐸𝑚𝑝𝑡𝑦𝐷𝑖𝑐𝑡 () +2: function GetModule(ns) +3: +if 𝑛𝑠 ∈ 𝑛𝑠𝑚𝑎𝑝 then +4: +return 𝑛𝑠𝑚𝑎𝑝 [𝑛𝑠] +5: +else +6: +𝑚𝑜𝑑 ← 𝑐𝑟𝑒𝑎𝑡𝑒𝑀𝑜𝑑 (𝑛𝑠) +7: +𝑛𝑠𝑚𝑎𝑝 [𝑛𝑠] ← 𝑚𝑜𝑑 +8: +return 𝑚𝑜𝑑 +9: +end if +10: end function +11: function EvalInNS(ns, code) +12: +𝑚𝑜𝑑 ← GetModule(ns) +13: +𝑖𝑠𝐸𝑥𝑝𝑟 ← IsExpr(code) +14: +if isExpr then +15: +return eval(code, mod) +16: +else +17: +exec(code, mod) +18: +return NULL +19: +end if +20: end function +21: function AddImport(from, to, name) +22: +𝑠 ← "$name=eval($name,from)" +23: +EvalInNS(to, s) +24: end function +25: function DeleteImport(from, name) +26: +𝑠 ← "del $name" +27: +EvalInNS(from, s) +28: end function +The EvalInNS function is responsible for loading the module +specified by the namespace and evaluating code with that mod- +ule instance active. It first loads or creates the module instance +by the GetModule function. Many languages treat expression and +statement separately, where expressions return values, while state- +ments do not. Thus the algorithm first checks whether this code +is an expression or statement. If it is an expression, the language’s +EVAL API is called with the code and module instance and returns +the expression results for display in the front-end. Otherwise, the +language’s EXEC API is called to evaluate the code in the module +instance for side-effect only and return NULL to the user. +Finally, AddImport and DeleteImport work by meta-programming: +a new program string is constructed to add or delete names in the +target namespace. The function AddImport receives the name to +import and the "from" and "to" namespace. The goal is to import the +function "name" from the "from" namespace and make it available in +the "to" namespace. A string is constructed to assign a variable with +the same name "name" with the evaluation of the name in the "from" +namespace (lines 22-23). The DeleteImport works by constructing +a "delete $name" code and evaluating the target namespace (lines +26-27). +We note that the EVAL and EXEC API with module awareness are +different across languages. Some languages provide native support, +while others might need to perform reflection-level operations, +such as manually passing in Python symbol maps. The code strings +for AddImport and DeleteImport are generally different across +languages too. We supply the core code of the four kernels we have +implemented in Figure 8, 9, 10, and 11. +3.3 +Pod Hierarchy Algorithm +This section formally describes the key algorithms in Algorithm 2 +that implement Codepod’s module system, the namespace rules, and +how the evaluation of pods/decks is handled in the pod hierarchy. +A pod is evaluated with the RunPod function, which calls EvalInNS +with the pod’s code content and namespace string. A deck is eval- +uated with the RunDeck function, which evaluates the subtree of +the deck. The function first evaluates all the child decks in a DFS +(depth-first search) manner, then evaluates all the pods in this deck. +1 +d = {} +2 +3 +def getmod(ns): +4 +if ns not in d: +5 +d[ns] = types.ModuleType(ns) +6 +d[ns]. __dict__[" +CODEPOD_GETMOD"] = getmod +7 +return d[ns] +8 +9 +def add_import(src , dst , name): +10 +return eval_func(""" +11 +{name}= getmod ({src}). +__dict__ ["{ name }"] +12 +""", dst) +13 +14 +def delete_import(ns, name): +15 +eval_func("""del {name}""", ns) +1 +def eval_func(code , ns): +2 +mod = getmod(ns) +3 +[stmt , expr] = code2parts(code) +4 +if stmt: +5 +exec(stmt , mod.__dict__) +6 +if expr: +7 +return eval(expr , mod. +__dict__) +Figure 8: Python Kernel Namespace Implementation +1 +var NSMAP = NSMAP || {}; +2 +function eval(code , ns, names) { +3 +if (!NSMAP[ns]) { +4 +NSMAP[ns] = {}; +5 +} +6 +for (let k of keys(NSMAP[ns])) { +7 +eval( +8 +`var ${k}=NSMAP["${ns}"].${k}` +9 +); +10 +} +11 +let res = eval(code); +12 +for (let name of names) { +13 +eval( +14 +`NSMAP["${ns}"].${name}=${name}` +15 +); +16 +} +17 +return res; +18 +} +1 +function addImport(from , to, name) { +2 +if (!NSMAP[from]) { +3 +NSMAP[from] = {}; +4 +} +5 +if (!NSMAP[to]) { +6 +NSMAP[to] = {}; +7 +} +8 +eval(` +9 +NSMAP["${to}"].${name}= +10 +NSMAP["${from}"].${name}`); +11 +} +12 +function deleteImport(ns, name) { +13 +if (!NSMAP[ns]) { +14 +NSMAP[ns] = {}; +15 +} +16 +eval( +17 +`delete NSMAP["${ns}"].${name}`) +; +18 +} +Figure 9: Javascript Kernel Namespace Implementation +1 +function isModuleDefined(names) +2 +mod = :(Main) +3 +for name in names +4 +name = Symbol(name) +5 +if !isdefined(eval(mod), name) +6 +return false +7 +end +8 +mod = :($mod.$name) +9 +end +10 +return true +11 +end +12 +function ensureModule(namespace) +13 +names = split(namespace , "/", +14 +keepempty=false) +15 +mod = :(Main) +16 +for name in names +17 +name = Symbol(name) +18 +if !isdefined(eval(mod), name) +19 +include_string(eval(mod), +20 +"module $name end") +21 +end +22 +mod = :($mod.$name) +23 +end +24 +return mod , eval(mod) +25 +end +1 +function eval(code , ns) +2 +_, mod = ensureModule(ns) +3 +include_string(mod , code) +4 +end +5 +function addImport(from , to, name) +6 +from_name , _ = ensureModule(from) +7 +_, to_mod = ensureModule(to) +8 +code = """ +9 +using $from_name: $name as CP$name +10 +$name=CP$name +11 +$name +12 +""" +13 +include_string(to_mod , code) +14 +end +15 +function deleteImport(ns, name) +16 +_, mod = ensureModule(ns) +17 +include_string(mod , "$name=nothing") +18 +end +Figure 10: Julia Kernel Namespace Implementation +The reason to evaluate child decks before child pods is that the child +decks might define public functions exposed to this deck; thus, the +pods must have those definitions in place before execution. The +reason to use DFS is to make sure the lowest-level pods are first +evaluated before moving up the hierarchy. Child pods are evaluated +sequentially. When running the child pods, the algorithm examines + +Codepod: A Namespace-Aware, Hierarchical Jupyter +for Interactive Development at Scale +Conference’23, Jan, 2023, Ames, IA, USA +1 +(compile-enforce-module-constants #f) +2 +3 +(define (ns- >submod ns) +4 +(let ([names (string-split ns "/")]) +5 +(when (not (empty? names)) +6 +(let ([one (string- >symbol +7 +(first names))] +8 +[two (map string- >symbol +9 +(rest names))]) +10 +‘(submod ',one +11 +,@two))))) +12 +(define (ns- >enter ns) +13 +(let ([mod (ns- >submod ns)]) +14 +(if +(void? mod) '(void) +15 +‘(dynamic-enter! ',mod)))) +16 +17 +(define (ns- >ensure-module ns) +18 +(let loop +19 +([names (string-split ns "/")]) +20 +(if (empty? names) +21 +'(void) +22 +‘(module +23 +,(string- >symbol +24 +(first names)) +25 +racket/base +26 +,(loop (rest names)))))) +1 +(define (add-import from to name) +2 +(let ([name (string- >symbol name)]) +3 +(eval (ns- >enter to)) +4 +(eval +5 +‘(define ,name +6 +(dynamic-require/expose +7 +',(ns- >submod from) +8 +',name))))) +9 +10 +(define (delete-import ns name) +11 +(eval (ns- >enter ns)) +12 +(namespace-undefine-variable! +13 +(string- >symbol name))) +14 +15 +(define (string- >sexp s) +16 +(call-with-input-string +17 +s +18 +(lambda (in) +19 +(read in)))) +20 +21 +(define (codepod-eval code ns) +22 +(eval (ns- >ensure-module ns)) +23 +(eval (ns- >enter ns)) +24 +(begin0 +25 +(eval +26 +(string- >sexp +27 +(~a "(begin " code ")"))) +28 +(enter! #f))) +Figure 11: Racket Kernel Namespace Implementation +Algorithm 2 Namespace-aware Runtime: Pod Hierarchy +1: function RunPod(pod) +2: +EvalInNS(pod.ns, pod.code) +3: end function +4: function RunTest(ns, code) +5: +for 𝑛𝑎𝑚𝑒 ← 𝑝𝑜𝑑.𝑝𝑎𝑟𝑒𝑛𝑡.𝑛𝑎𝑚𝑒𝑠 do +6: +AddImport(pod.parent.ns, pod.ns, name) +7: +end for +8: +EvalInNS(pod.ns, pod.code) +9: end function +10: function RunUtility(from, name) +11: +EvalInNS(pod.ns, pod.code) +12: +function dfs(parent) +13: +for 𝑐ℎ𝑖𝑙𝑑 ← 𝑝𝑎𝑟𝑒𝑛𝑡.𝑐ℎ𝑖𝑙𝑑_𝑑𝑒𝑐𝑘𝑠 do +14: +for 𝑛𝑎𝑚𝑒 ← 𝑝𝑜𝑑.𝑝𝑎𝑟𝑒𝑛𝑡.𝑛𝑎𝑚𝑒𝑠 do +15: +AddImport(pod.parent.ns, pod.ns, name) +16: +end for +17: +dfs(child) +18: +end for +19: +end function +20: +dfs(pod.parent) +21: end function +22: function RunTree(root) +23: +function dfs(parent) +24: +for 𝑐ℎ𝑖𝑙𝑑 ← 𝑝𝑎𝑟𝑒𝑛𝑡.𝑐ℎ𝑖𝑙𝑑_𝑑𝑒𝑐𝑘𝑠 do +25: +RunTree(child) +26: +end for +27: +end function +28: +dfs(pod) +29: +for 𝑝𝑜𝑑 ← 𝑟𝑜𝑜𝑡.𝑐ℎ𝑖𝑙𝑑_𝑝𝑜𝑑𝑠 do +30: +if pod.type is "Pod" then +31: +RunPod(pod) +32: +else if pod.type is "Test" then +33: +RunTest(pod) +34: +else if pod.type is "Utility" then +35: +RunUtility(pod) +36: +end if +37: +end for +38: end function +the type of the pods and calls the corresponding functions RunPod, +RunUtility, and RunTest for different types accordingly. +Hierarchical +Layout +Commnication +Protocol +Kernel +Runtime +CodeServer +API +Total +LOC +4.1k +1.8k +1k +1k +7.9k +Table 1: Codepod Implemenatation Statistics +The RunUtility function will first evaluate the pods in the names- +pace. Then, the public names are exported to the parent’s subtree +by traversing the parent’s subtree in a DFS manner and calling +AddImport for each of the namespaces in the sub-tree during tra- +versal. In this way, the name of the utility function is available in +all the decks of the parent’s sub-tree. The RunTest function will +loop through the parent deck’s public names and run AddImport +for all the names from the parent’s namespace to the testing pod’s +namespace, and then evaluate the test pods in the namespace where +the parent’s function definitions are available. +3.4 +Fallback Execution +Codepod requires the programming languages to support interpret- +ing in order to run and evaluate code interactively. With interactive +development becoming popular, there have been many interpreters +implementations for even compiled languages. For example, C++ +has a highly regarded interpreter, Cling. Another requirement is +that the language needs to support namespace and provide a way +to evaluate code blocks within the namespace. +Suppose the support of namespace-aware interactive develop- +ment is not fully supported due to the limitation of the language +interpreter. In that case, Codepod provides a fallback option: export- +ing code to files and invoking the language interpreter/compiler to +run the program as a whole. The downside of this approach is that +the variables are not persisted in memory across runs because each +invocation starts a new process. +4 +CASE STUDIES +4.1 +Implementation +We have implemented a fully working Codepod as a Web applica- +tion (front-end in React and backend server in Node.js) and cur- +rently implemented full namespace-aware runtime support for +four language kernels, including Python, JavaScript, Julia, and +Scheme/Racket. New kernels can be easily developed upon exist- +ing Jupyter notebook kernels. Codepod is open-sourced at https: +//example.com. +Codepod implementation contains 4 major parts, the LOC sta- +tistics is shown in Table 1. The hierarchical layout implements the +front-end hierarchical pods and tools. The communication proto- +col implements the RunTree/RunPod logic and how the front-end +communicates with the backend API server and language runtime +kernels. Kernel runtime implements the kernels and WebSocket +protocol message handling. Finally, CodeServer API implements the +API for retrieving and updating pods hierarchy from the front-end +and talking to the database. + +Conference’23, Jan, 2023, Ames, IA, USA +Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian +4.2 +Python and Julia Projects Statistics +In this section, we study several highly regarded open-source projects +and visually show how the representation of Codepod can poten- +tially help develop code projects. This case study aims to see how +functions are distributed in the files and directories and the calling +relations of the functions. This helps us evaluate how the Codepod +model can help with real-world open-source software projects. +The projects are obtained from GitHub’s top-rated Python and +Julia projects; the information about the projects is shown in Ta- +ble 2. We analyze the projects with the help of tree-sitter [1] parser +framework. We count only the source directory of the projects, +ignoring testing code and documents. Our study focuses on func- +tions as the code block granularity. Python projects might contain +classes, and we treat each method as a function. +In Table 2, we can see that software projects often contain a large +number of functions, and those functions are distributed in a large +number of files, possibly in a deep directory hierarchy of tens of +directories. For example, the you-get project contains 444 functions +and is distributed into 133 files in 8 directories. The LightGraph.jl +project contains 242 functions, distributed in 110 files within 27 +directories. It can be pretty challenging to grasp the hierarchy of +the functions by browsing through the files and directories. +We also count the number of internal and external functions of +each file. A file’s internal functions are defined as those only called +by the other functions in the same file. These internal functions are +helper functions that are used to implement the external functions +that act as the public interface of the file, called by other files. In +Table 2, we see that approximately half of the files are internal +and are used as building blocks of other functions. However, this +dependency information is not clear in a file because the functions +are considered linear within a file, and no clear hierarchy can be +effectively maintained. Thus potentially, Codepod can help to apply +hierarchical relations to the functions inside each file. +Figure 12: Statistics for top open source Python Projects +To understand the distributions of functions in each file and call- +ing relationships, we plot the in-degree,out-degree,LOC,#functions- +per-file of the Python and Julia Projects in Fig. 12 and Fig. 13, re- +spectively. The in/out-degrees of a function is defined by how many +calls to the functions and how many calls this function makes to +other functions. Both in/out-degrees are computed based on the call +Figure 13: Statistics for top open source Julia Projects +graphs over the functions defined in the project; thus, the statistics +do not include language or library API calls. +From the plot Fig. 12.(a) and Fig. 13.(a), we see that most func- +tions are being called no more than two times. This shows that +most functions are “local”, and only used to implement higher-level +functions. Also, in Fig. 12.(b) and Fig. 13.(b), we observe that the +out-degree is more than in-degree, and most functions have less +than five function calls to other functions. This is also consistent +with Codepod’s tree-based model: a deck in a tree node might have +multiple sub-decks. A pod defined in a deck might use the functions +defined in the sub-decks to implement higher-level functionality. +From the function per file plot Fig. 12.(d) and Fig. 13.(d), we can +see that the distribution of functions into files are not even: a large +number of files contain only 1 or 2 functions, while there can also be +a few very large file containing tens of functions. This data shows +that in order to maintain a cleaner hierarchy, developers might use +a separate file for each function, resulting in too many small files +and deep directories, which are relatively hard to maintain, edit +and reference using file browsers and file editors. Also, within the +large files with tens of functions, the hierarchy of those functions +cannot be cleanly maintained within a file, and Codepod could help +build a finer-granular hierarchy for the functions. +4.3 +Jupyter Project Statistics +In this study, we investigate how Jupyter notebooks are used in real- +world open projects. This study aims to see how code is distributed +among Jupyter notebooks and text files and how the Jupyter note- +books interact with the functions defined in text files regarding +calling relationships. We query GitHub APIs to find top Python +projects whose primary language is Python and whose secondary +language is Jupyter Notebook. The projects and statistics are shown +in Table 3. +In Table 3, we can see that there are typically more text files +than Jupyter notebooks. This is also the case for the total LOC +and number of functions in Jupyter notebooks vs. text files. In +fact, the LOC of the Jupyter notebook consists only 3.7% of the +codebase for these projects. This means that the majority of the +code is implemented in the text files. +To understand the calling relationships of the Jupyter notebooks +and text files, we calculate the percentage of internal functions + +ab-no +DOE +cookiecutter +locust +150 +unoo +requests +200 +100 +100 +50 +5 +10 +5 +1D +15 +(afuncbion indegree +bj functinoutdegree +60 +60 +40 +40 +no +20 +20 +0 +0 +0 +25 +50 +75 +100 +0 +5 +10 +15 +20 +(ciavglocperfunction +dyfunctionsperfileJuliaDB.jl +100 +HTTP.jI +100 +Flux.jl +un +LightGraphs.jl +50 +50 +5 +10 +0 +5 +10 +15 +al function indegree +(b) functin outdegree +30 +60 + 20 +40 +10 +20 +0 +0 +0 +25 +50 +75 +100 +5 +10 +15 +20 +(c) avg loc perfunction +(d) functionsper fileCodepod: A Namespace-Aware, Hierarchical Jupyter +for Interactive Development at Scale +Conference’23, Jan, 2023, Ames, IA, USA +#stars +#dirs +#files +#loc +#funcs +#internal funcs +description +soimort/you-get +41.6k +8 +133 +14707 +444 +273 +CMD-line utility to download media from Web +cookiecutter/cookiecutter +15.2k +0 +18 +2139 +51 +30 +CMD-line utility to create projects from templates +locustio/locust +17k +10 +59 +18671 +529 +144 +performance testing tool +psf/requests +45.9k +0 +18 +5183 +135 +73 +HTTP library +JuliaData/JuliaDB.jl +706 +0 +20 +3113 +106 +82 +Parallel analytical database +JuliaWeb/HTTP.jl +439 +0 +38 +7513 +65 +36 +HTTP client and server +FluxML/Flux.jl +3.2k +5 +33 +6408 +84 +41 +Machine Learning Framework +JuliaGraphs/LightGraphs.jl +675 +27 +110 +16963 +242 +101 +Network and graph analysis. +Table 2: Function Statistics in Open Source Projects (Python and Julia) +#stars +#file +(ipynb/files) +#loc +(ipynb/files) +#call +fs-to-nb +#call +nb-to-fs +#func +(ipynb/files) +#%internal +(ipynb/files) +description +blei-lab/edward +4.6k +14/42 +1340/5449 +0 +11 +10/102 +100%/93% +A probabilistic programming language +tqdm/tqdm +19.3k +2/30 +284/2257 +0 +9 +0/83 +NA/80% +A Fast, Extensible Progress Bar +google/jax +14.1k +3/196 +104/42879 +0 +8 +0/2418 +NA/38% +Autograd and Optimizing Compiler for ML +google-research/bert +29k +1/13 +322/4547 +0 +8 +7/92 +100%/88% +State-of-the-art NLP language model +quantopian/zipline +14.4k +2/183 +115/30087 +0 +3 +3/851 +100%/61% +Algorithmic Trading Library +Table 3: Jupyter Notebooks statistics in Open Source Projects +for files and Jupyter notebooks. The internal function is defined +the same as above, i.e., the functions that are only called from the +same file or Jupyter notebook. From the result, two projects contain +no function definitions in the Jupyter notebooks, and the other +three projects’ functions are 100% internal to the notebook files. In +contrast, the text files contain 38% to 93% internal functions. Also, +we calculate the number of calls from the Jupyter notebook to text +files and vice versa and show that there are no calls from source +text files to notebooks, but only calls from the Jupyter notebook to +the text files. This means that the Jupyter notebook is not used to +develop functions. The projects are implemented in text files, and +Jupyter notebooks are only used to call those files’ APIs and are +most likely for presentation and tutorial purposes. +5 +RELATED WORK +Running and debugging code is a major activity in software de- +velopment. There have been many tools to help the debugging +process. In the simplest form, developers write code in files using +some editors such as VIM and Emacs [13], and compile and run +files in the command line. The problem with this approach is that +it is non-interactive, and the program always needs to restart from +the beginning. An interactive development method is to launch +a Read-Eval-Print-Loop (REPL) [7, 14], type, load, and evaluate +code expressions in the REPL. Although REPL is interactive, the +code being evaluated is not editable, and users have to type code +into the REPL. It is also common to open a file editor and send a +code region to the REPL for evaluation. Integrated Development +Environments (IDEs) such as VSCode integrate file editors with +a file browser, command line, plugins, and debuggers. There also +exist editor plugins to navigate between the functions of a project. +However, those plugins still do not give a within-file hierarchy +to the code and depend on the programming language designs to +support the within-file module system, which only a handful of +languages support to various degrees. File-based approaches force +developers to maintain the correspondence between code and files, +which is tricky due to the significantly different granularity of code +blocks and files. The unbalance in granularity poses dilemmas to +developers: including too many code blocks into one file makes the +hierarchy hard to maintain, while including few code blocks into +one file creates many small files and deep directories that are also +hard to work with. +Figure 14: Interactive Development with Jupyter. Image from [11]. +A recent new paradigm is a web-based interactive notebook +called Jupyter Notebook [6]. The interface of Jupyter is shown in +Figure 14. The notebook consists of code cells. Each cell can be run +in arbitrary order, and the results are displayed under the cell. The +code output can be visualized, e.g., plotting a figure. Thus Jupyter +notebooks support literate programming that combines code, text, + +Fibonnaci +In [3]: +def fib(x): +if x <= 1: +Markdown +return x +Cells +return fib(x-1) + fib(x-2) +fib(10) +Out[3]: +55 +Output 1 +Code +Cells +Let's plot the numbers +In [8]: +from matplotlib import pyplot +%matplotlib inline +x = range(15) +y = [fib(n) for n in x] +pyplot.plot(x, y); +Execution +350 +300 +Counter +250 +200 + Output 2 +150 +100 +50 +0 +6 +10 +12 +14Conference’23, Jan, 2023, Ames, IA, USA +Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian +and execution results with rich media visualizations. However, the +Jupyter notebook cells are linear, and all the code blocks live in the +global namespace. Jupyter notebook lacks a module system that +is crucial for complex software. This makes it hard to develop a +large-scale software system. Thus Jupyter notebook is typically +used only for surface demo and visualization purposes. The real +code of the projects is still developed in text files with text editors +and external runtime. In order to define hierarchical code, users +have to write code into text files and import the module from text +files into the notebook. Such a Jupyter-file hybrid model has several +disadvantages. Changes in files are not in sync with the Jupyter +runtime. When a function definition in a file is changed, the Jupyter +runtime must be restarted, and the file must be reloaded for the +change to take effect. +Another programming paradigm is Visual Programming, e.g. Lab- +View [2], and Google’s Blockly [9], and Microsoft’s MakeCode [5]. +Such a visual programming paradigm has distinguished advantages: +it is visually clean, easy to learn, and impossible to make syntax er- +rors. However, all these visual programming systems use such block +style items down to each expression (e.g., a+b), which can be ver- +bose. Also, visual programming blocks live in a global namespace, +and there is no module system available for developing large-scale +software. +There exist code analyzers to generate a visual presentation of +code, such as Unified Modeling Language (UML) [3, 8]. Tools such +as call graph visualizers [4] are also developed to help to understand +a codebase. However, those visual presentations are not editable, +making them less useful during development. +6 +CONCLUSION +In this paper, we propose Codepod, a namespace-aware hierarchi- +cal interactive development environment. Codepod uses a novel +hierarchical code block model to represent code and abstracts away +files. We propose namespace rules that make it easy to organize +the pods and provide a consistent module system across languages. +Codepod provides an incremental evaluation runtime that helps +interactively develop a large-scale software project. +We hope Codepod can provide a novel way to drive the software +development process and inspire other research. In the future, we +will push Codepod forward with the contribution from the com- +munity, perform user studies to compare Codepod with VSCode +and Jupyter. It is interesting to integrate program analysis tools +into the Codepod model. We are also interested in integrating other +programming paradigms into the Codepod framework; for exam- +ple, it would be helpful to integrate graphical programming as a +“graphical pod” and mix graphical programs with plain-text code. +REFERENCES +[1] Tree-sitter: An incremental parsing system for programming tools. https://tree- +sitter.github.io/tree-sitter/. Accessed: 2021-07-30. +[2] Rick Bitter, Taqi Mohiuddin, and Matt Nawrocki. LabVIEW: Advanced program- +ming techniques. Crc Press, 2006. +[3] Grady Booch, Ivar Jacobson, James Rumbaugh, et al. The unified modeling +language. Unix Review, 14(13):5, 1996. +[4] David Callahan, Alan Carle, Mary W. Hall, and Ken Kennedy. Constructing the +procedure call multigraph. IEEE Transactions on Software Engineering, 16(4):483– +487, 1990. +[5] James Devine, Joe Finney, Peli de Halleux, Michał Moskal, Thomas Ball, and +Steve Hodges. Makecode and codal: intuitive and efficient embedded systems +programming for education. ACM SIGPLAN Notices, 53(6):19–30, 2018. +[6] Thomas Kluyver, Benjamin Ragan-Kelley, Fernando Pérez, Brian Granger, +Matthias Bussonnier, Jonathan Frederic, Kyle Kelley, Jessica Hamrick, Jason +Grout, Sylvain Corlay, Paul Ivanov, Damián Avila, Safia Abdalla, and Carol Will- +ing. Jupyter notebooks – a publishing format for reproducible computational +workflows. In F. Loizides and B. Schmidt, editors, Positioning and Power in +Academic Publishing: Players, Agents and Agendas, pages 87 – 90. IOS Press, 2016. +[7] John McCarthy, Michael I Levin, Paul W Abrahams, Daniel J Edwards, and +Timothy P Hart. LISP 1.5 programmer’s manual. MIT press, 1965. +[8] Nenad Medvidovic, David S Rosenblum, David F Redmiles, and Jason E Rob- +bins. Modeling software architectures in the unified modeling language. ACM +Transactions on Software Engineering and Methodology (TOSEM), 11(1):2–57, 2002. +[9] Erik Pasternak, Rachel Fenichel, and Andrew N Marshall. Tips for creating a +block language with blockly. In 2017 IEEE Blocks and Beyond Workshop (B&B), +pages 21–24. IEEE, 2017. +[10] Jeffrey M Perkel. Why jupyter is data scientists’ computational notebook of +choice. Nature, 563(7732):145–147, 2018. +[11] João Felipe Pimentel, Leonardo Murta, Vanessa Braganholo, and Juliana Freire. A +large-scale study about quality and reproducibility of jupyter notebooks. In 2019 +IEEE/ACM 16th International Conference on Mining Software Repositories (MSR), +pages 507–517. IEEE, 2019. +[12] Bernadette M Randles, Irene V Pasquetto, Milena S Golshan, and Christine L +Borgman. Using the jupyter notebook as a tool for open science: An empirical +study. In 2017 ACM/IEEE Joint Conference on Digital Libraries (JCDL), pages 1–2. +IEEE, 2017. +[13] Richard M Stallman. Emacs the extensible, customizable self-documenting dis- +play editor. In Proceedings of the ACM SIGPLAN SIGOA symposium on Text +manipulation, pages 147–156, 1981. +[14] L Thomas van Binsbergen, Mauricio Verano Merino, Pierre Jeanjean, Tijs van der +Storm, Benoit Combemale, and Olivier Barais. A principled approach to repl +interpreters. In Proceedings of the 2020 ACM SIGPLAN International Symposium on +New Ideas, New Paradigms, and Reflections on Programming and Software, pages +84–100, 2020. +[15] Jiawei Wang, Li Li, and Andreas Zeller. Better code, better sharing: on the need of +analyzing jupyter notebooks. In Proceedings of the ACM/IEEE 42nd International +Conference on Software Engineering: New Ideas and Emerging Results, pages 53–56, +2020. + diff --git a/9tE0T4oBgHgl3EQffwDm/content/tmp_files/load_file.txt b/9tE0T4oBgHgl3EQffwDm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d81e83e9fe93624cd952e41a3bb5e635eb6ad160 --- /dev/null +++ b/9tE0T4oBgHgl3EQffwDm/content/tmp_files/load_file.txt @@ -0,0 +1,701 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf,len=700 +page_content='Codepod: A Namespace-Aware, Hierarchical Jupyter for Interactive Development at Scale Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' of Computer Science, Iowa State University Ames, Iowa, USA {hebi,qxiao,jtian}@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='edu,forrest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='bao@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='com ABSTRACT Jupyter is a browser-based interactive development environment that has been popular recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jupyter models programs in code blocks, and makes it easy to develop code blocks interactively by running the code blocks and attaching rich media output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' How- ever, Jupyter provides no support for module systems and names- paces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Code blocks are linear and live in the global namespace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' therefore, it is hard to develop large projects that require modular- ization in Jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' As a result, large-code projects are still devel- oped in traditional text files, and Jupyter is only used as a surface presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We present Codepod, a namespace-aware Jupyter that is suitable for interactive development at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Instead of linear code blocks, Codepod models code blocks as hierarchical code pods, and provides a simple yet powerful module system for namespace-aware incremental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod is open source at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='com/codepod-io/codepod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' ACM Reference Format: Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod: A Namespace- Aware, Hierarchical Jupyter for Interactive Development at Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Pro- ceedings of (Conference’23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' ACM, New York, NY, USA, 10 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION Traditional software development is typically closely tied with file systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Developers write code into a set of files in the file-system hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, developers write functions in files using a text editor and invoke a compiler or an interpreter to run or evaluate the code in the files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Modern Integrated Development Environments (IDEs) provide a file system browser and integrate debuggers to help run and debug over the files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jupyter notebook [6] is a browser-based interactive development environment that has been widely adopted by many different com- munities, both in science and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jupyter notebooks support literate programming that combines code, text, and execution re- sults with rich media visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Juyter models the code as a sequence of "code cells".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This provides a clean separation between code blocks, whereas text editors do not partition code in the same text file but instead relying on developers and editor plugins to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Code cells can be interactively (re)-run and display results in rich media such as data visualization right beside the cell, providing de- velopers an interactive Read-Eval-Print-Loop (REPL) development experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jupyter has been popular recently in software devel- opment [10–12, 15], proving such interactive cycle is beneficial to software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Conference’23, Jan, 2023, Ames, IA, USA 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='nnnnnnn However, Jupyter falls short for module systems and namespaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Code blocks in Jupyter notebooks are linear and live in the global namespace, making it non-scalable for large software projects of hundreds of function definitions with potential naming conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' As a result, large code projects are still developed in traditional text files, and Jupyter is primarily used as a surface presentation of the projects, consisting of only a fraction of the entire codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Our case study in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 found that Jupyter notebook shares less than 5% of the code of real-world open-source projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' All functions defined in the Jupyter notebook are only accessed in the same notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' There are calls from Jupyter to the code in the text files, but no calls from text files to Jupyter code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The Jupyter-file hybrid development model has several disadvan- tages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Changes in files are not in sync with the Jupyter runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This effectively breaks the REPL interactive development functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The hybrid model still relies on text editors, external debuggers, and IDEs and thus still suffers from the drawbacks of file-based software development, which we will detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Although computers store information into files, organizing code into text files where information is linearly presented is counter- productive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Complex software requires proper abstraction and seg- mentation of code, typically by defining functions and hierarchical modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For simplicity, in this paper, we assume functions are the building blocks of software projects and refer to the functions when we talk about “code blocks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' File-based approaches force developers to maintain the correspondence between code and files, which differ significantly in granularity: code blocks are small in size, but large in amount, while files are typically long but few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The unbalance in granularity poses dilemmas to developers: including too many code blocks into one file makes the hierarchy hard to maintain, while including few code blocks into one file creates many small files and deep directories that are also hard to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Besides, programming languages typically design module systems around file systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=', a file is a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' It becomes tedious to reference and import from different modules scattered over multiple files and levels of directories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This is the case in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Among highly regarded open source projects, each project contains tens to hundreds of files, possibly with levels of different directories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For a file containing tens of functions, about half of the functions are internal to the file and are not called in other files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' To overcome the above disadvantages of both Jupyter and text- file-based development, we propose Codepod, a namespace-aware Jupyter for interactive software development at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod models a program as hierarchical code blocks and represents it accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Developers write each function as a code pod and place it at an appropriate hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Codepod, the code blocks are organized into a tree of code pods and decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A code pod resembles arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='02410v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='SE] 6 Jan 2023 Conference’23, Jan, 2023, Ames, IA, USA Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian a cell in Jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The partition of pods is maintained by grouping them into decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A deck can also contain child decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' All code of the entire project can be developed without needing files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In addition, Codepod features a simple yet powerful module sys- tem that abstracts over the native module system of programming languages to provide a consistent and straightforward evaluation model for different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod’s module system consists of five namespace rules, inspired by the hierarchical nature of code blocks and the access pattern among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (1) namespace separa- tion by default: in Codepod, each deck is a namespace, and the root deck is the global namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Pods in different decks are defined in separate namespaces and cannot see each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (2) public pods: a pod can be marked as "public" and is made available to its parent deck;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (3) utility pods: A pod or deck in Codepod can be marked as a “utility pod/deck”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Such a pod is visible to the parent deck node’s sub-tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (4) testing pods: a testing pod or deck can access its parent deck’s namespace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' and (5) explicit path: a pod is always accessible by specifying the full path within the tree, providing the compatibility for arbitrary imports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The detailed rationale of the rules is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Last but not least, Codepod provides a namespace-aware incre- mental evaluation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Codepod, every pod can be executed, and the evaluation happens in the appropriate namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Similar to Jupyter notebooks, the results are displayed right beside the code pod for easy debugging and intuitive interactive development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' When a pod is changed, the pod can be re-evaluated, and the up- dated definition is applied incrementally in that scope in the active runtime, and the new definition is visible to all other pods using it in the entire codebase without restarting the current runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We have implemented a fully working Codepod as a Web ap- plication and currently have implemented full namespace-aware runtime support for four language kernels: Python, JavaScript, Julia, and Scheme/Racket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' New kernels can be easily developed based on existing Jupyter notebook kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod is open-sourced at https://example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='com In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' we make the following contributions in this work: we propose Codepod,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' a novel namespace-aware interactive development environment we propose a simple yet powerful module system abstraction for Codepod we provide a fully working Codepod implementation with namespace-awareness and incremental runtime support for four programming languages,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' and make it open source we conduct case studies of real-world open-source projects to statistically show that our Codepod model will be useful for real-world development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2 HIERARCHICAL PODS In this section, we introduce the Codepod model and its namespace rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In the next section, we describe the incremental evaluation runtime and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 Codepod Interface In Codepod, code blocks are organized into a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In the tree, non- leaf nodes are decks, and leaf nodes are pods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus a deck may contain a list of pods and a list of sub-decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A pod is a text editor containing the real code, and a deck is the container of the pods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We will use “node” to refer to a node in the tree, which can be either a deck or a pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' An overview demo of the Codepod interface is shown in Figure 1, implementing a simplified Python regular expression compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The code is organized into a tree, which starts from the leftmost ROOT node, and grows to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The background level of grey of the deck indicates the level of the deck in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In order to define the interactions between code pods in the tree, Codepod provides simple yet powerful namespace rules abstracting different languages’ native module systems and providing a consis- tent module system for all languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In the following sections, we introduce the rules in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We will revisit this overview exam- ple in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='7 for the meaning of different kinds of pods after introducing the namespace rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A typical workflow using Codepod starts from an empty tree of a single ROOT deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Developers can create pods as a child of the deck and start to develop in the global namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' To develop hierarchical modules, developers can create a deck under the ROOT deck and create pods under the new deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Pods and decks can be moved from one node to another node in different levels to group the pods and re-order the code hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Pods and decks can be folded so that only the pods of interest are displayed during the development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A pod can be evaluated, and the possibly rich media result will be displayed under the pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 NS Rule 1: Namespace Separation In Codepod, the code blocks are organized into a tree of decks and pods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Each deck can contain multiple pods and child decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A pod contains the actual code, and a deck declares a namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A pod belongs to the namespace of its parent deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The first rule is the basic namespace separation: pods in the same namespace are visible to each other, but pods in different namespaces are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2, there are 5 decks, and thus 5 namespaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Deck-2, there are two pods defining functions a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Functions a and b can call each other without a problem because they are in the same deck and thus the same namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In all other four decks, the reference to either a or b will throw errors because they belong to different namespaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 NS Rule 2: Public Interface to Parent In order to build up the software, we have to establish connections between the definitions of code pods in different namespaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Soft- ware programs are often highly hierarchical: lower-level functions are composed together to build higher-level functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This is a natural fit to the Codepod model, where code blocks are ordered hierarchically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus in this rule, we allow public interfaces to be exposed from child decks to parent decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' More specifically, each pod can be marked as “public”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Such public pods are visible in the parent deck of the current deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 3, there are 3 decks, thus 3 namespaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The three namespaces are composed hierarchically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Deck-A is the parent of Deck-B, which is the parent of Deck-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Deck-C, there are four pods, defining four functions c1, c2, c3, and c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Those functions can see each other because they are in the same namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The pods for c1, c2, and c4 are marked public (indicated by highlight), while c3 is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In its parent deck Codepod: A Namespace-Aware, Hierarchical Jupyter for Interactive Development at Scale Conference’23, Jan, 2023, Ames, IA, USA Figure 1: Codepod overview example for Regular Expression code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Figure 2: NS Rule 1: separate namespace by default Figure 3: NS Rule 2: export to parent namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Yellow highlights indicate pods to be exported/exposed to parent decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (Deck-B), the call of c1, c2, c4 is allowed, meaning that they are available in this parent namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, the usage of c3 will raise an error because it is not exposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The public functions are exported only to the parent deck but not to the child decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, function b1 is defined in Deck- B, and the pod is marked public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This function b1 is visible to its parent deck, Deck-A, but not to its child deck, Deck-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Lastly, the public interface is only exposed to one level up the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' If the names are desired to be visible further up, the names can be further exposed up to the root deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, although c4 is marked as public, it is only visible to its immediate parent deck, Deck-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Calling c4 in the for pod for a2 in Deck-A will raise an error as c4 is not visible in Deck-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In the middle deck, Deck-B, the functions c1 and c2 are re-exported to the parent deck, and thus c1 and c2 are available in the top deck, Deck-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In summary, this “up-rule” allows users to mark a pod public and expose it to one-level deck up, and can be re-exported to upper levels explicitly until the root pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This namespace rule closely resembles the hierarchical nature of software and is natural to use this to build up complex functionalities from the ground up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='4 NS Rule 3: Utility Pods Although exposing pods from child decks to parent decks is natural for building software, it cannot cover all use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' One particular access pattern is utility functions that are supposed to be called in many other pods at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This is commonly used in real-world software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, many software projects will have a utils folder that implements utility functions such as string manipulation, general parsing, logging functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Such utility func- tions are used by other functions at different hierarchy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In the Codepod hierarchy, pods for such functions need to be children for all other pods calling the utility functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' thus, the model will no longer be a tree but a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, modeling code blocks as graphs is not as scalable as trees, and too many utility pods will CPiwTe3yqmmC sre parse Pattern parse class Pattern(): 1 def parse(): def closegroup(): pass 2 _parse_sub() 3 def opengroup(): pass 3 isstring() sre_compile 4 Tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='match() SubPattern class SubPattern(): compile _parse_sub 2 def closegroup(): pass 1 def compile(): 1 def _parse_sub(): m def append(): pass re 2 _code() 2 _parse() 4 def getwidth(): pass 3 parse() 3 Tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='match() compile 4 isstring() Tokenizer def re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='compile(): _parse 1 class Tokenizer(): compile() _code def _parse(): 2 def get(): pass 1 def _code(): 2 _parse_sub() 3 def match(): pass match 2 _compile() 3 _escape() def match(): 3 compile_info() 4 Pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='opengroup() _compile().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='search() CPQW6M4jqdqG +Test 5 SubPattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='getwidth() _compile_info Tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='get() 1 p = Pattern() search 1 def _compile_info(): 2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='opengroup("a") def search(): 2 SubPattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='getwidth() ROOT escape 3 print(p) 2 _compile().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='match() 1 def _escape(): _compile 1 _parse(p,"hello abc") Pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='checkgroup() _compile 1 def _compile(): 1 _escape(p,"a") def _compileO: 2 _simple() 2 sre_compile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='compile() 3 SubPattern CPbxCz8GETTx AUtility _simple 1 def _simple(): isstring 2 SubPattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='getwidth() 1 def isstring(obj): CPGGBRbcaBGb +Test 2 return isinstance( 1 print("Testing .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='."' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=') 3 obj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='(str,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='bytes)) 2 isstring("abc") isnumber 1 isstring(1) def isnumber(obj): 2 return isinstance( 3 obj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='(int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' float))Deck-2 Deck-4 a # ERROR not defined 1 def a(): 2 a() 2 # ok 3 return b() Deck-1 Deck-5 b 1 # ERROR not defined 1 def b(): 1 # ERROR not defined 2 a() 2 # ok 2 a() 3 return a() Deck-3 1 # ERROR not defined 2 a()Deck-B Deck-C Deck-A b1 def b1(): c1 1 al 2 # ok,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' cl exported def cl(): 1 def al(): 3 return cl() 2 return c1() + c3() 2 # ok 3 b1() cl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='c2 c2 4 # ok 1 # dummy pod def c2(): 5 c1() 2 # re-export cl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='c2 2 c3() a2 b3 c3 1 def a2(): 1 def b3(): 1 def c3(): 2 # ERROR not defined 2 # ERROR R not defined 2 pass 3 b4() 3 c3() c4 4 # ERROR not defined c4() b4 5 1 def c4(): 1 def b4(): 2 pass c4()Conference’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Ames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' USA Hebi Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Forrest Sheng Bao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Qi Xiao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jin Tian Figure 4: NS Rule 3: Utility pods/decks (indicated in green icon Utility) make it impossible to layout the pod hierarchy cleanly in a 2D space without many intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus, we design a “utility rule”: a deck/pod can be marked as a utility deck/pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Such a utility pod is meant to provide utility functions to the parent deck’s sub-tree, and thus all the public functions in the utility deck are visible in the parent deck’s whole sub-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The utility pods are also namespace-aware: it is only visible to the parent deck, but not the grand-parent deck and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus the utility decks can also be hierarchically ordered to build utility functions at different abstraction levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' As a special case, a utility deck under the root deck defines global utility functions that can be accessed throughout the entire codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 4, there are three regular decks and two utility decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The public functions utils_b1 and utils_b2 defined in utility deck B are visible in its parent deck B’s sub-tree, including decks B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Another utility deck, A, is defined in the upper level and has a greater visible scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='5 NS Rule 4: Testing Pods Figure 5: NS Rule 4: Testing pods/decks (indicated in green icon Test) Another essential pattern in interactive software development is to test whether the functions work as expected by writing some testing code and observing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Such testing code must access the functions being tested, thus having to be in the same namespace or the parent namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, either option has problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' On the one hand, testing code might create variables, introduce helper functions, and produce side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus they should be in a separate namespace to avoid polluting the namespace of the functions under testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' On the other hand, placing the testing deck as the parent deck of the function under testing is not logically natural because it does not provide upper-level functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Therefore, we allow a deck/pod to be marked as a testing deck/- pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A testing deck is placed as a child deck in the same namespace of the functions being tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Although the testing deck/pod is a child-namespace, it can access the definitions visible within its parent deck, thus is able to call and test the function of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The testing pods are also namespace-aware: it can only access the function definitions in its parent deck, but not the grand-parent deck or siblings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus the testing decks can also be hierarchically ordered to build testing functions at different abstraction levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 5, there are three regular decks and two testing decks, and one testing pod inside the regular deck A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The code pods in the testing deck are visible within the same testing deck, allowing for a testing setup like defining variables x and y and using them in other pods in the same testing deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The pods in the testing deck run in a separate namespace, thus will not pollute other namespaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A testing deck can access functions defined in its parent deck and can thus call and test whether the function yields expected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A testing pod is similar to a testing deck, running in a separate namespace, and has access to the function definitions in the deck it belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='6 NS Rule 5: Explicit Access via Full Path Figure 6: NS Rule 5: explicit imports by full path Finally, the 5th rule is the “brute-force rule”: a pod can always be accessible by specifying the full path within the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In other words, all pods are accessible via an explicit full path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This provides compatibility for arbitrary imports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This is considered the last resort and is ideally not needed but can be helpful in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The path can be either a relative path connecting two pods or the absolute path from the root deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In order to specify the path, the decks have to be named.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Codepod,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' an unnamed deck receives a UUID as the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='Deck-C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def c1(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_bl() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='Deck-B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def c2(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def b(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_b2() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='# OK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_al() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='Deck-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_a2() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='# OK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='UtilityDeck-B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='AUtility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def al(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_bl() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_al() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_b2() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_bl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_a2() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 def util_b(): pass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def a() : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='# ERROR not defined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_bi() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 def util_b2(): pass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='UtilityDeck-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='AUtility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 def util_al(): pass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='util_a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 def util a2(): passDeck-C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def c1(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='Deck-B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='pass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='b1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def b1(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def c2(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='Deck-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='return c() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='pass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def b2(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='TestDeck-B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='+Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='def a() : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='print("Testing b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='.") 3 2 assert bl() == 1 4 assert b2() == 2 5 4 assert cl() == 3 1 print("Testing a + Test assert a() == 3 TestDeck-A +Test 1 # test context setup 2x=2 3y=3 1 # x and y are available 2 print("Testing a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='.") 3 assert a(x) == yC C 1 # explicit relative import 2 import d from .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='./D B 3 def c() : 4 ()p 1 2 A D 1 d 2 def d(): 2 CPrFHVDyXTrC # explicit absolute import 2 import c from /A/B/C 3 def f() : 4 c()Codepod: A Namespace-Aware, Hierarchical Jupyter for Interactive Development at Scale Conference’23, Jan, 2023, Ames, IA, USA name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Most of the time, developers do not need to specify names to the decks, as the first four rules will make the modules system usable without specifying names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Named decks are also helpful as a document for naming important module hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 6, there are 5 decks in the codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Deck- D, a function d is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Deck-C, the function d is not accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, it is still able to be imported by a relative path .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='./D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' As another example, in the bottom deck, a full absolute path /A/B/C is used to access the function c defined in Deck-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='7 Discussion In summary, based on the hierarchical pod model, Codepod pro- vides a simple yet powerful module system including five rules: namespace separation, public pods, utility pods, testing pods, and full-path explicit access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' These rules are highly hierarchical, and therefore are well suited for building hierarchical software projects from the ground up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This module system abstracts over different programming languages’ native module systems and provides a con- sistent module system across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The following section will discuss the runtime system and algorithms to support the Codepod module system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Let us revisit the Codepod example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 1, and see how these namespace rules are useful in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This ex- ample implements a simplified Python regular expression com- piler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The functions are ordered into the decks re, sre_compile and sre_parse decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' sre_parse is the basic buiding block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' It contains a child deck that defines three internal classes, Pattern, SubPattern, Tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' These classes are only used in sre_parse module and not exposed to the upper level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The sre_parse mod- ule defines several helper functions including _parse_sub, _parse, _escape, and they are used to build a parse function which is ex- posed to parent module sre_compile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Similarly, module sre_compile defines some internal helper functions that are composed to pro- vide compile to the parent re module to build the top level API compile, match and search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The general functions isstring and isnumber are defined in a utility deck and are accessed through the modules sre_compile and sre_parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Finally, the testing pods at different hierarchy make it easy to test and debug the functions at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Ordering code blocks in Codepod is natural in building the dif- ferent levels of abstractions, and the hierarchy of code is close to the call graph, and is more cleanly maintained compared to file editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The Codepod implementation maintains a clear code hi- erarchy and makes it easy to develop the project interactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In comparison, the file-based implementation using VSCode would spread the functions into files, and within the file, the hierarchy of the functions is not clearly maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Writing all the functions into a Jupyter notebook is challenging due to the lack of namespace support, and the code hierarchy cannot be maintained within a single global namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='8 Version Control Implementing a version control system requires tremendous effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' There already exists mature file-based version control systems such as git and svn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We re-use the git version control system to apply version control to Codepod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Specifically, the code pods are first CodePod Front-end Runtime Kernel RunCode Request Complete EvalInNS(code, ns) AddImport(from, to, name) DeleteImport(from, name) Jupyter Protocol CodePod added protocol Figure 7: Kernel Communication Protocols exported to files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Then git version control is applied to the generated files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For front-end rendering, we query the git version control system for the diff between two versions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=', the current changes or specific git commit changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Then the diff results are parsed to show pod-level diffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 3 INCREMENTAL RUNTIME Codepod develops an intuitive, effective, consistent cross-language scope-aware incremental runtime abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Codepod, the run- time loads the code pods in the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' When a pod in some scope is changed, the updated definition can be applied incrementally in that scope in the active runtime without restarting the runtime, and the new definition is visible for other pods depending on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 Runtime Kernel Communication We build the Codepod Kernel communication based on the Jupyter Message Queue protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The Jupyter kernel protocols and our added protocols are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In its simplest form, Jupyter kernel messaging supports eval and complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We add the fol- lowing protocol to the messaging queue: EvalInNS, AddImport, DeleteImport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The protocol EvalInNS instructs the langauge ker- nel to evaluate code in a specific namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The AddImport proto- col makes a function “name” defined in “from” namespace available in the “to” namespace, and DeleteImport undo the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The algorithm for the language kernel is given in next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 Language Kernel Algorithm The language kernel needs to support four functions to work with Codepod: GetModule, EvalInNS, AddImport, DeleteImport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Most languages do not natively support all these operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We imple- ment a thin wrapper of these functions as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' GetModule is a function that gets the module instance for a spe- cific namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Since we need to refer to the same module given the same name, we need to maintain a mapping from namespace names to the module instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In line 1, the nsmap object is created as a global variable, initialized with an empty dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The Get- Module function will query this map, and if the module already exists, return the module instance (lines 3-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' If the module is not found, create a new module by calling the language’s createMod API, record the module in the nsmap, and return the module (lines 6-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Conference’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Ames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' USA Hebi Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Forrest Sheng Bao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Qi Xiao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jin Tian Algorithm 1 Namespace-aware Runtime: Language Kernel 1: 𝑛𝑠𝑚𝑎𝑝 ← 𝐸𝑚𝑝𝑡𝑦𝐷𝑖𝑐𝑡 () 2: function GetModule(ns) 3: if 𝑛𝑠 ∈ 𝑛𝑠𝑚𝑎𝑝 then 4: return 𝑛𝑠𝑚𝑎𝑝 [𝑛𝑠] 5: else 6: 𝑚𝑜𝑑 ← 𝑐𝑟𝑒𝑎𝑡𝑒𝑀𝑜𝑑 (𝑛𝑠) 7: 𝑛𝑠𝑚𝑎𝑝 [𝑛𝑠] ← 𝑚𝑜𝑑 8: return 𝑚𝑜𝑑 9: end if 10: end function 11: function EvalInNS(ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' code) 12: 𝑚𝑜𝑑 ← GetModule(ns) 13: 𝑖𝑠𝐸𝑥𝑝𝑟 ← IsExpr(code) 14: if isExpr then 15: return eval(code,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' mod) 16: else 17: exec(code,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' mod) 18: return NULL 19: end if 20: end function 21: function AddImport(from,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' to,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' name) 22: 𝑠 ← "$name=eval($name,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='from)" 23: EvalInNS(to,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' s) 24: end function 25: function DeleteImport(from,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' name) 26: 𝑠 ← "del $name" 27: EvalInNS(from,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' s) 28: end function The EvalInNS function is responsible for loading the module specified by the namespace and evaluating code with that mod- ule instance active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' It first loads or creates the module instance by the GetModule function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Many languages treat expression and statement separately, where expressions return values, while state- ments do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus the algorithm first checks whether this code is an expression or statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' If it is an expression, the language’s EVAL API is called with the code and module instance and returns the expression results for display in the front-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Otherwise, the language’s EXEC API is called to evaluate the code in the module instance for side-effect only and return NULL to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Finally, AddImport and DeleteImport work by meta-programming: a new program string is constructed to add or delete names in the target namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The function AddImport receives the name to import and the "from" and "to" namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The goal is to import the function "name" from the "from" namespace and make it available in the "to" namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A string is constructed to assign a variable with the same name "name" with the evaluation of the name in the "from" namespace (lines 22-23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The DeleteImport works by constructing a "delete $name" code and evaluating the target namespace (lines 26-27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We note that the EVAL and EXEC API with module awareness are different across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Some languages provide native support, while others might need to perform reflection-level operations, such as manually passing in Python symbol maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The code strings for AddImport and DeleteImport are generally different across languages too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We supply the core code of the four kernels we have implemented in Figure 8, 9, 10, and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 Pod Hierarchy Algorithm This section formally describes the key algorithms in Algorithm 2 that implement Codepod’s module system, the namespace rules, and how the evaluation of pods/decks is handled in the pod hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A pod is evaluated with the RunPod function, which calls EvalInNS with the pod’s code content and namespace string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A deck is eval- uated with the RunDeck function, which evaluates the subtree of the deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The function first evaluates all the child decks in a DFS (depth-first search) manner, then evaluates all the pods in this deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 1 d = {} 2 3 def getmod(ns): 4 if ns not in d: 5 d[ns] = types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ModuleType(ns) 6 d[ns].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' __dict__[" CODEPOD_GETMOD"] = getmod 7 return d[ns] 8 9 def add_import(src , dst , name): 10 return eval_func(""" 11 {name}= getmod ({src}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' __dict__ ["{ name }"] 12 """, dst) 13 14 def delete_import(ns, name): 15 eval_func("""del {name}""", ns) 1 def eval_func(code , ns): 2 mod = getmod(ns) 3 [stmt , expr] = code2parts(code) 4 if stmt: 5 exec(stmt , mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='__dict__) 6 if expr: 7 return eval(expr , mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' __dict__) Figure 8: Python Kernel Namespace Implementation 1 var NSMAP = NSMAP || {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 2 function eval(code , ns, names) { 3 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='NSMAP[ns]) { 4 NSMAP[ns] = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 5 } 6 for (let k of keys(NSMAP[ns])) { 7 eval( 8 `var ${k}=NSMAP["${ns}"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='${k}` 9 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 10 } 11 let res = eval(code);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 12 for (let name of names) { 13 eval( 14 `NSMAP["${ns}"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='${name}=${name}` 15 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 16 } 17 return res;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 18 } 1 function addImport(from , to, name) { 2 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='NSMAP[from]) { 3 NSMAP[from] = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 4 } 5 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='NSMAP[to]) { 6 NSMAP[to] = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 7 } 8 eval(` 9 NSMAP["${to}"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='${name}= 10 NSMAP["${from}"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='${name}`);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 11 } 12 function deleteImport(ns, name) { 13 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='NSMAP[ns]) { 14 NSMAP[ns] = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 15 } 16 eval( 17 `delete NSMAP["${ns}"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='${name}`) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 18 } Figure 9: Javascript Kernel Namespace Implementation 1 function isModuleDefined(names) 2 mod = :(Main) 3 for name in names 4 name = Symbol(name) 5 if !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='isdefined(eval(mod), name) 6 return false 7 end 8 mod = :($mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='$name) 9 end 10 return true 11 end 12 function ensureModule(namespace) 13 names = split(namespace , "/", 14 keepempty=false) 15 mod = :(Main) 16 for name in names 17 name = Symbol(name) 18 if !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='isdefined(eval(mod), name) 19 include_string(eval(mod), 20 "module $name end") 21 end 22 mod = :($mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='$name) 23 end 24 return mod ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' eval(mod) 25 end 1 function eval(code ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' ns) 2 _,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' mod = ensureModule(ns) 3 include_string(mod ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' code) 4 end 5 function addImport(from ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' to,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' name) 6 from_name ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' _ = ensureModule(from) 7 _,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' to_mod = ensureModule(to) 8 code = """ 9 using $from_name: $name as CP$name 10 $name=CP$name 11 $name 12 """ 13 include_string(to_mod ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' code) 14 end 15 function deleteImport(ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' name) 16 _,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' mod = ensureModule(ns) 17 include_string(mod ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' "$name=nothing") 18 end Figure 10: Julia Kernel Namespace Implementation The reason to evaluate child decks before child pods is that the child decks might define public functions exposed to this deck;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' thus, the pods must have those definitions in place before execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The reason to use DFS is to make sure the lowest-level pods are first evaluated before moving up the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Child pods are evaluated sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' When running the child pods, the algorithm examines Codepod: A Namespace-Aware, Hierarchical Jupyter for Interactive Development at Scale Conference’23, Jan, 2023, Ames, IA, USA 1 (compile-enforce-module-constants #f) 2 3 (define (ns- >submod ns) 4 (let ([names (string-split ns "/")]) 5 (when (not (empty?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=" names)) 6 (let ([one (string- >symbol 7 (first names))] 8 [two (map string- >symbol 9 (rest names))]) 10 ‘(submod ',one 11 ,@two))))) 12 (define (ns- >enter ns) 13 (let ([mod (ns- >submod ns)]) 14 (if (void?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=" mod) '(void) 15 ‘(dynamic-enter!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' \',mod)))) 16 17 (define (ns- >ensure-module ns) 18 (let loop 19 ([names (string-split ns "/")]) 20 (if (empty?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=" names) 21 '(void) 22 ‘(module 23 ,(string- >symbol 24 (first names)) 25 racket/base 26 ,(loop (rest names)))))) 1 (define (add-import from to name) 2 (let ([name (string- >symbol name)]) 3 (eval (ns- >enter to)) 4 (eval 5 ‘(define ,name 6 (dynamic-require/expose 7 ',(ns- >submod from) 8 ',name))))) 9 10 (define (delete-import ns name) 11 (eval (ns- >enter ns)) 12 (namespace-undefine-variable!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 13 (string- >symbol name))) 14 15 (define (string- >sexp s) 16 (call-with-input-string 17 s 18 (lambda (in) 19 (read in)))) 20 21 (define (codepod-eval code ns) 22 (eval (ns- >ensure-module ns)) 23 (eval (ns- >enter ns)) 24 (begin0 25 (eval 26 (string- >sexp 27 (~a "(begin " code ")"))) 28 (enter!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' #f))) Figure 11: Racket Kernel Namespace Implementation Algorithm 2 Namespace-aware Runtime: Pod Hierarchy 1: function RunPod(pod) 2: EvalInNS(pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ns, pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='code) 3: end function 4: function RunTest(ns, code) 5: for 𝑛𝑎𝑚𝑒 ← 𝑝𝑜𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='𝑝𝑎𝑟𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='𝑛𝑎𝑚𝑒𝑠 do 6: AddImport(pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ns, pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ns, name) 7: end for 8: EvalInNS(pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ns, pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='code) 9: end function 10: function RunUtility(from, name) 11: EvalInNS(pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ns, pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='code) 12: function dfs(parent) 13: for 𝑐ℎ𝑖𝑙𝑑 ← 𝑝𝑎𝑟𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='𝑐ℎ𝑖𝑙𝑑_𝑑𝑒𝑐𝑘𝑠 do 14: for 𝑛𝑎𝑚𝑒 ← 𝑝𝑜𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='𝑝𝑎𝑟𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='𝑛𝑎𝑚𝑒𝑠 do 15: AddImport(pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ns, pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='ns, name) 16: end for 17: dfs(child) 18: end for 19: end function 20: dfs(pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='parent) 21: end function 22: function RunTree(root) 23: function dfs(parent) 24: for 𝑐ℎ𝑖𝑙𝑑 ← 𝑝𝑎𝑟𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='𝑐ℎ𝑖𝑙𝑑_𝑑𝑒𝑐𝑘𝑠 do 25: RunTree(child) 26: end for 27: end function 28: dfs(pod) 29: for 𝑝𝑜𝑑 ← 𝑟𝑜𝑜𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='𝑐ℎ𝑖𝑙𝑑_𝑝𝑜𝑑𝑠 do 30: if pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='type is "Pod" then 31: RunPod(pod) 32: else if pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='type is "Test" then 33: RunTest(pod) 34: else if pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='type is "Utility" then 35: RunUtility(pod) 36: end if 37: end for 38: end function the type of the pods and calls the corresponding functions RunPod, RunUtility, and RunTest for different types accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Hierarchical Layout Commnication Protocol Kernel Runtime CodeServer API Total LOC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='8k 1k 1k 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='9k Table 1: Codepod Implemenatation Statistics The RunUtility function will first evaluate the pods in the names- pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Then, the public names are exported to the parent’s subtree by traversing the parent’s subtree in a DFS manner and calling AddImport for each of the namespaces in the sub-tree during tra- versal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In this way, the name of the utility function is available in all the decks of the parent’s sub-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The RunTest function will loop through the parent deck’s public names and run AddImport for all the names from the parent’s namespace to the testing pod’s namespace, and then evaluate the test pods in the namespace where the parent’s function definitions are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='4 Fallback Execution Codepod requires the programming languages to support interpret- ing in order to run and evaluate code interactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' With interactive development becoming popular, there have been many interpreters implementations for even compiled languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, C++ has a highly regarded interpreter, Cling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Another requirement is that the language needs to support namespace and provide a way to evaluate code blocks within the namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Suppose the support of namespace-aware interactive develop- ment is not fully supported due to the limitation of the language interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In that case, Codepod provides a fallback option: export- ing code to files and invoking the language interpreter/compiler to run the program as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The downside of this approach is that the variables are not persisted in memory across runs because each invocation starts a new process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 4 CASE STUDIES 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1 Implementation We have implemented a fully working Codepod as a Web applica- tion (front-end in React and backend server in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='js) and cur- rently implemented full namespace-aware runtime support for four language kernels, including Python, JavaScript, Julia, and Scheme/Racket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' New kernels can be easily developed upon exist- ing Jupyter notebook kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod is open-sourced at https: //example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod implementation contains 4 major parts, the LOC sta- tistics is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The hierarchical layout implements the front-end hierarchical pods and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The communication proto- col implements the RunTree/RunPod logic and how the front-end communicates with the backend API server and language runtime kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Kernel runtime implements the kernels and WebSocket protocol message handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Finally, CodeServer API implements the API for retrieving and updating pods hierarchy from the front-end and talking to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Conference’23, Jan, 2023, Ames, IA, USA Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2 Python and Julia Projects Statistics In this section, we study several highly regarded open-source projects and visually show how the representation of Codepod can poten- tially help develop code projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This case study aims to see how functions are distributed in the files and directories and the calling relations of the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This helps us evaluate how the Codepod model can help with real-world open-source software projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The projects are obtained from GitHub’s top-rated Python and Julia projects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' the information about the projects is shown in Ta- ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We analyze the projects with the help of tree-sitter [1] parser framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We count only the source directory of the projects, ignoring testing code and documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Our study focuses on func- tions as the code block granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Python projects might contain classes, and we treat each method as a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Table 2, we can see that software projects often contain a large number of functions, and those functions are distributed in a large number of files, possibly in a deep directory hierarchy of tens of directories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' For example, the you-get project contains 444 functions and is distributed into 133 files in 8 directories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The LightGraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl project contains 242 functions, distributed in 110 files within 27 directories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' It can be pretty challenging to grasp the hierarchy of the functions by browsing through the files and directories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We also count the number of internal and external functions of each file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A file’s internal functions are defined as those only called by the other functions in the same file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' These internal functions are helper functions that are used to implement the external functions that act as the public interface of the file, called by other files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Table 2, we see that approximately half of the files are internal and are used as building blocks of other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, this dependency information is not clear in a file because the functions are considered linear within a file, and no clear hierarchy can be effectively maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus potentially, Codepod can help to apply hierarchical relations to the functions inside each file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Figure 12: Statistics for top open source Python Projects To understand the distributions of functions in each file and call- ing relationships, we plot the in-degree,out-degree,LOC,#functions- per-file of the Python and Julia Projects in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 13, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The in/out-degrees of a function is defined by how many calls to the functions and how many calls this function makes to other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Both in/out-degrees are computed based on the call Figure 13: Statistics for top open source Julia Projects graphs over the functions defined in the project;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' thus, the statistics do not include language or library API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' From the plot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (a), we see that most func- tions are being called no more than two times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This shows that most functions are “local”, and only used to implement higher-level functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Also, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (b), we observe that the out-degree is more than in-degree, and most functions have less than five function calls to other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This is also consistent with Codepod’s tree-based model: a deck in a tree node might have multiple sub-decks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A pod defined in a deck might use the functions defined in the sub-decks to implement higher-level functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' From the function per file plot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (d) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' (d), we can see that the distribution of functions into files are not even: a large number of files contain only 1 or 2 functions, while there can also be a few very large file containing tens of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This data shows that in order to maintain a cleaner hierarchy, developers might use a separate file for each function, resulting in too many small files and deep directories, which are relatively hard to maintain, edit and reference using file browsers and file editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Also, within the large files with tens of functions, the hierarchy of those functions cannot be cleanly maintained within a file, and Codepod could help build a finer-granular hierarchy for the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3 Jupyter Project Statistics In this study, we investigate how Jupyter notebooks are used in real- world open projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This study aims to see how code is distributed among Jupyter notebooks and text files and how the Jupyter note- books interact with the functions defined in text files regarding calling relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We query GitHub APIs to find top Python projects whose primary language is Python and whose secondary language is Jupyter Notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The projects and statistics are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Table 3, we can see that there are typically more text files than Jupyter notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This is also the case for the total LOC and number of functions in Jupyter notebooks vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' text files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In fact, the LOC of the Jupyter notebook consists only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='7% of the codebase for these projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This means that the majority of the code is implemented in the text files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' To understand the calling relationships of the Jupyter notebooks and text files, we calculate the percentage of internal functions ab-no DOE cookiecutter locust 150 unoo requests 200 100 100 50 5 10 5 1D 15 (afuncbion indegree bj functinoutdegree 60 60 40 40 no 20 20 0 0 0 25 50 75 100 0 5 10 15 20 (ciavglocperfunction dyfunctionsperfileJuliaDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl 100 HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jI 100 Flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl un LightGraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl 50 50 5 10 0 5 10 15 al function indegree (b) functin outdegree 30 60 20 40 10 20 0 0 0 25 50 75 100 5 10 15 20 (c) avg loc perfunction (d) functionsper fileCodepod: A Namespace-Aware, Hierarchical Jupyter for Interactive Development at Scale Conference’23, Jan, 2023, Ames, IA, USA #stars #dirs #files #loc #funcs #internal funcs description soimort/you-get 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='6k 8 133 14707 444 273 CMD-line utility to download media from Web cookiecutter/cookiecutter 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2k 0 18 2139 51 30 CMD-line utility to create projects from templates locustio/locust 17k 10 59 18671 529 144 performance testing tool psf/requests 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='9k 0 18 5183 135 73 HTTP library JuliaData/JuliaDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl 706 0 20 3113 106 82 Parallel analytical database JuliaWeb/HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl 439 0 38 7513 65 36 HTTP client and server FluxML/Flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='2k 5 33 6408 84 41 Machine Learning Framework JuliaGraphs/LightGraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='jl 675 27 110 16963 242 101 Network and graph analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Table 2: Function Statistics in Open Source Projects (Python and Julia) #stars #file (ipynb/files) #loc (ipynb/files) #call fs-to-nb #call nb-to-fs #func (ipynb/files) #%internal (ipynb/files) description blei-lab/edward 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='6k 14/42 1340/5449 0 11 10/102 100%/93% A probabilistic programming language tqdm/tqdm 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='3k 2/30 284/2257 0 9 0/83 NA/80% A Fast, Extensible Progress Bar google/jax 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='1k 3/196 104/42879 0 8 0/2418 NA/38% Autograd and Optimizing Compiler for ML google-research/bert 29k 1/13 322/4547 0 8 7/92 100%/88% State-of-the-art NLP language model quantopian/zipline 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='4k 2/183 115/30087 0 3 3/851 100%/61% Algorithmic Trading Library Table 3: Jupyter Notebooks statistics in Open Source Projects for files and Jupyter notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The internal function is defined the same as above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=', the functions that are only called from the same file or Jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' From the result, two projects contain no function definitions in the Jupyter notebooks, and the other three projects’ functions are 100% internal to the notebook files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In contrast, the text files contain 38% to 93% internal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Also, we calculate the number of calls from the Jupyter notebook to text files and vice versa and show that there are no calls from source text files to notebooks, but only calls from the Jupyter notebook to the text files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This means that the Jupyter notebook is not used to develop functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The projects are implemented in text files, and Jupyter notebooks are only used to call those files’ APIs and are most likely for presentation and tutorial purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 5 RELATED WORK Running and debugging code is a major activity in software de- velopment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' There have been many tools to help the debugging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In the simplest form, developers write code in files using some editors such as VIM and Emacs [13], and compile and run files in the command line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The problem with this approach is that it is non-interactive, and the program always needs to restart from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' An interactive development method is to launch a Read-Eval-Print-Loop (REPL) [7, 14], type, load, and evaluate code expressions in the REPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Although REPL is interactive, the code being evaluated is not editable, and users have to type code into the REPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' It is also common to open a file editor and send a code region to the REPL for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Integrated Development Environments (IDEs) such as VSCode integrate file editors with a file browser, command line, plugins, and debuggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' There also exist editor plugins to navigate between the functions of a project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, those plugins still do not give a within-file hierarchy to the code and depend on the programming language designs to support the within-file module system, which only a handful of languages support to various degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' File-based approaches force developers to maintain the correspondence between code and files, which is tricky due to the significantly different granularity of code blocks and files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The unbalance in granularity poses dilemmas to developers: including too many code blocks into one file makes the hierarchy hard to maintain, while including few code blocks into one file creates many small files and deep directories that are also hard to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Figure 14: Interactive Development with Jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Image from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A recent new paradigm is a web-based interactive notebook called Jupyter Notebook [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The interface of Jupyter is shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The notebook consists of code cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Each cell can be run in arbitrary order, and the results are displayed under the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The code output can be visualized, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=', plotting a figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=" Thus Jupyter notebooks support literate programming that combines code, text, Fibonnaci In [3]: def fib(x): if x <= 1: Markdown return x Cells return fib(x-1) + fib(x-2) fib(10) Out[3]: 55 Output 1 Code Cells Let's plot the numbers In [8]: from matplotlib import pyplot %matplotlib inline x = range(15) y = [fib(n) for n in x] pyplot." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='plot(x, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Execution 350 300 Counter 250 200 Output 2 150 100 50 0 6 10 12 14Conference’23, Jan, 2023, Ames, IA, USA Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian and execution results with rich media visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, the Jupyter notebook cells are linear, and all the code blocks live in the global namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jupyter notebook lacks a module system that is crucial for complex software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' This makes it hard to develop a large-scale software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Thus Jupyter notebook is typically used only for surface demo and visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The real code of the projects is still developed in text files with text editors and external runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In order to define hierarchical code, users have to write code into text files and import the module from text files into the notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Such a Jupyter-file hybrid model has several disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Changes in files are not in sync with the Jupyter runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' When a function definition in a file is changed, the Jupyter runtime must be restarted, and the file must be reloaded for the change to take effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Another programming paradigm is Visual Programming, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Lab- View [2], and Google’s Blockly [9], and Microsoft’s MakeCode [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Such a visual programming paradigm has distinguished advantages: it is visually clean, easy to learn, and impossible to make syntax er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, all these visual programming systems use such block style items down to each expression (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=', a+b), which can be ver- bose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Also, visual programming blocks live in a global namespace, and there is no module system available for developing large-scale software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' There exist code analyzers to generate a visual presentation of code, such as Unified Modeling Language (UML) [3, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Tools such as call graph visualizers [4] are also developed to help to understand a codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' However, those visual presentations are not editable, making them less useful during development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we propose Codepod, a namespace-aware hierarchi- cal interactive development environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod uses a novel hierarchical code block model to represent code and abstracts away files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We propose namespace rules that make it easy to organize the pods and provide a consistent module system across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Codepod provides an incremental evaluation runtime that helps interactively develop a large-scale software project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We hope Codepod can provide a novel way to drive the software development process and inspire other research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In the future, we will push Codepod forward with the contribution from the com- munity, perform user studies to compare Codepod with VSCode and Jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' It is interesting to integrate program analysis tools into the Codepod model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' We are also interested in integrating other programming paradigms into the Codepod framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' for exam- ple, it would be helpful to integrate graphical programming as a “graphical pod” and mix graphical programs with plain-text code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' REFERENCES [1] Tree-sitter: An incremental parsing system for programming tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' https://tree- sitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='io/tree-sitter/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Accessed: 2021-07-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [2] Rick Bitter, Taqi Mohiuddin, and Matt Nawrocki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' LabVIEW: Advanced program- ming techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Crc Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [3] Grady Booch, Ivar Jacobson, James Rumbaugh, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' The unified modeling language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Unix Review, 14(13):5, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [4] David Callahan, Alan Carle, Mary W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Hall, and Ken Kennedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Constructing the procedure call multigraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' IEEE Transactions on Software Engineering, 16(4):483– 487, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [5] James Devine, Joe Finney, Peli de Halleux, Michał Moskal, Thomas Ball, and Steve Hodges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Makecode and codal: intuitive and efficient embedded systems programming for education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' ACM SIGPLAN Notices, 53(6):19–30, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [6] Thomas Kluyver, Benjamin Ragan-Kelley, Fernando Pérez, Brian Granger, Matthias Bussonnier, Jonathan Frederic, Kyle Kelley, Jessica Hamrick, Jason Grout, Sylvain Corlay, Paul Ivanov, Damián Avila, Safia Abdalla, and Carol Will- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Jupyter notebooks – a publishing format for reproducible computational workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Loizides and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Schmidt, editors, Positioning and Power in Academic Publishing: Players, Agents and Agendas, pages 87 – 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' IOS Press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [7] John McCarthy, Michael I Levin, Paul W Abrahams, Daniel J Edwards, and Timothy P Hart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' LISP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content='5 programmer’s manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' MIT press, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [8] Nenad Medvidovic, David S Rosenblum, David F Redmiles, and Jason E Rob- bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Modeling software architectures in the unified modeling language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' ACM Transactions on Software Engineering and Methodology (TOSEM), 11(1):2–57, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [9] Erik Pasternak, Rachel Fenichel, and Andrew N Marshall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Tips for creating a block language with blockly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In 2017 IEEE Blocks and Beyond Workshop (B&B), pages 21–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [10] Jeffrey M Perkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Why jupyter is data scientists’ computational notebook of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Nature, 563(7732):145–147, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [11] João Felipe Pimentel, Leonardo Murta, Vanessa Braganholo, and Juliana Freire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A large-scale study about quality and reproducibility of jupyter notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In 2019 IEEE/ACM 16th International Conference on Mining Software Repositories (MSR), pages 507–517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [12] Bernadette M Randles, Irene V Pasquetto, Milena S Golshan, and Christine L Borgman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Using the jupyter notebook as a tool for open science: An empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In 2017 ACM/IEEE Joint Conference on Digital Libraries (JCDL), pages 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [13] Richard M Stallman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Emacs the extensible, customizable self-documenting dis- play editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Proceedings of the ACM SIGPLAN SIGOA symposium on Text manipulation, pages 147–156, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [14] L Thomas van Binsbergen, Mauricio Verano Merino, Pierre Jeanjean, Tijs van der Storm, Benoit Combemale, and Olivier Barais.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' A principled approach to repl interpreters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Proceedings of the 2020 ACM SIGPLAN International Symposium on New Ideas, New Paradigms, and Reflections on Programming and Software, pages 84–100, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' [15] Jiawei Wang, Li Li, and Andreas Zeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' Better code, better sharing: on the need of analyzing jupyter notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} +page_content=' In Proceedings of the ACM/IEEE 42nd International Conference on Software Engineering: New Ideas and Emerging Results, pages 53–56, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE0T4oBgHgl3EQffwDm/content/2301.02410v1.pdf'} diff --git a/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf b/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..93c3978e43299598a1ddc6bc685efd55f5248b2b --- /dev/null +++ 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Improved Hubble constant measurement from lensing time delays using +spatially resolved stellar kinematics of the lens galaxy +Anowar J. Shajib,1, 2,⋆ Pritom Mozumdar,3 Geoff C.-F. Chen,4 Tommaso Treu,4 Michele Cappellari,5 Shawn Knabel,4 +Sherry H. Suyu,6, 7, 8 Vardha N. Bennert,9 Joshua A. Frieman,1, 2, 10 Dominique Sluse,11 Simon Birrer,12, 13, 14 +Frederic Courbin,15 Christopher D. Fassnacht,3 Lizvette Villafaña,4 Peter R. Williams4 +(Affiliations can be found after the references) +Received xxx, xxxx; accepted xxx, xxxx +ABSTRACT +Strong-lensing time delays enable measurement of the Hubble constant (H0) independently of other traditional methods. The main limitation to +the precision of time-delay cosmography is mass-sheet degeneracy (MSD). Some of the previous TDCOSMO analyses broke the MSD by making +standard assumptions about the mass density profile of the lens galaxy, reaching 2% precision from seven lenses. However, this approach could +potentially bias the H0 measurement or underestimate the errors. In this work, for the first time, we break the MSD using spatially resolved +kinematics of the lens galaxy in RXJ1131−1231 obtained from the Keck Cosmic Web Imager spectroscopy, in combination with previously +published time delay and lens models derived from Hubble Space Telescope imaging. This approach allows us to robustly estimate H0, effectively +implementing a maximally flexible mass model. Following a blind analysis, we estimate the angular diameter distance to the lens galaxy Dd = +865+85 +−81 Mpc and the time-delay distance D∆t = 2180+472 +−271 Mpc, giving H0 = 77.1+7.3 +−7.1 km s−1 Mpc−1 – for a flat Λ cold dark matter cosmology. The +error budget accounts for all uncertainties, including the MSD inherent to the lens mass profile and the line-of-sight effects, and those related to the +mass–anisotropy degeneracy and projection effects. Our new measurement is in excellent agreement with those obtained in the past using standard +simply parametrized mass profiles for this single system (H0 = 78.3+3.4 +−3.3 km s−1 Mpc−1) and for seven lenses (H0 = 74.2+1.6 +−1.6 km s−1 Mpc−1), or for +seven lenses using single-aperture kinematics and the same maximally flexible models used by us (H0 = 73.3+5.8 +−5.8 km s−1 Mpc−1). This agreement +corroborates the methodology of time-delay cosmography. +Key words. cosmology: distance scale – gravitational lensing: strong – Galaxy: kinematics and dynamics – Galaxies: elliptical and lenticular, cD +– Galaxies: individual: RXJ1131−1231 +1. Introduction +The Hubble constant, H0, the current value of the Universe’s ex- +pansion rate, is a crucial cosmological parameter that also sets +the extragalactic distance scale. Recently, tension has emerged +between early- and late-Universe estimates of H0 (e.g., Freed- +man 2021; Abdalla et al. 2022). The temperature and polarisa- +tion fluctuations in the cosmic microwave background (CMB) +provide an estimate of the Hubble parameter at the last scatter- +ing surface H(z ≈ 1100), which can be extrapolated to the cur- +rent epoch using the Λ cold dark matter (ΛCDM) cosmology. +The CMB measurements from Planck give H0 = 67.4 ± 0.5 km +s−1 Mpc−1 (Planck Collaboration 2020) and H0 = 67.6 ± 1.1 km +s−1 Mpc−1(Aiola et al. 2020). In the local Universe, H0 can be +estimated using the cosmic distance ladder, which uses luminos- +ity distances of type Ia supernovae (SNe Ia) with their absolute +brightness calibrated using different classes of stars. The Super- +nova H0 for the Equation of State of the dark energy (SH0ES) +team uses Cepheids and parallax distances for this calibration, +and they find H0 = 73.04 ± 1.04 km s−1 Mpc−1 (Riess et al. +2022). This value is in 5σ tension with the Planck CMB-based +measurements. If this difference is not due to systematic errors +in either of these measurements (e.g., Efstathiou 2021), then this +tension could point to new physics beyond the ΛCDM cosmo- +logical model (e.g., Knox & Millea 2020). +⋆ NFHP Einstein Fellow +To determine whether this “Hubble tension” is due to sys- +tematics or new physics, multiple independent methods to mea- +sure H0 are needed (e.g., Verde et al. 2019; Di Valentino et al. +2021; Freedman 2021). The Carnegie–Chicago Hubble Project +uses the tip of the red giant branch (TRGB) to calibrate the +SNe Ia absolute brightness and measures H0 = 69.6 ± 1.9 km +s−1 Mpc−1 (Freedman et al. 2019, 2020). This TRGB-calibrated +measurement is statistically consistent with both the SH0ES +measurement and the CMB-based measurements. However, sev- +eral independent local probes strengthen the “H0 tension” by +measuring values consistent with the SH0ES value. For exam- +ple, the Megamaser Cosmology Project (MCP) estimates H0 = +73.9 ± 3.0 km s−1 Mpc−1(Pesce et al. 2020), the surface bright- +ness fluctuation (SBF) method measures H0 = 73.7 ± 0.7 ± 2.4 +km s−1 Mpc−1 (Blakeslee et al. 2021), and the Tully–Fisher- +relation-based method calibrated with Cepheids measures H0 = +75.1 ± 0.2 ± 0.3 km s−1 Mpc−1 (Kourkchi et al. 2020). +Strong-lensing time delays provide an independent measure- +ment of H0 (Refsdal 1964; for an up-to-date review, see Bir- +rer et al. 2022b; Treu et al. 2022, for a historical perspective, +see Treu & Marshall 2016). In strong lensing, a background +source appears as multiple images due to the gravitational de- +flection of photons by a massive foreground galaxy or galaxy +cluster. The photons that were emitted at the same time from +the background source arrive in different images with a rela- +tive time delay. This time delay carries cosmological informa- +Article number, page 1 of 21 +arXiv:2301.02656v1 [astro-ph.CO] 6 Jan 2023 + +A&A proofs: manuscript no. ms +tion through a combination of angular diameter distances in- +volved in the lensing system. This combination is referred to +as the “time-delay distance", which is inversely proportional to +H0 (Refsdal 1964; Schneider et al. 1992; Suyu et al. 2010). The +Time-Delay COSMOgraphy (TDCOSMO) collaboration has an- +alyzed seven time-delay lenses to measure H0 with 2% error, +H0 = 74.2±1.6 km s−1 Mpc−1 assuming a power-law or compos- +ite (i.e., stars and Navarro–Frenk–White (NFW) halo; Navarro +et al. 1996, 1997) mass profile for the lensing galaxies (Mil- +lon et al. 2020b). The TDCOSMO collaboration encompasses +the COSmological MOnitoring of GRAvItational Lenses (COS- +MOGRAIL; Courbin et al. 2005; Millon et al. 2020a), the H0 +Lenses in COSMOGRAIL’s Wellspring (H0LiCOW; Suyu et al. +2010, 2013; Bonvin et al. 2017; Birrer et al. 2019; Rusu et al. +2020; Wong et al. 2020), the Strong-lensing High Angular Reso- +lution Programme (SHARP; Chen et al. 2019), and the STRong- +lensing Insights into the Dark Energy Survey (STRIDES; Treu +et al. 2018; Shajib et al. 2020) collaborations. +The simple parametric lens models, e.g., the power law, +adopted in the TDCOSMO analyses are “industry standard” con- +sistent with non-lensing measurements. The TDCOSMO collab- +oration has performed various systematic checks on the adopted +lens modeling procedure. These checks find potential system- +atic biases to be lower than the acceptable limit (∼1%) from +the choice of mass model parametrization (i.e., power law or +composite, Millon et al. 2020b), from ignoring dark substruc- +tures in the lens galaxy’s halo (Gilman et al. 2020), from ig- +noring disky or boxy-ness in the baryonic distribution (Van de +Vyvere et al. 2022a), from using different lens modeling soft- +ware (Shajib et al. 2022a), and from ignoring potential isoden- +sity twists and ellipticity gradients in the lens galaxy (Van de +Vyvere et al. 2022b). However, a significant source of potential +systematics could arise due to the relatively simple parametriza- +tion of the lens mass profile (Kochanek 2020). The well-known +mass-sheet degeneracy (MSD) does not allow one to constrain +the mass profile shape of the deflector galaxy from lens imaging +observables alone (Falco et al. 1985; Schneider & Sluse 2013, +2014). Non-lensing observables, such as the deflector galaxy’s +velocity dispersion or the source’s unlensed intrinsic brightness, +are required to break the mass-sheet degeneracy and simultane- +ously constrain H0 and the mass profile shape (Treu & Koop- +mans 2002; Shajib et al. 2018; Yıldırım et al. 2020, 2021; Birrer +et al. 2020, 2022a). +The TDCOSMO collaboration has redesigned the experi- +ment to mitigate this systematic by relaxing the simple paramet- +ric assumptions in the mass profile and constraining the profile +shape solely from stellar velocity dispersion measurements of +the lensing galaxies (Birrer et al. 2020). Relaxing the assump- +tion on the mass profile leads to an increase in the H0 uncer- +tainty from 2 to 8% – which is dominated by the uncertainty of +the measured velocity dispersion – giving H0 = 74.5+5.6 +−6.1 km s−1 +Mpc−1. One approach to improving the precision is to incorpo- +rate prior information on the mass profile shape from the mea- +sured velocity dispersions of a larger sample of external lenses +without measured time delays. Assuming that the Sloan Lens +ACS (SLACS) survey’s lens galaxies are drawn from the same +population as the TDCOSMO lens galaxies and using their ve- +locity dispersions to constrain the mass profile shape, the un- +certainty on H0 improves to 5%, giving H0 = 67.4+4.1 +−3.2 km s−1 +Mpc−1(Birrer et al. 2020). Note that this estimate is statistically +consistent within 1σ with the larger 8% H0 measurement above. +However, the shift could also arise from systematic differences, +e.g., a difference between the parent populations of time-delay +and non-time-delay lenses (Gomer et al. 2022). Such differ- +ences could arise, for example, from evolutionary effects, as the +SLACS sample is at lower redshift than the TDCOSMO lenses +(see, e.g., Sonnenfeld et al. 2015, for a discussion of the evolu- +tion of mass density profiles of massive elliptical galaxies). +Spatially resolved velocity dispersion measurements of lens +galaxies for systems with measured time delays are critical to +drastically improving the H0 precision, given the limited sam- +ple size of time-delay lenses (Shajib et al. 2018; Yıldırım et al. +2021). The spatially resolved nature of the measured veloc- +ity dispersion is especially powerful in simultaneously break- +ing the MSD and the mass-anisotropy degeneracy (Cappellari +2008; Barnabè et al. 2009, 2012; Collett et al. 2018; Shajib et al. +2018). Spatially resolved velocity dispersion measurements for +∼40 time-delay lens galaxies will yield an independent ≲2% +H0 measurement without any mass profile assumption (Birrer +& Treu 2021). Additional constraints from velocity dispersion +measurements of non-time-delay lens galaxies or magnification +information for standardizable lensed type Ia supernovae can +further improve the uncertainty to ≲ 1% (Birrer & Treu 2021; +Birrer et al. 2022a). +In this paper, we measure the spatially resolved velocity dis- +persion for the lens galaxy in the strongly lensed quasar system +RXJ1131−1231using the Keck Cosmic Web Imager (KCWI) in- +tegral field spectrograph on the W. M. Keck Observatory (Mor- +rissey et al. 2012, 2018). and constrain H0 without any mass +profile assumption from this single time-delay lens system. This +is the first application of spatially resolved velocity dispersion +from a time-delay lens to measure H0. This lens system was pre- +viously used to measure H0 by combining the observed imaging +data, single-aperture velocity dispersion, time delays, and anal- +ysis of the line-of-sight environment (Suyu et al. 2013, 2014). +However, these previous studies assumed simple parametriza- +tions for the mass profile, such as a power law or a combination +of the NFW profile and the stellar profile with constant mass- +to-light, which is the industry standard in modeling of galaxy- +scale lenses (Shajib et al. 2022b). Birrer et al. (2016) marginal- +ized over the MSD effect for the system RXJ1131−1231 to con- +strain H0 using a single-aperture velocity dispersion measure- +ment. Here, we allow the maximal freedom in the MSD by in- +troducing one free parameter on top of the simply parametrized +mass profile constrained by lens modeling, which is completely +degenerate with H0. +This paper is organized as follows. In Section 2, we describe +the observational strategy and data reduction. In Section 3, we +describe the procedures to extract the spatially resolved kine- +matics map from the KCWI data. In Section 4, we briefly review +the lensing and dynamical formalisms and how we combine the +two to mitigate the MSD in our analysis. Then in Section 5, we +describe our dynamical models and present results. We infer the +cosmological parameters from our analysis in the Section 6. We +discuss our results in Section 7 and conclude the paper in Sec- +tion 8. +We performed the cosmological inference blindly in this pa- +per. The measurement of velocity dispersion was not blinded. +However, we blinded the cosmological and other model parame- +ters directly related to cosmological parameters in the dynamical +modeling. Before unblinding, this analysis went through an in- +ternal collaboration-wide review and code review. After all the +coauthors had agreed that the necessary systematic checks were +satisfactorily performed, we froze the analysis and unblinded on +5 January 2023. All the sections in this paper except for the final +discussion in Section 7 and summary in Section 8 were written +before unblinding. After unblinding, we only made minor edits +Article number, page 2 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +N +E +1" +A +B +C +D +G +S +Fig. 1. HST/ACS image of RXJ1131−1231 in the F814W band. The +four quasar images are labeled with A, B, C, and D. The central deflec- +tor is marked with G, of which we are measuring the spatially resolved +velocity dispersion. An arrow points to the nearby satellite S, which we +mask out in the velocity dispersion measurement.. The North and East +directions and 1¨scale are also illustrated. +for clarity and grammatical corrections in the previous sections +and added the unblinded numbers where relevant in the abstract, +main text, and plots. +2. Observations and data reduction +In this section, we provide a brief description of the lens sys- +tem RXJ1131−1231 (Section 2.1), the spectroscopic observation +with KCWI (Section 2.2), and the data reduction procedure (Sec- +tion 2.3). +2.1. Description of lens system +The quadruply imaged quasar lens system RXJ1131−1231 was +discovered by Sluse et al. (2003). The deflector in this system +is an elliptical galaxy with redshift zd = 0.295, and the source +redshift is zs = 0.657 (Sluse et al. 2003). Due to its low red- +shifts, the system is relatively bright and large in angular size. +The Einstein ring in this system contains intricate features, pro- +viding a wealth of information to constrain the lens mass model +(see Figure 1). Due to its early discovery and information-rich +features, this system is one of the most studied lensed quasar +systems. The time delays for this system were measured by +Tewes et al. (2013). Suyu et al. (2013, 2014) performed cosmo- +graphic analyses of this system. These authors combined simply +parametrized lens models based on the high-resolution imaging +from the HST’s Advanced Camera for Surveys (ACS) instru- +ment (HST-GO 9744; PI: Kochanek), the measured time delays, +single-aperture velocity dispersion, and external convergence es- +timate to infer H0 = 80.0+4.5 +−4.7 km s−1 Mpc−1. However, such sim- +ply parametrized lens models implicitly break the MSD. Birrer +et al. (2016) performed an independent mass modeling of this +system while marginalizing the MSD with a prior on the source +size. These authors found the prior choice on the anisotropy in +the dynamical modeling to be the dominant systematic in infer- +ring H0. +2.2. KCWI Spectroscopy +We +obtained +integral +field +unit +(IFU) +spectroscopy +of +RXJ1131−1231 on 16 May and 7 June 2021 with the KCWI in- +strument on the Keck Observatory (Morrissey et al. 2012, 2018). +We chose KCWI with the small IFU slicer and the low-resolution +blue grating (BL) with a field-of-view (FoV) of 8′′.4 × 20′′.4. The +spectral resolution is R ≈ 3600, corresponding to an instrumental +dispersion σinst ∼ 35 km s−1. The reciprocal dispersion is 0.5 Å +per pixel. The observed wavelength range 3600–5600 Å covers +the Ca H&K lines with wavelengths λλ3933, 3968 Å at the red- +shift of the lens galaxy (zd = 0.295). We primarily use these lines +to determine the stellar velocity dispersion. The redshifted 4304 +Å G-band is beyond the observed range, so it is not accessible +with the KCWI for the RXJ1131−1231 system. +We aligned the FoV’s longer side with the North direction +(i.e., PA = 0◦) and dithered the individual exposures by 9′′ along +the North-South direction. As the extent of the RXJ1131−1231 +system is smaller than the FoV, each exposure contained the en- +tire lens system within the FoV. In different exposures, the lens +system occupied the upper or lower portion of the FoV. Thus, +the sky in an exposure with the system occupying the upper por- +tion can be subtracted using another exposure with the system +occupying the lower portion, and vice versa. We obtained six ex- +posures with a total integration time of 10,560 s on 16 May and +three with a total integration time of 5,400 s on 7 June. There- +fore, the total exposure time is texp = 15, 960 s. The airmass +ranged from 1.2 to 1.48 over the integrating period. +2.3. Data Reduction +We use the official Python-based data reduction pipeline1 (DRP) +to reduce our data. The DRP converts the 2D raw data captured +on the detector into a 3D datacube. It performs geometry cor- +rection, differential atmospheric refraction correction, and wave- +length calibration and produces a final standard-star-calibrated +3D datacube for each exposure. The calibration with the stan- +dard star corrects for instrumental response and scales the data +to flux units (Morrissey et al. 2018). We use the final output file +with the suffix “_icubes” for further analysis. +We stack the dithered datacubes through drizzling (Fruchter +& Hook 2002). Since the exposures are obtained on different +dates, the world coordinate system information is not accurate +enough to determine the relative positions of the dithered expo- +sures. We follow Chen et al. (2021b) to determine the relative po- +sitions by simultaneously fitting the point spread function (PSF) +to the four quasar image positions. To perform the drizzling on +the datacubes, we repurpose the drizzling routine of the DRP for +OSIRIS, another IFU spectrograph on the Keck Observatory2. +For the drizzling process, we set pixfrac = 0.7 as recom- +mended to reduce correlated uncertainties between the drizzled +pixels (Avila et al. 2015). We calculate the drizzled weight image +and ensure that the ratio of RMS/median < 0.2 in the region of +interest so that the trade-off is balanced between improving the +1 developed by Luca Rizzi, Don Neill, Max Brodheim; https:// +kcwi-drp.readthedocs.io/ +2 https://github.com/Keck-DataReductionPipelines/ +OsirisDRP +Article number, page 3 of 21 + +A&A proofs: manuscript no. ms +image resolution and increasing the background noise (Gonzaga +et al. 2012). The rectangular pixel size 0′′.1457′′ × 0′′.3395 of the +KCWI is kept the same in the drizzled output. We transform the +datacube to have square pixels of size 0′′.1457 × 0′′.1457 through +resampling while conserving the total flux. We converted the pix- +els into square sizes for the convenience of Voronoi binning the +spectra using the software vorbin as described in Section 3.2. +We directly estimate the PSF from the observed data. We +produce a 2D image from the datacube by summing along the +wavelength axis (see Figure 2). We create a model for this KCWI +image using a high-resolution template from the HST imaging +(Figure 1) that has a pixel size 0′′.05 and PSF full width at half +maximum (FWHM) 0′′.10. In the model, the template is con- +volved with a Gaussian PSF with a free FWHM parameter, and +the positioning of the template on the KCWI image grid is fitted +with two additional free parameters. By optimizing the model, +we estimate that the PSF FWHM is 0′′.96. +3. Kinematics maps +This section describes our procedure to obtain the final kine- +matics map. We use the pPXF package3 to fit the spectra with +a library of stellar templates and extract the velocity dispersion +(Cappellari 2017, 2022). In Section 3.1, we describe the stel- +lar templates used for the analysis. In Section 3.2, we present +the measurement of the spatially-resolved kinematics map of the +lens galaxy. In Section 3.3, we test the systematics of the velocity +dispersion measurement. +3.1. Library of Stellar Templates +The popularly used template libraries Medium-resolution +Isaac Newton Telescope library of empirical spectra (MILES; +Sánchez-Blázquez et al. 2006) and INDO-US templates (Valdes +et al. 2004) are both too low resolution to fit our datasets. The +KCWI’s instrumental resolution of R ≈ 3600 leads to σinst ∼ 35 +km s−1 for a Gaussian line spread function (LSF)4. MILES +has a resolution of σtemplate ∼ 64 km s−1 (i.e., R ∼ 2000), +and the INDO-US templates have an approximately constant- +wavelength resolution of 1.2 Å, which corresponds to σtemplate = +39 km s−1 over the Ca H&K wavelength range. Therefore, we +choose the X-shooter Spectral Library (XSL), which contains +628 stars covering three segments, including UVB, Vis, and NIR +bands (Gonneau et al. 2020). As our data cover the rest-frame +blue/UV range, we only use the UVB segment to fit the data, +where its resolution is R ∼ 9700 and σtemplate ∼ 13 km s−1. +3.2. Measuring the velocity dispersion +We choose a cutout centred on the lens system with 6′′.235 × +6′′.235 (43 pixels × 43 pixels) to initiate the analysis (see Fig- +ure 2). We estimate the lens galaxy light’s signal-to-noise ra- +tio (S/N) in each spatial pixel (hereafter, spaxel) within this ini- +tial cutout. We then select a region with sufficient S/N from the +lens galaxy and relatively low quasar contamination for measur- +ing the velocity dispersion (the yellow contour in Figure 2’s left +panel). We perform Voronoi binning within this selected region +3 https://pypi.org/project/ppxf/ +4 We quantitatively verified that the shape of the instrumental LSF is +Gaussian (cf. Figure 28 of Morrissey et al. 2018). Thus, the treatment of +the instrumental LSF in pPXF is self-consistent and avoids any system- +atic bias due to inconsistent definitions of the LSF’s FWHM (Robertson +2013). +to preserve the maximal spatial resolution and reduce the bias in +the lower-S/N region (Cappellari & Copin 2003). We elaborate +on these steps below. +To estimate the lens galaxy’s S/N in each spaxel, we first si- +multaneously fit the quasar and the lens galaxy in each spaxel to +calculate the signal from each of them. We perform this fitting +within the wavelength range 3400–4300 Å. As the four quasar +images surround the lens galaxy, each spaxel receives a differ- +ent contribution from the quasar light. We take spectra at the +central spaxel of image A as the quasar template, ignoring the +lens galaxy’s small contribution. Later in Section 3.3, we also +choose the quasar template from images B and C to account for +the associated systematic uncertainty, i.e., the potential impact +of chromatic microlensing that may change the contrast between +the line and the continuum (e.g., Sluse et al. 2007). +We determine a single optimal template spectrum for the lens +galaxy template. For this purpose, we binned the spectra from +spaxels within a circular region of radius 0′′.5 centered on the +lens galaxy and fit it with pPXF using the 628 stellar templates +from the XSL and the quasar template. We also include a Leg- +endre polynomial of degree 3 as a component in the fitting to ac- +count for any residual gradient in the continuum. pPXF chooses +39 of the stellar templates and builds the optimal template by +taking a weighted linear combination of them. See Figure 3 for +the weighted distribution of spectral types of the full template li- +brary and that of the 39 stars selected by pPXF. Among those +stars in the XSL with stellar classes specified by the Simbad +database (Wenger et al. 2000), G-type stars are selected with +the highest total weight, consistent with the fact that massive +elliptical galaxy spectra are dominated by G and K-type stars. +In the pPXF fitting procedure, the stellar templates are broad- +ened, corresponding to a freely varying velocity dispersion, but +the velocity dispersion does not broaden the quasar template. +Once the optimal galaxy template is constructed, we use this +template and the quasar template to fit the spectrum of each +spaxel individually. We use this optimal template to fit the galaxy +spectra in individual spaxels instead of the full template library +to avoid large spurious fluctuations in the measured velocity dis- +persion from spaxel to spaxel. We show the decomposition of +the spectra from one example spaxel into different components +after fitting with pPXF in Figure 2. We calculate the signal of the +lens galaxy’s spectrum in each spaxel by subtracting the mod- +eled quasar component from the observed spectra. The noise is +estimated by adding in quadrature the Poisson noise of the total +signal and the background noise estimated from an empty patch +of the sky. The noise values are multiplied by +√ +2 to account +for the fact that the square pixels are created from the rectan- +gular pixels about double the size through resampling. We esti- +mate the S/N using the restframe wavelength range 3985–4085 +Å, slightly above the Ca H&K absorption lines in wavelength +(see the purple shaded region in Figure 2). +To perform Voronoi binning before the velocity dispersion +measurement, we select the spaxels within a radius of 1′′.55 from +the lens galaxy center that avoid the brightest spaxels contain- +ing images A, B, and C and the lensed arcs. We also exclude a +circular region around image D with radius 0′′.5. To avoid any po- +tential bias due to contamination from the satellite galaxy S, we +exclude the spaxel at its position (∆RA = 0′′.09, ∆Dec = 0′′.54 +from the galaxy center, Suyu et al. 2013). We also exclude pixels +with S/N < 1 Å−1. In the end, the spaxels within the selected re- +gion have S/N > 1.4 Å−1 (see Figure 2 for the selected region). +5 For reference, 1′′.5 corresponds to 6.6 kpc at zd = 0.295 for a fiducial +flat ΛCDM cosmology with H0 = 70 km s−1 Mpc−1 and Ωm = 0.3. +Article number, page 4 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +N +E +1" +KCWI data of RXJ1131 1231 +3400 +3600 +3800 +4000 +4200 +Restframe wavelength (�) +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +Data +Best fit model (galaxy + quasar) +Best fit quasar model +Data quasar +Residual (= data model) +0 +200 +400 +600 +800 +1000 +1200 +Flux (arbitrary unit) +Fig. 2. Left: 2D representation (median-collapsed) of the 3D KCWI datacube for RXJ1131−1231. The yellow contour traces the region with +1′′.5 radial extent from the center selected for stellar kinematic measurement. A circular region with 0′′.5 radius around image D and the spaxel +containing the satellite S are excluded from this selected region. All the individual spaxels within this region have continuum S/N > 1.4 Å−1 for +the lens galaxy’s light within 3985–4085 Å (the purple shaded range in the right panel). Right: The spectra (grey) from an example pixel (grey +box in the left panel) and the estimate of the signal from the lens galaxy’s spectra (orange) after removing the contribution from the quasar light +(blue). The full model of the spectra is presented with the red line, and the model’s residual is plotted in emerald color. The vertical purple shaded +region marks where we compute the continuum S/N. +O +B +A +F +G +K +M +L +S +C +~ +Stellar type +0.0 +0.5 +1.0 +1.5 +Normalized density +All +Set 1 +Set 2 +Set 3 +Fig. 3. Distribution of the stellar spectral types in the XSL according +to the Simbad database. Unspecified stars are grouped in the ‘∼’ class. +The dark grey color represents the full library of 628 stars. Set 1 (or- +ange) refers to the 39 stars selected by pPXF out of the full library to +construct an optimal template 1. Set 2 (blue) refers to 32 stars selected +from a random half of the full library and set 3 (emerald) refers to 33 +stars selected from the other half. Sets 2 and 3 have 15 and 17 stars, re- +spectively, in common with Set 1. The alternating light grey and white +vertical regions divide the spectral classes for easier visualization. +We perform Voronoi binning using vorbin6 given the estimated +S/N values for each spaxel. In Figure 4, we show the 41 Voronoi +bins obtained by setting the target S/N ≈ 23 Å−1 for each bin. +This target S/N was chosen so that the resultant S/N ≳ 20 Å−1 +for each bin, which is standard practice (Figure 4, only bin 16 +has S/N ≈ 18 Å−1). +For each Voronoi bin, we measure the velocity dispersion by +fitting the binned spectra using pPXF using the optimal galaxy +template described above, the quasar template, and the additive +Legendre polynomial to model any slight gradient in the popula- +tion. A few examples of pPXF fit of the binned spectra are shown +in Figure 5. +6 https://pypi.org/project/vorbin/ +3.3. Estimation of systematic uncertainty +To estimate the systematic uncertainties in the velocity disper- +sion measurement, we consider a range of plausible choices in +the extraction procedure: the degrees of the additive Legendre +polynomial used to correct the template continuum shape be- +tween 2 to 4; the quasar template obtained from images A, B, and +C; the fitted wavelength range chosen from 3300–4200 Å, 3350– +4250 Å, and 3400–4300 Å; and three sets of template spectra +used in the fitting. The first set of template spectra contains the +complete XSL of 628 stars. The second set contains half of the +entire sample that is randomly selected, and the third set con- +tains the other half. The numbers of stars selected by pPXF in +the three sets are 39, 32, and 33, respectively. Sets 2 and 3 have +15 and 17 stars, respectively, in common with Set 1. Figure 3 +shows the distribution of spectral types in all three sets and the +entire library. We do not take the quasar template from image D +as it is much fainter than the other images, and thus the galaxy +contribution in the brightest spaxel on image D is non-negligible. +Taking a combination of all of these choices yields 81 different +setups. We illustrate the shift in the extracted velocity dispersion +maps for one change of setting at a time in Figures 6 and 7. +We estimate the variance-covariance matrix of the binned ve- +locity dispersions from these 81 setups. To do this, we generate +1,000 random realizations of the measured velocity dispersion +map for each of the 81 setups using the corresponding statistical +uncertainty. We create the variance-covariance matrix from the +81,000 realizations combined from all the setups. In this way, +the diagonal terms of the variance-covariance matrix encode +the total variance from systematic and statistical uncertainties, +and the off-diagonal terms encode the systematic covariances. +For example, if all 81 setups hypothetically provided the same +velocity dispersion map and uncertainty, then the off-diagonal +terms would be zero, and the diagonal terms would reflect only +the statistical uncertainties. We show the systematic variance- +covariance relative to the statistical variance in Figure 8. The +systematic variance is subdominant relative to the statistical vari- +ance (with a median of 0.47 of the ratio between systematic and +statistical covariances along the diagonal) except for bins 29 and +31. These two bins are closest to quasar images A and C, and +Article number, page 5 of 21 + +A&A proofs: manuscript no. ms +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +Map of Voronoi bins +0 +5 +10 +15 +20 +25 +30 +35 +40 +Bin number +16 +18 +20 +22 +24 +26 +28 +30 +S/N (� +1) +Voronoi bins +Target S/N +Fig. 4. Left: Voronoi binning of the selected spaxels within 1′′.5 from the galaxy center that avoid lensed arcs, quasar images, and the satellite +galaxy S. The different colors illustrate the regions for each Voronoi bin in a cartographic manner for easier visualization, with the bin number +specified within each bin. We perform the binning with a target S/N ≈ 23 Å−1 for each bin, which results in 41 bins in total. Right: Resultant S/N +for each Voronoi bin (red points). +3400 +3600 +3800 +4000 +4200 +Restframe wavelength (˚A) +0.00 +0.02 +0.04 +0.06 +Flux (arbitrary unit) +bin 1: σlos = 282 ± 7 km s−1 +3400 +3600 +3800 +4000 +4200 +Restframe wavelength (˚A) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +Flux (arbitrary unit) +bin 6: σlos = 284 ± 11 km s−1 +3400 +3600 +3800 +4000 +4200 +Restframe wavelength (˚A) +0.0 +0.1 +0.2 +0.3 +Flux (arbitrary unit) +bin 20: σlos = 274 ± 13 km s−1 +3400 +3600 +3800 +4000 +4200 +Restframe wavelength (˚A) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Flux (arbitrary unit) +bin 37: σlos = 263 ± 16 km s−1 +Data +Best fit model (galaxy + quasar) +Best fit quasar model +Fig. 5. pPXF fitting to the spectra from four examples of Voronoi bins. The bin number and the measured velocity dispersion for the corresponding +bin are specified in each panel. The grey line presents the full spectra, the red line traces the best-fit model, and the blue line shows the quasar +component in the best-fit model. +thus largely susceptible to the choice of quasar template (see +Figures 6 and 7.) +We show the velocity dispersion and mean velocity maps av- +eraged over the 81 setups in Figure 9. We estimate a systematic +velocity of 182 km s−1 using the pafit7 software program (Kra- +jnovi´c et al. 2006) and subtract it from the mean velocity map. +The systematic velocity is the result of a slight deviation in the +true redshift from the fiducial value. The mean velocity map does +7 https://pypi.org/project/pafit/ +Article number, page 6 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +not show any significant evidence of ordered rotation above the +systematic and statistical noise levels. Thus it is consistent with +the lens galaxy being a slow rotator. We use this systematic- +averaged velocity dispersion map and the variance-covariance +matrix estimated above when computing the likelihood function +for dynamical modeling in Section 5. +To test the impact of our choice for the Voronoi binning +scheme, we adopt an alternative target S/N ≈ 28 Å−1 for each +bin, which results in 27 bins. We similarly produce another +set of 81 model setups in this binning scheme and produce +the variance-covariance matrix for these binned velocity disper- +sions. We test the systematic impact of this different binning +scheme on the cosmological measurement later in Section 5.2. +We show the difference in the extracted kinematics between the +two binning schemes in Figure 10. +4. Overview of lens and dynamical modeling +This section reviews the theoretical formalism of lens and dy- +namical modeling. +4.1. Lensing observables and modeling +We briefly review the strong lensing formalism in the context +of time-delay cosmography in Section 4.1.1, describe the mass- +sheet transform (MST) in Section 4.1.2, and explain the internal +and external components of the MST in Section 4.1.3. +4.1.1. Strong lensing formalism +In the thin lens approximation applicable in this case, lensing +observables are described using the surface mass density Σ(R) +projected from the 3D mass density distribution ρ(r) in the lens +galaxy. Formally, the lensing observables depend on the dimen- +sionless convergence defined as +κ(θ) ≡ Σ(θ) +Σcr +, +(1) +which is the surface mass density normalized by the critical den- +sity +Σcr ≡ +c2Ds +4πGDdDds +. +(2) +Here, c is the speed of light, G is the gravitational constant, Ds +is the angular diameter distance between the observer and the +source, Dd is the angular diameter distance between the observer +and the lens galaxy, and Dds is the angular diameter distance +between the lens galaxy and the source. The on-sky deflection +angle α(θ) relates to the convergence as +κ(θ) = 1 +2∇ · α(θ). +(3) +The time delay between two quasar images labeled A and B is +given by +∆tAB = D∆t +c +�(θA − ς)2 +2 +− (θB − ς)2 +2 +− ψ(θA) + ψ(θB) +� +, +(4) +where θA is the angular position of image A, ς is the source’s +angular position, ψ(θ) is the lensing potential, and the time-delay +distance D∆t is defined as +D∆t ≡ (1 + zd)DdDs +Dds +. +(5) +4.1.2. Description of the MST +The MST is a mathematical transform of the convergence pro- +file that leaves invariant all the imaging observables, such as the +image positions and the flux ratios (Falco et al. 1985; Schneider +& Sluse 2014). This transform scales the convergence and the +unknown source position as +κ → κ′ = λMSTκ + (1 − λMST), +ς → ς′ = λMSTς. +(6) +where λMST is the transformation parameter. The predicted time +delay ∆t scales under the transform as +∆t → ∆t′ = λMST∆t. +(7) +Then, the inferred time-delay distance D∆t and the Hubble con- +stant H0 based on the observed time delays will change as +D′ +∆t = D∆t +λMST +, +H′ +0 = λMSTH0. +(8) +However, the MST changes the predicted velocity dispersion, +thus measuring it breaks the MSD. Notably, the MST also +rescales the lensing magnifications. Thus, standardizable candles +can also be used to break the MSD (Bertin & Lombardi 2006; +Birrer et al. 2022a) provided that microlensing and millilensing +can be mitigated (e.g., Yahalomi et al. 2017; More et al. 2017; +Foxley-Marrable et al. 2018). +4.1.3. Internal and external MST +We can express the “true” (i.e., physically present) lensing mass +distribution as +κtrue = κgal + κext, +(9) +where κgal is the mass distribution of the central lens galaxy +(or galaxies) that is (are) considered in the lens modeling, and +κext is called the external convergence, which approximates the +projected mass distribution of line-of-sight structures as a mass +sheet. Since limθ→∞ κgal = 0 has to be satisfied, we find that +limθ→∞ κtrue = κext, hence the interpretation of κext as the lensing +mass far from (or, “external” to) the central deflector(s). +All the lensing observables including imaging observables +result from κtrue. However, since only the central galaxies are +usually considered in lens modeling with imaging observables, +the lens model provides κ′ +model with limθ→∞ κ′ +model = 0. This +κ′ +model is an MST of κtrue for λMST = 1/(1 − κext) as +κ′ +model = κgal + κext +1 − κext ++ 1 − +1 +1 + κext += +κgal +1 − κext +. +(10) +Lens +mass +models +are +usually +described +with +simply +parametrized models, such as the power law or a combi- +nation of the NFW profile and the observed stellar distribution. +In that case, the assumption of a simple parametric form +implicitly breaks the MSD. Therefore, the simply parametrized +model κmodel can be expressed as another approximate MST of +the κ′ +model as +κ′ +model ≈ λintκmodel + (1 − λint)κs(θ), +(11) +where λint is called the internal MST parameter, and κs is a “vari- +able” mass sheet with limθ→∞ κs(θ) = 0 to ensure that both +Article number, page 7 of 21 + +A&A proofs: manuscript no. ms +Range: 3350 4250 � +Range: 3300 4200 � +Polynomial degree 2 +Polynomial degree 4 +Stellar template set 2 +Stellar template set 3 +Quasar B +Quasar C +20 +0 +20 +20 +0 +20 +20 +0 +20 +20 +0 +20 +20 +0 +20 +20 +0 +20 +20 +0 +20 +20 +0 +20 +Fig. 6. Absolute difference in km s−1 between the extracted velocity dispersion from two setups that differ by one setting. The baseline setup has +the range: 3400–4300 Å, polynomial degree: 3, stellar template set 1, and quasar template from image A. The different setting for each case is +specified at the top of each panel. +Range: 3350 4250 � +Range: 3300 4200 � +Polynomial degree 2 +Polynomial degree 4 +Stellar template set 2 +Stellar template set 3 +Quasar B +Quasar C +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +Fig. 7. Same as Figure 6, but the difference is normalized by the statistical uncertainty of the baseline setup. +limθ→∞ κ′ +model = 0 and limθ→∞ κmodel = 0 are satisfied. How- +ever, for Equation (11) to be an approximate MST, the variable +mass-sheet needs to satisfy κs(θ) ≃ 1 within the central region +that lensing observables are sensitive to (θ ≲ 2θE, Schneider & +Sluse 2013). This can be achieved with the formulation (Blum +et al. 2020) +κs(θ) = +θ2 +s +θ2 + θ2s +, +(12) +where θs ≫ θE is a scale radius where the variable mass-sheet +smoothly transitions from 1 − λint to 0. This approximate MST +converges to the pure MST in the limit θs → ∞. Thus, the actual +mass distribution of the central deflector(s) relates to the mod- +eled mass distribution as +κgal ≈ (1 − κext) [λintκmodel + (1 − λint)κs(θ)] . +(13) +The external convergence κext can be estimated by using rel- +ative number counts of line-of-sight galaxies near the central de- +Article number, page 8 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +10 +20 +30 +40 +Bin number +5 +10 +15 +20 +25 +30 +35 +40 +Bin number +Fractional systematic covariance: +xy +h +diag( +2 +stat) +i +xy +2 +stat,x +10 +0 +0 +10 +0 +Fig. 8. Illustration of the systematic covariance relative to the statistical +covariance. Σ is the variance-covariance matrix of the Voronoi-binned +velocity dispersions (with target S/N ≈ 23 Å−1 for each bin), σstat,x is +the statistical uncertainty in bin number x from our fiducial setup, and +diag(σstat) is a diagonal matrix. Note that we assume no covariance in +the statistical uncertainty from each setup for kinematic measurement. +Thus the off-diagonal terms in the variance-covariance matrix purely +represent the systematic covariance. Most diagonal terms are < 1 (with +a median of 0.47), showing that the systematic variances are subdom- +inant to the statistical variances except for bins 29 and 31. These bins +are close to images A and C. Thus, they are largely susceptible to the +choice of the quasar template, as seen in Figures 6 and 7. +Velocity dispersion +Velocity dispersion uncertainty +Mean velocity +Mean velocity uncertainty +225 +250 +275 +300 +los (km s +1) +20 +40 +los (km s +1) +50 +0 +50 +vmean (km s +1) +20 +30 +vmean (km s +1) +Fig. 9. Maps of extracted velocity dispersion (top row) and mean ve- +locity (bottom row) in Voronoi bins along with the corresponding un- +certainties (right column). The Voronoi binning was tuned to achieve +S/N ≈ 23 Å−1 for each bin. The illustrated maps (left column) corre- +spond to the average values after combining 81 model setups, and the +uncertainty maps correspond to the square root of the diagonal of the +variance-covariance matrices. A systematic velocity of 182 km s−1 was +subtracted from the mean velocity map. +los (km s +1) +los/ +los +20 +0 +20 +2 +0 +2 +Fig. 10. Absolute (left) and uncertainty-normalized (right) difference in +the extracted velocity dispersion between two Voronoi binning schemes. +The two binning schemes are obtained by setting the target S/N to 23 +Å−1 and 28 Å−1 for each bin. We take the case with target S/N ≈ 23 Å−1 +for each bin as the baseline in our analysis. +flector(s) (e.g., Suyu et al. 2010; Greene et al. 2013; Rusu et al. +2017; Buckley-Geer et al. 2020), or by using weak lensing of +distant galaxies by the line-of-sight mass distribution (e.g., Ti- +hhonova et al. 2018). The measured velocity dispersion then +constrains the internal MST parameter λint (Birrer et al. 2020; +Yıldırım et al. 2021). +4.2. Dynamical modeling +In this section, we describe the Jeans anisotropic multi- +Gaussian-expansion (JAM) framework to model our dynamical +observable, which is the spatially resolved stellar velocity dis- +persion measured in Section 3. The orbital motions of the stars, +i.e., the distribution function f(x, v) of position x and velocity v, +in the galactic potential Φ is described by the steady-state col- +lisionless Boltzmann equation (Binney & Tremaine 1987, Eq. +4-13b) +3 +� +i=1 +� +vi +∂f +∂xi +− ∂Φ +∂xi +∂f +∂vi +� += 0. +(14) +We assume an axisymmetric case (i.e., ∂Φ/∂φ = ∂f/∂φ = 0 +with φ being the polar angle in the spherical coordinate sys- +tem), a spherically aligned velocity ellipsoid, and the anisotropy +for each Gaussian component in the multi-Gaussian expan- +sion (MGE; Emsellem et al. 1994; Cappellari 2002) to be spa- +tially constant. Slow rotators such as the deflector galaxy in +RXJ1131−1231 are in general expected to be weakly triaxial or +oblate but never flat and instead quite close to spherical in their +central parts (e.g., Cappellari 2016). For this reason, we expect +the spherical alignment of the velocity ellipsoid of jamsph (Cap- +pellari 2020) to provide a better approximation to the galaxy dy- +namics than the cylindrical alignment jamcyl solution (Cappellari +2008). Then, the above equation gives two Jeans equations in +spherical coordinates (Jeans 1922; Bacon et al. 1983; de Zeeuw +et al. 1996; Cappellari 2020) +∂ +� +ζ⟨v2 +r⟩ +� +∂r ++ +(1 + β)ζ⟨v2 +r⟩ − ζ⟨v2 +φ⟩ +r += −ζ ∂Φ +∂r , +(1 − β) +∂ +� +ζ⟨v2 +r⟩ +� +∂θ ++ +(1 − β)ζ⟨v2 +r⟩ − ζ⟨v2 +φ⟩ +tan θ += −ζ ∂Φ +∂θ , +(15) +Article number, page 9 of 21 + +A&A proofs: manuscript no. ms +where the following notations are used +ζ⟨vpvq⟩ ≡ +� +vpvq fd3v, +β ≡ 1 − ⟨v2 +θ⟩ +⟨v2r⟩. +(16) +Here, β is the anisotropy parameter, and the velocity dispersion +ellipsoid is assumed to be spherically aligned, giving ⟨vrvθ⟩ = 0. +The line-of-sight second moment ⟨v2 +los⟩ is the integral given +by +S ⟨v2 +los⟩(x, y) = +� ∞ +−∞ +dz ζ⟨v2 +z⟩. +(17) +where S (x, y) is the surface density of the dynamical tracer. +Given that there is no evidence of significant ordered rotation +and the only significantly nonzero velocities are likely due to +systematic errors (see Figure 9), we assume ⟨vlos⟩ = 0 and define +⟨v2 +los⟩ = σ2 +los. The observed line-of-sight velocity dispersion is +given a luminosity-weighted integral as +� +σ2 +los +� +obs = +� +⟨v2 +los⟩ +� +obs = +� +ap dxdy I⟨v2 +los⟩ ⊗ PSF +� +ap dxdy I ⊗ PSF +, +(18) +where the symbol “⊗ PSF” denotes a convolution with the PSF. +In the equation above, we have chosen the surface brightness +profile I(x, y) as a substitute for the surface density S (x, y) of +the dynamical tracer since the constant factor between surface +brightness and surface number density cancels out in this ex- +pression. +We use the dynamical modeling software jampy8 to compute +the observed velocity dispersion by solving the Jeans equation +from Equation (15) for a given 3D potential Φ(r) and anisotropy +profile β(r). Specifically, we use the jam_axi_proj() routine +with the keyword align=‘sph’. See Cappellari (2008, 2020) +for a detailed formalism in computing Equation (18) by jampy. +4.3. Cosmological inference from combining dynamical and +lensing observables +We parametrize the 3D potential Φ(r) using the lens model pa- +rameters ξmass and the internal MST parameter λint to conve- +niently use the lens model posterior from Suyu et al. (2013) as a +mass model prior in the dynamical modeling. Thus from Equa- +tion (13), the surface mass density for our dynamical model is +given by +Σ(θ) = Σcr(1 − κext) [λint κmodel(θ) + (1 − λint)κs(θ)] . +(19) +We include D∆t and Dd as free parameters in our model, which +give the critical density Σcr as +Σcr = +c2 +4πG +D∆t +(1 + zd)D2 +d += +c2 +4πG +Dmodel +∆t +(1 + zd)(1 − κext)λintD2 +d +, +(20) +where Dmodel +∆t +is the time-delay distance predicted by the lens +mass model κmodel(θ) for the time delays observed by Tewes et al. +(2013). +We approximate the surface mass density Σ(θ) with an MGE +(Emsellem et al. 1994; Cappellari 2002; Shajib 2019) using the +8 https://pypi.org/project/jampy/ +software program mgefit9. jampy deprojects the MGE compo- +nents into an oblate or prolate spheroid with an inclination angle +i (Cappellari 2002). The deprojected 3D mass density provides +the 3D potential Φ for the kinematic computation. We also take +the MGE of the surface brightness I(x, y) for deprojection to 3D +with the inclination angle i for the kinematic computation by +jampy. +The combination of lens imaging observables and the stellar +kinematics is sensitive to λint(1 − κext)Ds/Dds (Birrer et al. 2016; +Chen et al. 2021a). We apply a prior on κext using the estimated +κext distribution from Suyu et al. (2014) to help break the de- +generacy in distributing the total MSD into external and internal +components. +4.4. Bayesian framework +According to Bayes’ theorem, the posterior of the model param- +eters Ξ = {ξmass, ξlight, Dmodel +∆t +, i, κext, λint, Dd, β} as +p(Ξ | D) ∝ p(D | Ξ) p(Ξ), +(21) +where p(D | Ξ) is the likelihood given data D and p(Ξ) is the +prior. In this study, the data D is the measured velocity disper- +sions in Voronoi bins (Figure 9). The observational information +from the published time delays, lens models using HST imag- +ing, and the line-of-sight effects (Tewes et al. 2013; Suyu et al. +2013, 2014) is incorporated by adopting those previous posteri- +ors as the prior on our model parameters. The likelihood of the +observed velocity dispersion vector σlos ≡ [σ1, . . . , σNbin], with +Nbin being the number of Voronoi bins, is given by +L(σlos | Ξ) ∝ exp +� +−1 +2σT +losΣ−1σlos +� +, +(22) +where Σ is the variance-covariance matrix. Specific priors used +in this Bayesian framework are given in Section 5. We ob- +tain the posterior probability distribution function (PDF) of +the model parameters using the Markov-chain Monte Carlo +(MCMC) method using the affine-invariant ensemble sampler +emcee (Goodman & Weare 2010; Foreman-Mackey et al. 2013). +We ensure the MCMC chains’ convergence by running the +chains for ≳20 times the autocorrelation length after the chains +have stabilized (Foreman-Mackey et al. 2013). +5. Dynamical models +We first describe our baseline dynamical model in Section 5.1 +and then perform various checks on systematics in Section 5.2. +5.1. Baseline dynamical model +This subsection describes the baseline settings in our dynamical +model, namely the specific parametrization of the mass model +(Section 5.1.1), the dynamical tracer profile (Section 5.1.2), the +probability of oblate or prolate axisymmetry (Section 5.1.3), the +inclination angle (Section 5.1.4), and the choice of anisotropy +profile (Section 5.1.5). +5.1.1. Parametrization of the mass model +We adopt the power-law mass model as our baseline model. In +this model, the mass profile is defined with Einstein radius θE, +9 https://pypi.org/project/mgefit/ +Article number, page 10 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +logarithmic slope γ, projected axis ratio qm, and position angle +ϕmass. The convergence profile κmodel in Equation (19) for the +power-law model is given by +κpl +model(θ1, θ2) = 3 − γ +2 +������������ +θE +� +qmθ2 +1 + θ2 +2/qm +������������ +γ−1 +(23) +Here, the coordinates (θ1, θ2) are rotated by ϕmass from the (RA, +Dec) coordinate system. We adopt the lens model posterior from +Suyu et al. (2013) as a prior in our dynamical model. For sim- +plicity, we set the position angle ϕmass the same as the observed +position angle of light ϕlight. We use the estimated κext distribu- +tion for the power-law model as the prior (see Figure 3 of Suyu +et al. 2014). +We set θs = 12′′ (≃ 7.5θE) in the approximate mass-sheet κs +(Equation 12) so that the imaging constraints alone cannot dif- +ferentiate the power-law mass profile and its approximate MST +from Equation (11). We obtain this lower limit by running the +jupyter notebook that produces Figure 3 of (Birrer et al. 2020).10 +However, we adjusted the fiducial lens model parameters in the +notebook to match with those for RXJ1131−1231. We take a uni- +form prior for the internal MST parameter λint ∼ U(0.5, 1.13). +The upper limit of 1.13 is set by the requirement that the trans- +formed mass profile under the approximate MST must be mono- +tonic so that the MGE can approximate the transformed pro- +file sufficiently well (Shajib et al. 2019). Previous studies also +found similar or more restrictive upper limits for λint to satisfy +the physical requirement of non-negative density (Birrer et al. +2020; Yıldırım et al. 2021). +The appropriate number of MGE components for the mass +or light profile is automatically chosen by jampy with a maxi- +mum of 20 components. We check that the MGE approximates +the input mass or light profile very well (with a maximum 1% +deviation at < 10′′ and maximum 10% deviation between 10′′– +50′′). These deviations from the density profile have an oscilla- +tory pattern due to the MGE approximation’s nature, except near +the end of the fitted ranges. Thus the deviation in the integrated +mass profile often averages out in the line-of-sight integration up +to a very large radius. We perform the MGE fitting up to 100′′. +Thus, the large mismatch between the MGE approximation and +the original profile occurs largely outside the integration limit +∼ 70′′. The chosen number of maximum Gaussian components +is not a dominant source of numerical error. Setting this maxi- +mum number to a very high value, such as 100, shifts the com- +puted velocity dispersion by only < 0.5% within the observed +region, which is insignificant compared to the 1% numerical sta- +bility targeted by jampy. +5.1.2. Dynamical tracer profile +We update the light profile fitting for the lens galaxy from Suyu +et al. (2013) using a larger HST image cutout than that therein, +which did not contain the full extent of the lens galaxy’s light +profile (see Figure 11). The lensed arcs and quasar images are +first subtracted from the cutout using the prediction of the best- +fit lens model from Suyu et al. (2013). We use the software pack- +age lenstronomy11 to fit the residual light distribution attributed +to the lens galaxy (Birrer & Amara 2018; Birrer et al. 2021). +10 https://github.com/TDCOSMO/hierarchy_analysis_2020_ +public/blob/6c293af582c398a5c9de60a51cb0c44432a3c598/ +MST_impact/MST_pl_cored.ipynb +11 https://github.com/lenstronomy/lenstronomy +Table 1. Values of the light model parameters for the double Sérsic +model in our fitting of a large cutout and those from Suyu et al. (2013). +The position angle ϕlight is defined as East of North. +Parameter +This analysis +Suyu et al. (2013) +Sérsic profile 1 +I0 (e−1 s−1 pixel−1) +32.8 ±0.1 +36.4 ±0.4 +θeff (′′) +2.437 ±0.005 +2.49 ±0.01 +ns +1.10 ±0.01 +0.93 ±0.03 +ql +0.865 ±0.001 +0.878 ±0.004 +Sérsic profile 2 +I0 (e−1 s−1 pixel−1) +441 ±7 +356 ±12 +θeff (′′) +0.300 ±0.003 +0.362 ±0.009 +ns +1.60 ±0.02 +1.59 ±0.03 +ql +0.847 ±0.002 +0.849 ±0.004 +ϕlight (◦) +120.5 ±0.3 +121.6 ±0.5 +Following Suyu et al. (2013), we use the double Sérsic model to +fit the light profile, which is a superposition of two concentric +Sérsic profiles. The Sérsic profile is defined as +I(θ1, θ2) = I0 exp +������������� +−bn +������������ +� +qlθ2 +1 + θ2 +2/ql +θeff +������������ +1/ns ++ bn +������������� +, +(24) +where I0 the amplitude, ql is the axis ratio, θeff is the effective +radius, ns is the Sérsic index, and bn = 1.999n − 0.327 is a nor- +malizing factor so that θeff becomes the half-light radius (Sérsic +1968). The coordinates (θ1, θ2) are rotated by ϕlight from the (Ra, +Dec) coordinate system. +We first mask circular regions at the quasar image positions +due to slightly saturated pixels producing significant residuals in +the subtracted cutout (see Figure 11). We then iteratively mask +the other pixels with significant residuals above statistical expec- +tations to effectively perform an outlier rejection while preserv- +ing the shape of a Gaussian tail. For each iteration of this pro- +cess, we take a discrepancy threshold, which we decrease from +5σ to 2σ with step size 0.5σ across these iterations. We then +randomly mask a subset of the pixels with residuals more than +the discrepancy level at the given iteration such that the number +of remaining pixels with such high residuals is statistically ex- +pected. The final masked area after the iterations is illustrated in +Figure 11. We tabulate the best-fit light model parameters in Ta- +ble 1 and compare them with those from Suyu et al. (2013). The +circularized half-light radius for our best-fit model is θeff = 1′′.91, +which is slightly larger than the value θeff = 1′′.85 from Suyu +et al. (2013) based on the same imaging data but from a smaller +cutout (illustrated in 11). We then take the MGE of the fitted +double Sérsic profile as the light distribution I(x, y) in our dy- +namical modeling. We propagate the uncertainties and covari- +ances from the light profile fitting into the dynamical modeling. +To do that, we sample from the multivariate normal distribution +corresponding to all the light model parameters for each call of +the likelihood function within the MCMC process and then take +the MGE of the light profile given the sampled parameters. +5.1.3. Oblate or prolate shape of the axisymmetry +The oblateness, prolateness, or triaxiality of a slow rotator +galaxy can, in principle, be constrained from the kinematic mis- +alignment angle ∆ϕkin ≡ +���ϕkin − ϕlight +���. However, we do not +Article number, page 11 of 21 + +A&A proofs: manuscript no. ms +N +E +1" +Data +1" +Reconstructed +E +N +1" +Normalized Residuals +E +N +4 +3 +2 +1 +0 +1 +log10 flux +4 +3 +2 +1 +0 +1 +log10 flux +3 +2 +1 +0 +1 +2 +3 +(data model) / noise +Fig. 11. Fit of the lens galaxy’s surface brightness profile. Left: the HST/ACS imaging in the F814W filter of the lens system RXJ1131−1231 +with the quasar images and the lensed arcs subtracted using the prediction from the best-fit lens model from Suyu et al. (2013), thus leaving +only the lens galaxy’s light to be fitted. The orange circle shows the large circular region considered for fitting in our analysis, and the yellow +square shows the smaller cutout used for lens modeling by Suyu et al. (2013). The cyan annulus contains the region where pixels were fitted to +reconstruct the source by Suyu et al. (2013). Thus the lensed arcs from the quasar host galaxy were subtracted only within this annulus. The red +contours mark quasar image positions with significant residuals due to saturated pixels, which we mask. Middle: The fitted light profile with a +double Sérsic model. The black pixels correspond to masked pixels. The additional masked pixels within the orange circle not described above +are randomly selected through an iterative process that performs outlier rejection while preserving the Gaussian tail (see Section 5.1.2 for details). +Right: Normalized residual of the best-fit model. +detect any significant rotational pattern in the vmean map (Fig- +ure 9). Thus, the uncertainty for the constrained kinematic major +axes is too large to be meaningful, and we cannot directly con- +strain this galaxy’s oblateness from the data. Instead, we obtain +the probability of oblateness from a population prior based on +189 slow rotator elliptical galaxies that are in the Sloan Digi- +tal Sky Survey’s (SDSS’s) Mapping Nearby Galaxies at APO +(MaNGA) sample (Abolfathi et al. 2018; Graham et al. 2018). +We take the distribution of ∆ϕkin for this sample of slow rota- +tors (Li et al. 2018), where ∆ϕkin = 0◦ corresponds to a purely +oblate shape, and ∆ϕkin = 90◦ corresponds to a purely prolate +shape. Li et al. (2018) find two distinct peaks in the distribution +at ∆ϕkin = 0◦ and ∆ϕkin = 90◦ (see Figure 12). We, therefore, +fit the data points with a double Gaussian profile with the means +set at ∆ϕkin = 0◦ and ∆ϕkin = 90◦ (see the fit in Figure 12). +Although the slow rotators with 0◦ < ∆ϕkin < 90◦ have triaxial +shapes, we choose only oblate or prolate axisymmetric shapes +in our dynamical modeling for computational simplicity. There- +fore, we take ∆ϕkin < 45◦ as the oblate case and ∆ϕkin > 45◦ as +the prolate case. We obtain the prior probability p(oblate)pop of +the galaxy being oblate as +p(oblate)pop = +� 45◦ +0◦ +d(∆ϕkin) p(∆ϕkin)pop ≃ 0.65, +(25) +and thus p(prolate)pop = 1 − p(oblate)pop ≃ 0.35. +The jampy software package, by default, adopts the oblate +case for deprojection. We implement the prolate case in jampy +by setting qprolate = 1/q > 1 and switching the x and y axes in the +input coordinate system. Due to the switching of x and y axes, +σ parameters of the MGEs for mass and light models need to be +scaled as σprolate = qσ. +5.1.4. Inclination +The observed axis ratio of light ql,obs = 0.850 ± 0.002 relates to +ql,int through the inclination angle i as +q2 +l,obs = q2 +l,int sin2 i + cos2 i. +(26) +We impose a prior on the intrinsic axis ratio ql,int from a sam- +ple of massive elliptical galaxies in the SDSS with stellar mass +0 +20 +40 +60 +80 +Kinematic misalignment, +kin ( +◦) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Probability density (�90) +oblate +prolate +Li et al. (2018)'s population +double Gaussian fit +Fig. 12. Population prior on kinematic misalignment angle ∆ϕkin ≡ +���ϕkin − ϕlight +��� for a sample of slow rotator elliptical galaxies from the +SDSS’s MaNGA dataset (Li et al. 2018). Here, ∆ϕkin = 0◦ corresponds +to a purely oblate shape, and ∆ϕkin = 90◦ corresponds to a purely prolate +shape. The vertical dashed grey lines mark ∆ϕkin = 0◦, 45◦, and 90◦. The +red points with error bars show the measurements from Li et al. (2018). +We fit this distribution with a double Gaussian model (blue line) with +the means fixed to ∆ϕkin = 0◦ and ∆ϕkin = 90◦. We take ∆ϕkin < 45◦ +as the oblate case and ∆ϕkin > 45◦ as the prolate case. Integrating the +double Gaussian model from 0◦ to 45◦ gives the prior probability of +oblateness p(oblate)pop ≃ 0.65. +10.8 < log10(M⋆/M⊙) < 11.5 at 0.04 < z < 0.08 (Chang et al. +2013). The distribution of ql,int by Chang et al. (2013) is differ- +ent for oblate and prolate assumptions. Therefore, we adopt the +specific prior corresponding to the oblate or the prolate case (see +Figure 13). +5.1.5. Anisotropy profile +We investigate two choices to parametrize the anisotropy pro- +file. The first choice is a single spatially constant β = 1 − σ2 +θ/σ2 +r +value for all the light MGE components. Numerically, we sam- +Article number, page 12 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +0.2 +0.4 +0.6 +0.8 +1.0 +Intrinsic axis ratio of light, ql,int +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Probablity density +oblate case +prolate case +Fig. 13. Prior on the intrinsic axis ratio ql,int of light for oblate (solid +line) and prolate (dashed line) cases from Chang et al. (2013). The pri- +ors correspond to massive elliptical galaxies from the SDSS survey at +0.04 < z < 0.08 with 10.8 < log10(M⋆/M⊙) < 11.5. +ple σθ/σr with a uniform prior (σθ/σr) ∼ U(0.78, 1.14). This +range of σθ/σr allows −0.31 < β < 0.38. We adopt this range +using the β values of eight slow rotator galaxies measured by +Cappellari et al. (2007, see Figure 2). These measurements of +β by Cappellari et al. (2007) are from Schwarzschild modeling +of data with one of the highest S/N values in the literature, al- +lowing to constrain the Gauss–Hermite moments up to order six. +Applying the student’s t-distribution on the sample mean of this +small sample, we find the 95% confidence interval of the pop- +ulation mean for β to be [-0.10, 0.17] and the standard devia- +tion to be 0.16. These values infer that 95% of the population is +contained within β ∈ [−0.31, 0.38], which we take as the bound- +aries of our prior range. The second choice of the anisotropy +profile has two free parameters: the inner light MGE compo- +nents with σ < rbreak = θeff = 1′′.91 are assigned one value for +(σθ/σr)inner and the outer light MGE components with σ ≥ rbreak +are assigned another independent value of (σθ/σr)outer. Thus, +this parametrization with two free parameters allows radial vari- +ability in the anisotropy profile. Both the inner and outer ratios +have uncorrelated uniform priors (σθ/σr) ∼ U(0.78, 1.14). For +these two choices of parametrization, we compute the Bayesian +information criterion (BIC) given by +BIC ≡ k log(Nbin) − 2 log ˆL, +(27) +where k is the number of free model parameters, Nbin is the num- +ber of data points, and ˆL is the maximum likelihood. We approx- +imate ˆL from the highest likelihood value sampled in the MCMC +chain. The single-parameter β model provides the lowest BIC +value excluding the two-parameter β model with ∆BIC ≈ 3.7 +(i.e., positively excluded; Raftery 1995). We check that the dif- +ference between the highest and the second highest likelihood +values among the MCMC samples is ≪ ∆BIC, thus this ∆BIC +value is robust against our approximation of ˆL from the high- +est likelihood value in the sampled chain. The non-detection +of varying anisotropy in our data is consistent with that ob- +served in nearby elliptical galaxies, as even high-S/N SAURON +data for a large sample of galaxies are accurately described by +JAM models with constant anisotropy, as used here, within the +noise of the kinematics (e.g., Cappellari et al. 2013). We com- +pare the posterior distributions of the model parameters for the +two anisotropy models in Figure 14. An example of a best-fit +kinematic model and the corresponding residual with the single- +parameter β model and oblate axisymmetry is illustrated in Fig- +ure 15. The reduced χ2 +ν value is 0.83 with ν = 41 degrees of +freedom. The distribution of residuals is similar to a normal dis- +tribution expected from a perfect model for data with Gaussian +noise, illustrating that our model is appropriate for the data. We +show the range of velocity dispersion radial profiles sampled by +our model in Figure 16 and compare it with the radially averaged +measurements of the velocity dispersion. This illustration shows +that our model reproduces the uncertainty range of the measure- +ment. +5.2. Checking potential systematics due to modeling choices +In this section, we perform several checks on potential system- +atics for different choices in the dynamical model setup. +5.2.1. Comparison between power-law and composite mass +models +In addition to the power-law mass model, Suyu et al. (2014) also +adopted a composite mass model individually describing the lens +galaxy’s dark matter and baryonic components. The dark matter +distribution was modeled with an elliptical NFW profile in the +potential. The parameters in this profile are the normalization +of the NFW component κs, the NFW scale radius rscale, and the +mass axis ratio qm. The baryonic component was modeled with +a mass-follow-light profile with a free mass-to-light ratio (M/L) +parameter. Thus, this mass model parametrization has one more +free parameter than the power-law model. See Suyu et al. (2014) +for parametric definitions of these profiles. We implement this +composite mass profile as κcomp +model in Equation (19) and adopt the +model posterior from Suyu et al. (2014) as a prior in our model. +We appropriately convert the ellipticity defined in the potential +by Suyu et al. (2014) to an ellipticity defined in the convergence +in our model. We take the MGE of this composite surface density +model as done for the power-law surface density model. How- +ever, since the dark matter and baryonic components have differ- +ent ellipticities, we take the MGE of each component separately +to preserve the ellipticity information in deprojection. Specifi- +cally, We take the MGE of the approximate MST with λint of +the dark matter profile and the MGE of an accordingly rescaled +baryonic profile, which effectively results in the total mass pro- +file being transformed as the approximate MST with λint. +This mass model with one more free parameter than the +power-law model has a higher BIC score with ∆BIC = 3.8. Thus, +the BIC excludes the composite model with positive evidence +(Raftery 1995). The median values of Dd from the power-law +and composite mass models differ by 0.9% (0.07σ, Figure 17), +and the median D∆t values differ by 1.26% (0.06σ). Therefore, +we conclude that our power-law mass model with an additional +degree of freedom to scale with the MST robustly describes the +observed data. +5.2.2. Comparison between prolate and oblate axisymmetry +We compare the inferred Dd between the purely oblate and +purely prolate cases in the deprojected 3D spheroidal shape of +the mass and light models (Figure 18). The median Dd values +from these two cases differ by 3.6% (0.3σ), and the median +D∆t values differ by 0.94% (0.04σ). Our final distance poste- +rior is the combination of oblate and prolate cases, with weights +p(oblate)pop = 0.65 and 1 − p(oblate)pop = 0.35, respectively. +Article number, page 13 of 21 + +A&A proofs: manuscript no. ms +1.64 +E ( ) +0.0 +0.2 +blinded Dd +0.0 +0.5 +blinded D t +0.4 +0.0 +blinded +int +0.85 +1.00 +( +/ r)outer +0.85 +1.00 +/ r +0.1 +0.2 +ext +60 +80 +i ( +◦) +0.75 +0.77 +qm +1.9 +2.0 +2 +0.76 +qm +60 +80 +i ( +◦) +0.07 +0.23 +ext +0.87 +1.05 +/ r +0.87 +1.05 +( +/ r)outer +0.1 +blinded +int +0.0 0.5 +blinded D t +0.0 +0.3 +blinded Dd +different parameters for inner and outer MGE components +same for all MGE components +Fig. 14. Constraints from axisymmetric JAM modeling on the power-law mass model parameters (θE, γ, and qm), internal MST parameter λint, +external convergence κext, anisotropy profile parameter(s), and the cosmological distances D∆t and Dd. assuming two anisotropy parametrizations: +(i) one single constant β ≡ 1 − (σθ/σr)2 for all light MGE components (orange contours), and (ii) one free (σθ/σr)inner ≡ (σθ/σr) for light MGE +components with σ < rbreak = θeff = 1′′.91 and another free (σθ/σr)outer for light MGE components with σ > rbreak(blue contours). The blinded +parameters are blinded as pblinded ≡ p/⟨p⟩ − 1 so that the distributions only reveal fractional uncertainties. The darker and lighter shaded regions in +the 2D plots trace 68% and 95% credible regions, respectively. The mass model parameters Einstein radius θE, power-law slope γ, axis ratio q, and +position angle PA are additionally constrained through a prior from the imaging data from Suyu et al. (2013). The two anisotropy parametrizations +provide equally good fits to the kinematics data. However, the BIC selects the constant-β anisotropy model over the other one with one additional +free parameter (∆BIC value is 3.5). +Thus, this difference between the oblate and prolate cases is +marginalized in our final cosmological distance posterior. +We also compare the predictions from axisymmetric and +spherical mass models in Figure 18. The median Dd from the +spherical model matches very well with the axisymmetric pro- +late model, but the median D∆t differs by 2.0% (0.08σ). The +galaxy is only mildly elliptical in projection (ql ∼ 0.85), and +the resulting axisymmetric models are not very flat. For this rea- +son, the relatively small difference between the axisymmetric +and spherical models is not surprising. +Article number, page 14 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +Data +Model +Residual +2.5 +0.0 +2.5 +(data model)/noise +0.0 +0.2 +0.4 +density +Residual distribution +300 +250 +los (km s +1) +300 +250 +los (km s +1) +2 +0 +2 +(data model)/noise +Fig. 15. Observed velocity dispersion map in Voronoi bins (first panel), the best-fit dynamical model with a power-law mass model, constant β +anisotropy profile, and oblate shape (second panel), the normalized residual for the best-fit dynamical model (third panel), and the distribution of +the normalized residual (orange, fourth panel). The reduced χ2 quantity is χ2 +ν = 0.83 with degrees of freedom ν = 41. The grey dashed line in the +fourth panel shows a normal distribution expected for residuals from a perfect model to the data with Gaussian noise. The residual distribution for +41 points is similar to this Gaussian distribution. +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 + ( ) +240 +260 +280 +300 +Velocity dispersion (km s +1) +oblate +prolate +Radially binned measurements +Fig. 16. Radial profile of the line-of-sight velocity dispersion. The red +points are radially binned values from the 2D maps, with the horizontal +error bars illustrating the widths of the annuli. The lines show the radial +profiles for random samples from the dynamical model posterior. The +radial profile of the model is averaged over the major, minor, and inter- +mediate axes. The solid purple lines correspond to 65 random samples +for the oblate case, and the dashed green lines correspond to 35 ran- +dom samples for the prolate case. Note that the model was fit to the 2D +kinematics data. However, we illustrate the 1D radial profile only for +visualization. +5.2.3. Comparison between Voronoi binning schemes +Here, we compare the Voronoi binning schemes with two +choices for the target S/N in each bin: ≈ 23 Å−1 and ≈ 28 +Å−1. The two cases match very well with only a 0.21% differ- +ence (0.02σ) in the median values of Dd (Figure 19) and 0.28 +% difference (0.01σ) in the median D∆t values. As a result, we +conclude that our choice of the Voronoi binning scheme is not a +significant source of systematic error in our analysis. +Based on the systematics tests performed above, we adopt a +robust final distance posterior from the model with the power- +law parametrization for the mass profile that the approximate in- +ternal MST is applied to. We marginalize the oblate and prolate +axisymmetrical cases by combining the posteriors from these +two choices with weights of 0.65 and 0.35, respectively. In the +next section, we present the unblinded values from the distance +posterior and infer the value of H0 from it. +0.87 +1.05 +/ r +0.2 +0.0 +0.2 +blinded Dd +0.0 +0.5 +blinded D t +0.4 +0.0 +blinded +int +0.1 +blinded +int +0.0 0.5 +blinded D t +0.2 +0.2 +blinded Dd +Power-law mass model +Composite mass model +Fig. 17. Comparison of the constrained Dd from power-law (blue con- +tours) and composite (orange contours) mass models. The blinded pa- +rameters are blinded as pblinded ≡ p/⟨p⟩−1 so that the distributions only +reveal fractional uncertainties. The darker and lighter shaded regions in +the 2D plots trace 68% and 95% credible regions, respectively. +6. Cosmological inference +In this section, we infer cosmological parameters from the joint +distribution of Dd and D∆t, accounting for their covariance. The +unblinded point estimates of these distances are Dd = 865+85 +−81 +Mpc (a 9.6% measurement) at zd = 0.295, and D∆t = 2180+472 +−271 +Mpc (a 17% measurement) for zs = 0.657. +We infer H0 and Ωm from our distance posterior for a flat +ΛCDM cosmology (see Figure 20, left panel). We leave the ex- +ploration of more exotic cosmologies based on our distance pos- +terior for future studies. We approximate the likelihood function +L(H0, Ωm | Dd, D∆t) of the cosmological parameters using a 2D +Gaussian kernel density estimate (KDE) from the 2D distance +posterior. We adopt two choices of prior for Ωm: one is a uni- +form prior Ωm ∼ U(0.05, 0.5), and the other is a Gaussian prior +Ωm ∼ N(0.334, 0.018) from the Pantheon+ analysis of type Ia +Article number, page 15 of 21 + +A&A proofs: manuscript no. ms +0.87 +1.05 +/ r +0.3 +0.0 +blinded Dd +0.0 +0.5 +blinded D t +0.4 +0.0 +blinded +int +0.4 +0.1 +blinded +int +0.0 +0.5 +blinded D t +0 +blinded Dd +spherical +axisymmetric prolate +axisymmetric oblate +Fig. 18. Comparison of the constrained Dd between oblate (blue) and +prolate (blue) cases of the deprojected spheroidal shape in the dynam- +ical model. The blinded parameters are blinded as pblinded ≡ p/⟨p⟩ − 1 +so that the distributions only reveal fractional uncertainties. The darker +and lighter shaded regions in the 2D plots trace 68% and 95% credible +regions, respectively. +supernovae relative distances (Brout et al. 2022). We infer the +posterior joint PDF of H0 and Ωm by performing MCMC sam- +pling using emcee, given the likelihood function and prior choice. +We infer H0 = 77.1+7.3 +−7.1 km s−1 Mpc−1(a 9.4% measurement) +with the uniform Ωm-prior, and H0 = 76.0+7.3 +−6.6 km s−1 Mpc−1 +(a 9.1% measurement) with the Pantheon+ Ωm-prior (solid con- +tours in the right panel of Figure 20). We show the D∆t–Dd re- +gion allowed by our priors in the left panel of Figure 20, which +also shows the region allowed by our distance posterior that +provides information for the cosmological inference. Other cos- +mological models beyond flat ΛCDM (e.g., Bonvin et al. 2017; +Wong et al. 2020) or combining other cosmological probes in a +cosmology-independent manner (e.g., Taubenberger et al. 2019) +can utilize the additional cosmological information contained by +our full 2D posterior outside the regions probed by our cosmo- +logical priors. +For comparison, we also perform cosmological inference us- +ing only the 1D posterior of Dd (dashed contours in right panel +of Figure 20). This gives H0 = 75.5+8.3 +−7.2 km s−1 Mpc−1 (a 10.3% +measurement) for the uniform Ωm-prior, and H0 = 74.4+8.1 +−6.2 km +s−1 Mpc−1 (a 9.6% measurement) for the Pantheon+ Ωm-prior. +The Dd-only constraints are lower by ∼1.4% (0.15σ) than that +from the full 2D distance posterior (for the uniform Ωm-prior). +This slight difference arises from the projection difference of the +2D posterior along the Dd direction and along the narrow track +allowed by our choice of cosmological priors. +7. Discussion +We now compare our results with previous works (Section 7.1), +discuss the improvement of the constraint in this paper over +0.87 +1.05 +/ r +0.2 +0.0 +0.2 +blinded Dd +0.0 +0.5 +blinded D t +0.4 +0.0 +blinded +int +0.1 +blinded +int +0.0 +0.5 +blinded D t +0.2 +0.2 +blinded Dd +target S/N + 23 � +1 +target S/N + 28 � +1 +Fig. 19. Comparison of the constrained Dd between two choices of the +target S/N for each bin in the Voronoi binning scheme. The blinded pa- +rameters are blinded as pblinded ≡ p/⟨p⟩−1 so that the distributions only +reveal fractional uncertainties. The darker and lighter shaded regions in +the 2D plots trace 68% and 95% credible regions, respectively. +single-aperture stellar kinematics (Section 7.2), and describe the +limitations of this work (Section 7.3). +7.1. Comparison with previous time-delay H0 measurements +Our measured value H0 = 77.1+7.3 +−7.1 km s−1 Mpc−1 is consistent +with previous measurements from lensing time delays with dif- +ferent treatments of the MSD (see Figure 21). These previous +studies can be divided into two approaches: the first breaks the +MSD by assuming simple parametric mass profiles such as the +power law or composite (i.e., NFW halo and stars with constant +mass-to-light ratio), and the second breaks the MSD based solely +on stellar kinematics. Our study belongs to the second approach +by allowing the freedom in the model to be maximally degener- +ate with H0 and constraining it solely from the spatially resolved +stellar kinematics. However, it is illustrative to compare our re- +sult with the first approach to discuss the validity of their mass +model assumptions. +Following the first approach, Suyu et al. (2013, 2014) mea- +sured H0 = 80.0+4.5 +−4.7 km s−1 Mpc−1 from this same system +RXJ1131−1231 with simple parametric mass profiles using HST +imaging. Chen et al. (2019) combined the HST imaging and +adaptive-optics-assisted imaging from the Keck Telescope to +measure H0 = 78.3+3.4 +−3.3 km s−1 Mpc−1 ˙Although these studies +used single-aperture stellar kinematics, the MSD was already +broken by the assumption of parametric mass profiles, and the +single-aperture velocity dispersion helped tighten the constraint +and made the inferred H0 values from the power-law and com- +posite models more consistent(Suyu et al. 2014). Our measured +value – albeit with a larger uncertainty due to the maximal free- +dom allowed in the mass model – has a median value very close +to these previous measurements. Such a good agreement in the +medians suggests that these previous studies’ simple parametric +Article number, page 16 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +2000 +3000 +D t (Mpc) +600 +800 +1000 +Dd (Mpc) +800 +1000 +Dd (Mpc) +Our measurement +60 +80 +100 +H0 (km s +1 Mpc +1) +0.0 +0.2 +0.4 +m +0.0 +0.2 +0.4 +m +m ∼U(0.05,0.5) +m ∼N(0.334,0.018) +from Dd only +from Dd only +Fig. 20. Left: Final 2D posterior of the time-delay distance D∆t and the angular diameter distance Dd (emerald contour). The darker and lighter +shaded regions in the 2D plots trace 68% and 95% credible regions, respectively. We infer H0 and Ω from this distance posterior accounting for +the covariance in a flat ΛCDM cosmology. We take a wide uniform prior on H0 ∼ U(0, 150) km s−1 Mpc−1. The blue-shaded region corresponds +to a uniform prior Ωm ∼ U(0.05, 0.5) and the orange-shaded region corresponds to a Gaussian prior Ωm ∼ N(0.334, 0.018) from the Pantheon+ +analysis of type Ia supernovae relative distances (Brout et al. 2022). Right: Posterior PDF of H0 and Ωm in flat ΛCDM cosmology. We constrain +H0 to 9.4% and 9.1% precision for the uniform and Pantheon+ Ωm-priors, respectively. We show the cosmological parameter posterior from only +the 1D Dd posterior with dashed contours with colors matching the associated Ω prior. In this case, the H0 precision is 10.3% and 9.6% for the +uniform and Gaussian priors, respectively. The Dd-only constraint on the H0 is lower by ∼1.4% (0.15σ) than the constraint from the full 2D +posterior, for the uniform Ωm-prior. +mass models are close to the ground truth, and no bias is de- +tected within the precision afforded by the data. Future spatially +resolved velocity dispersion measurements for more time-delay +lens systems or better quality data for this system (e.g., from +the James Webb Space Telescope) will allow us to make a more +definitive statement on the validity of the parametric mass model +assumptions. +Following the second approach, Birrer et al. (2016) ana- +lyzed this same system RXJ1131−1231 using HST imaging and +single-aperture velocity dispersion. These authors marginalized +the effect of MSD by incorporating a source on the prior but +found that the H0 posterior strongly depends on the shape of +the anisotropy prior. These authors use two different choices for +this prior to find H0 = 74.5+8.0 +−7.8 km s−1 Mpc−1 and H0 = 86.6+6.8 +−6.9 +km s−1 Mpc−1. This large difference illustrates that single aper- +ture velocity dispersion imposes only a weak constraint on the +anisotropy profile and, thus, on the MSD. This result highlights +the need for spatially resolved velocity dispersion, such as the +one presented in this study. Our measured H0 has a precision +of 9% while allowing the data to constrain the MSD effect that +is maximally degenerate with H0, illustrating the power of spa- +tially resolved kinematics in constraining the anisotropy profile +and the MSD, despite the seeing-limited nature of our data. In +the future, exquisite data from the James Webb Space Telescope +(JWST) will provide an even more dramatic improvement (4% +H0 precision forecasted, Yıldırım et al. 2021). +We also compare our result with the measured values of H0 +from the current TDCOSMO sample of seven time-delay lenses. +With the power-law mass model assumptions, the combination +of seven time-delay lenses gives a 2% measurement with H0 = +74.2+1.6 +−1.6 km s−1 Mpc−1 (Wong et al. 2020; Millon et al. 2020b). +However, relaxing this mass profile assumption and constrain- +ing the MSD solely from the single-aperture stellar kinematics +of the TDCOSMO sample leads to a 9% uncertainty on the re- +sultant H0 = 74.5+5.6 +−6.1 km s−1 Mpc−1. In this study, we achieve +the same 9% precision from a single system, highlighting the +superb constraining power of spatially resolved kinematics over +single-aperture ones. +It is also worth comparing with the result obtained by Birrer +et al. (2020) when combining the seven TDCOSMO lenses with +information obtained from the external SLACS sample of non- +time-delay lenses, H0 = 67.4+4.1 +−3.2 km s−1 Mpc−1. Given the un- +certainties, our new measurement is not statistically inconsistent +with that result, although the difference is clearly important from +a cosmological standpoint. With the data in hand, we cannot con- +clude whether (a) the difference is real and the SLACS sam- +ple cannot, therefore, be combined with the TDCOSMO sam- +ple, or whether (b) it is due to a statistical fluctuation. This study +demonstrates that, as we gather more and better data for spatially +resolved kinematics and external samples of non-lenses, we will +soon be able to conclude whether the difference is real or not. +In the context of the “Hubble tension”, our new measurement +strengthens the tension by reaffirming the previously obtained +time-delay H0 measurements that agreed with other local mea- +surement values, e.g., from SH0ES (Riess et al. 2022). Although +the 9% uncertainty in H0 from our measurement alone is not suf- +ficient to resolve the tension, it demonstrates that time-delay cos- +mography can provide a powerful independent perspective with +the help of future data from telescopes such as Keck, JWST, and +the extremely large telescopes (e.g., see forecasts from Shajib +Article number, page 17 of 21 + +A&A proofs: manuscript no. ms +et al. 2018; Yıldırım et al. 2021; Birrer & Treu 2021). We cannot +help noticing that the median of our measurement is somewhat +higher than the mean of the local values (∼73 km s−1 Mpc−1). +However, the difference is not significant, given the uncertain- +ties. Therefore our likely explanation is that the difference origi- +nates from the inevitable statistical fluctuation pertaining to this +system, as the initial H0 measurements using simple parametric +assumption all provided such higher values (Suyu et al. 2013, +2014; Birrer et al. 2016; Chen et al. 2019). We conclude by +stressing that some dispersion around the mean is, of course, ex- +pected, and indeed Millon et al. (2020b) shows that the seven +TDCOSMO lenses scatter around the mean by an amount con- +sistent with the estimated errors. +7.2. Improvement from the spatial resolution of the stellar +kinematics +We investigate the improvement in constraints provided by the +spatially resolved nature of the stellar kinematics presented in +this paper over the unresolved or single-aperture case. Suyu +et al. (2013) presents a single-aperture measurement of the line- +of-sight velocity dispersion σlos = 323 ± 20 km s−1 obtained +within a 0′′.81 × 0′′.7 aperture with a 0′′.7 seeing. This mea- +surement was from the Low-Resolution Imaging Spectrometer +(LRIS; Oke et al. 1995) on the Keck Observatory. The probed +wavelength range was ∼3900–4700 Å, which probes mostly the +redward range of the Ca H&K lines with a little overlap with +the range probed by our data (i.e., 3300–4200 Å). If we take a +luminosity-weighted-sum of the spatially resolved velocity dis- +persion map within the same 0′′.81×0′′.7 aperture, we get 288±5 +km s−1, which is 1.7σ (11%) lower than the previous single- +aperture measurement. Although the 1.7σ difference is not sta- +tistically significant, some parts of it can be due to potential sys- +tematics in the kinematic extraction procedure or due to differ- +ent wavelength ranges probed. It is generally considered that the +minimum error, considering systematics, on velocity dispersion +measurements is 5%, even for very high-S/N data. +However, to illustrate the improvement in precision from the +spatially resolved nature of the velocity dispersion presented in +this study, we take a fiducial single-aperture measurement value +of 288 ± 18 km s−1. This mean value is from the luminosity- +weighted sum within the single aperture mentioned above, and +the 18 km s−1 uncertainty comes from applying the 6% uncer- +tainty of the 323 ± 20 km s−1 measurement on the fiducial mean. +We take the galaxy’s major axis to align with the rectangular +aperture’s longer side. Rotating the aperture by 90◦ only changes +the predicted velocity dispersion integrated within the aperture +by ≲ 0.1%, which is unsurprising given the mild ellipticity +(ql ∼ 0.85) of the galaxy and the 0′′.96 seeing. We compare the +key dynamical model parameters between the spatially resolved +and single-aperture cases in Figure 22. As expected, the internal +MST parameter λint and the anisotropy profile parameter σθ/σr +are almost completely unconstrained in the case of the single- +aperture stellar kinematics due to the mass-anisotropy degener- +acy (Treu & Koopmans 2002; Courteau et al. 2014). However, +the angular diameter distance Dd can be constrained to 15.7% +precision, largely by the anisotropy prior (cf. the 9.6% constraint +on Dd from the spatially resolved data). This single-aperture pre- +cision level on Dd agrees very well with the 17.9% precision +on Dd (= 810+160 +−130 Mpc) obtained by Jee et al. (2019) from the +same system RXJ1131−1231 based on the previously available +single-aperture stellar kinematics mentioned above. The Hubble +constant H0 can be inferred to 12.5% precision with the uniform +Ωm prior from the full 2D posterior of the fiducial single-aperture +case. Although the improvement in H0 precision (by ∼3%) from +the spatially resolved kinematics does not appear to be dramatic, +this is due to the fact that the projection of D∆t–Dd posterior +along the narrow track allowed by our chosen prior happens to +give a small difference between the two cases. The improvement +could have appeared more drastic if the full 2D posterior had a +different orientation from the prior region. In reality, the full cos- +mological information (illustrated by the area enclosed within +the 95% contour) contained by the single-aperture data is much +less than that from the spatially resolved data presented in this +study (see the D∆t–Dd contours in Figure 22). +7.3. Limitations of this study +One limitation of our study is the data quality. Although our data +are the first of their kind from a cutting-edge ground-based facil- +ity such as the Keck Observatory, there are opportunities to ob- +tain better-quality data. The KCWI instrument is seeing-limited. +Thus the S/N on the lensing galaxy is degraded by contamina- +tion from the nearby quasars, and the spatial resolution of the ve- +locity dispersion map is limited by the seeing. Adaptive-optics- +assisted IFU spectroscopy from the ground or observations from +space, e.g., with the JWST, can deliver exquisite spatially re- +solved data for improved H0 precision in the future (Yıldırım +et al. 2020, 2021). +Future data with higher spatial resolution will be particu- +larly powerful in constraining the anisotropy profile better. Our +measurement has only weak constraints on the anisotropy pro- +file, which is largely bounded by the adopted uniform prior (see +Figure 14). This prior is obtained from a sample of eight local +massive ellipticals with one of the highest quality spatially re- +solved kinematics. However, this is a small sample size. A tighter +anisotropy prior from larger samples of massive ellipticals, even +better if they are from a redshift range that matches with the one +for our system, will be helpful to mitigate further the degener- +acy induced by the anisotropy profile, i.e., the mass-anisotropy +degeneracy (Treu & Koopmans 2002; Courteau et al. 2014). +8. Conclusion +We measured the spatially resolved stellar velocity dispersion of +the lens galaxy in RXJ1131−1231 using the KCWI IFU spectro- +graph on the Keck Observatory. We combined the new spatially +resolved stellar kinematics with previously obtained lens mod- +els derived from HST imaging data, observed time delays, and +estimated line-of-sight lensing effects (i.e., the external conver- +gence) to infer H0. Combining the spatially resolved velocity +dispersion with lens imaging and time delays simultaneously al- +leviates the MSD in the measured D∆t and additionally measures +the angular diameter distance Dd. +In order to prevent conscious or unconscious experimenter +bias, we blindly performed the dynamical modeling and the cos- +mographic inference. We unblinded the H0 value after all the co- +authors had agreed on the modeling choices after various checks +on systematics, and the analysis was frozen. The main conclu- +sions from our study are as follows: +– The 2D distance posterior of Dd and D∆t gives H0 = 77.1+7.3 +−7.1 +km s−1 Mpc−1 for a uniform prior on Ωm ∼ U(0.05, 0.5), and +H0 = 76.0+7.3 +−6.6 km s−1 Mpc−1for a Gaussian prior on Ωm from +the Pantheon+ analysis (Brout et al. 2022). +– Our 9.4% measurement from a single system with spatially +resolved kinematics provides a similar precision as, and is in +Article number, page 18 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +60 +70 +80 +90 +H0 (km s +1 Mpc +1) +Probability density +RXJ1131 1231 with the MSD broken assuming simple +parametric mass models (Chen et al. 2019) +Seven lenses with the MSD broken assuming simple +parametric mass models (Millon et al. 2020, +TDCOSMO-I) +Seven lenses with the MSD broken using single- +aperture kinematics (Birrer et al. 2020, TDCOSMO-IV) +RXJ1131 1231 with the MSD broken using spatially +resolved kinematics (this work) +Fig. 21. Comparison of our 9.4% H0 measurement (red, 77.1+7.3 +−7.1 km s−1 Mpc−1) from the single system RXJ1131−1231 with previous mea- +surements from Chen et al. (2019, blue), Millon et al. (2020b, grey), and Birrer et al. (2020, emerald). The distributions show the H0 posteriors +as described in the figure legend, and the points with error bars mark the mean and 68% credible intervals of the corresponding posterior with +matching color. For the same flexible mass models, our analysis on a single system provides a similar precision on H0 with that from seven lenses +with only single-aperture stellar kinematics (emerald, 73.3+5.8 +−5.8km s−1 Mpc−1). Moreover, the median value of our measurement falls very close to +those from previous analyses on the same system but with simple parametric assumption on the mass model breaking the MSD (blue, 78.3+3.4 +−3.3km +s−1 Mpc−1, cf. also 80.0+4.5 +−4.7 km s−1 Mpc−1by Suyu et al. 2014). +excellent agreement with, the current TDCOSMO sample of +seven time-delay lenses based only on single-aperture stellar +kinematics (H0 = 74.5+5.6 +−6.1 km s−1 Mpc−1, Birrer et al. 2020). +Note that the system RXJ1131−1231 analyzed here is part of +that sample of seven. +– The median value of H0 from our analysis is very close to the +previously inferred values assuming simple parametric mass +models (e.g., H0 = 78.3+3.4 +−3.3 km s−1 Mpc−1, Chen et al. 2019). +Thus we do not detect any potential bias in those mass profile +assumptions within the precision afforded by our data. +– Our measurement is in excellent agreement with that ob- +tained by Millon et al. (2020a), based on the standard as- +sumption of simply parametrized forms for the mass density +profile of the lens to break the MSD (74.2+1.6 +−1.6 km s−1 Mpc−1). +In conclusion, our study provides an important validation of +previous work by our collaboration on the determination of H0 +from time-delay cosmography (summarized in Figure 21). This +analysis also showcases the power of spatially resolved kinemat- +ics in breaking the degeneracies that limit the H0 precision when +mass profile assumptions on the galaxy density profile are re- +laxed. As the first application of such methodology performed +on real data, this study stands as an important proof of concept +to pioneer future studies on many more time-delay lens systems. +In the broader context of the "Hubble tension", the mea- +surement presented here is on the high end of the distribution. +However, the precision is not yet sufficient to rule out the val- +ues below 70km s−1 Mpc−1, generally favored by early universe +probes (Abdalla et al. 2022). Larger samples of time-delay lenses +with spatially resolved kinematics are needed to reach a conclu- +sive answer without making assumptions about the mass den- +sity profile of the deflectors. With JWST data already scheduled +and ground-based data similar to those presented here for 7 sys- +tems, ∼ 3% precision should be attainable in a relatively short +time scale (Birrer & Treu 2021). Beyond that, a sample of ∼40 +lensed quasars or supernovae with spatially resolved kinemat- +ics can provide the ∼1.2% precision on H0 that is necessary to +resolve or confirm the “Hubble tension” at 5σ confidence level, +with maximally flexible models, thanks to spatially resolved stel- +lar kinematics (Birrer & Treu 2021). +Acknowledgements. We thank Elizabeth Buckley-Geer, Thomas E. Collett, +Philip J. Marshall, and Chiara Spiniello for useful discussions and comments that +improved this study and the manuscript. Support for this work was provided by +NASA through the NASA Hubble Fellowship grant HST-HF2-51492 awarded to +AJS by the Space Telescope Science Institute, which is operated by the Associ- +ation of Universities for Research in Astronomy, Inc., for NASA, under contract +NAS5-26555. TT and GCFC acknowledge support by NSF through grants NSF- +AST-1906976 and NSF-AST-1836016, and from the Moore Foundation through +grant 8548. PM and CDF acknowledge support for this work from the National +Science Foundation under Grant No. AST-1907396 SHS thanks the Max Planck +Society for support through the Max Planck Research Group and the Max Planck +Fellowship. SHS is supported in part by the Deutsche Forschungsgemeinschaft +(DFG, German Research Foundation) under Germany’s Excellence Strategy - +EXC-2094 - 390783311. This project has received funding from SNSF and from +the European Research Council (ERC) under the European Union’s Horizon +2020 research and innovation programme (COSMICLENS : grant agreement No +787886). VNB gratefully acknowledges assistance from National Science Foun- +dation (NSF) Research at Undergraduate Institutions (RUI) grant AST-1909297. +Note that findings and conclusions do not necessarily represent views of the NSF. +This work used computational and storage services associated with the Hoffman2 +Shared Cluster provided by UCLA Institute for Digital Research and Education’s +Research Technology Group. +The data presented herein were obtained at the W. M. Keck Observatory, which is +operated as a scientific partnership among the California Institute of Technology, +the University of California and the National Aeronautics and Space Adminis- +tration. The Observatory was made possible by the generous financial support +of the W. M. Keck Foundation. The authors wish to recognize and acknowledge +the very significant cultural role and reverence that the summit of Maunakea has +always had within the indigenous Hawaiian community. We are most fortunate +to have the opportunity to conduct observations from this mountain. +This research made use of jampy (Cappellari 2008, 2020), pPXF (Cappellari +2017, 2022), pafit (Krajnovi´c et al. 2006), vorbin (Cappellari & Copin 2003), +mgefit (Cappellari 2002) , lenstronomy (Birrer & Amara 2018; Birrer et al. +Article number, page 19 of 21 + +A&A proofs: manuscript no. ms +0.87 +1.05 +/ r +600 +1000 +Dd (Mpc) +2000 +4000 +D t (Mpc) +0.6 +0.8 +1.0 +int +0.7 +1.0 +int +3000 +D t (Mpc) +900 +Dd (Mpc) +spatially resolved +single aperture +Fig. 22. +Comparison of the distance constraints between spatially +resolved velocity dispersion and single-aperture velocity dispersion. +Here, the integrated velocity dispersion is taken as the fiducial value +of 287 ± 18 km s−1 to match the mean of our spatially resolved mea- +surement, but the uncertainty of a single-aperture velocity dispersion +measurement (Suyu et al. 2013). The darker and lighter shaded regions +in the 2D plots trace 68% and 95% credible regions, respectively. The +single-aperture velocity dispersion cannot constrain the anisotropy pro- +file parameter σθ/σr and the internal MST parameter λint, with both +limited by the prior. As a result, the D∆t–Dd posterior is constrained +much more weakly. +2021), numpy (Oliphant 2015), scipy (Jones et al. 2001), astropy (Astropy Col- +laboration 2013, 2018), jupyter (Kluyver et al. 2016), matplotlib (Hunter 2007), +seaborn (Waskom et al. 2014), emcee (Foreman-Mackey et al. 2013), and getdist +(https://github.com/cmbant/getdist). +References +Abdalla, E., Abellán, G. F., Aboubrahim, A., et al. 2022, Journal of High Energy +Astrophysics, 34, 49 +Abolfathi, B., Aguado, D. S., Aguilar, G., et al. 2018, The Astrophysical Journal +Supplement Series, 235, 42 +Aiola, S., Calabrese, E., Maurin, L., et al. 2020, J. Cosmology Astropart. Phys., +2020, 047–047 +Astropy Collaboration. 2013, A&A, 558, A33 +Astropy Collaboration. 2018, AJ, 156, 123 +Avila, R., Koekemoer, A., Mack, J., & Fruchter, A. 2015, Optimizing pixfrac in +Astrodrizzle: An example from the Hubble Frontier Fields, Tech. rep. +Bacon, R., Simien, F., & Monnet, G. 1983, Astronomy and Astrophysics, Vol. +128, p. 405-410 (1983), 128, 405 +Barnabè, M., Czoske, O., Koopmans, L. V. E., et al. 2009, Monthly Notices of +the Royal Astronomical Society, 399, 21 +Barnabè, M., Dutton, A. A., Marshall, P. J., et al. 2012, Monthly Notices of the +Royal Astronomical Society, 423, 1073 +Bertin, G. & Lombardi, M. 2006, ApJ, 648, L17 +Binney, J. & Tremaine, S. 1987, Galactic dynamics +Birrer, S. & Amara, A. 2018, Physics of the Dark Universe, 22, 189 +Birrer, S., Amara, A., & Refregier, A. 2016, J. Cosmology Astropart. Phys., 8, +020 +Birrer, S., Dhawan, S., & Shajib, A. J. 2022a, ApJ, 924, 2 +Birrer, S., Millon, M., Sluse, D., et al. 2022b, Time-Delay Cosmography: Mea- +suring the Hubble Constant and other cosmological parameters with strong +gravitational lensing +Birrer, S., Shajib, A. J., Galan, A., et al. 2020, A&A, 643, A165 +Birrer, S., Shajib, A. J., Gilman, D., et al. 2021, JOSS, 6, 3283 +Birrer, S. & Treu, T. 2021, A&A, 649, A61 +Birrer, S., Treu, T., Rusu, C. E., et al. 2019, MNRAS, 484, 4726 +Blakeslee, J. P., Jensen, J. B., Ma, C.-P., Milne, P. A., & Greene, J. E. 2021, ApJ, +911, 65 +Blum, +K., +Castorina, +E., +& +Simonovi´c, +M. +2020, +arXiv +e-prints, +arXiv:2001.07182 +Bonvin, V., Courbin, F., Suyu, S. H., et al. 2017, MNRAS, 465, 4914 +Brout, D., Scolnic, D., Popovic, B., et al. 2022, The Astrophysical Journal, 938, +110 +Buckley-Geer, E. J., Lin, H., Rusu, C. E., et al. 2020, MNRAS, 498, 3241 +Cappellari, M. 2002, MNRAS, 333, 400 +Cappellari, M. 2008, MNRAS, 390, 71 +Cappellari, M. 2016, ARA&A, 54, 597 +Cappellari, M. 2017, MNRAS, 466, 798 +Cappellari, M. 2020, Monthly Notices of the Royal Astronomical Society, 494, +4819 +Cappellari, M. 2022, Full spectrum fitting with photometry in ppxf: non- +parametric star formation history, metallicity and the quenching boundary +from 3200 LEGA-C galaxies at redshift z 0.8 +Cappellari, M. & Copin, Y. 2003, Monthly Notices of the Royal Astronomical +Society, 342, 345 +Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418 +Cappellari, M., Scott, N., Alatalo, K., et al. 2013, MNRAS, 432, 1709 +Chang, Y.-Y., van der Wel, A., Rix, H.-W., et al. 2013, The Astrophysical Journal, +773, 149 +Chen, G. C. F., Fassnacht, C. D., Suyu, S. H., et al. 2019, MNRAS, 490, 1743 +Chen, G. C.-F., Fassnacht, C. D., Suyu, S. H., et al. 2021a, A&A, 652, A7 +Chen, G. C. F., Treu, T., Fassnacht, C. D., et al. 2021b, Monthly Notices of the +Royal Astronomical Society, 508, 755 +Collett, T. E., Oldham, L. J., Smith, R. J., et al. 2018, Science, 360, 1342 +Courbin, F., Eigenbrod, A., Vuissoz, C., Meylan, G., & Magain, P. 2005, 225, +297 +Courteau, S., Cappellari, M., de Jong, R. S., et al. 2014, Reviews of Modern +Physics, 86, 47 +de Zeeuw, P. T., Evans, N. W., & Schwarzschild, M. 1996, Monthly Notices of +the Royal Astronomical Society, 280, 903 +Di Valentino, E., Mena, O., Pan, S., et al. 2021, Classical and Quantum Gravity, +38, 153001 +Efstathiou, G. 2021, MNRAS, 505, 3866 +Emsellem, E., Monnet, G., Bacon, R., & Nieto, J.-L. 1994, A&A, 285, 739 +Falco, E. E., Gorenstein, M. V., & Shapiro, I. I. 1985, ApJ, 289, L1 +Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, +306 +Foxley-Marrable, M., Collett, T. E., Vernardos, G., Goldstein, D. A., & Bacon, +D. 2018, Monthly Notices of the Royal Astronomical Society, 478, 5081 +Freedman, W. L. 2021, ApJ, 919, 16 +Freedman, W. L., Madore, B. F., Hatt, D., et al. 2019, ApJ, 882, 34 +Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57 +Fruchter, A. S. & Hook, R. N. 2002, Publications of the Astronomical Society of +the Pacific, 114, 144 +Gilman, D., Birrer, S., & Treu, T. 2020, A&A, 642, A194 +Gomer, M. R., Sluse, D., Van de Vyvere, L., Birrer, S., & Courbin, F. 2022, +A&A, 667, A86 +Gonneau, A., Lyubenova, M., Lançon, A., et al. 2020, A&A, 634, A133 +Gonzaga, S., Hack, W., Fruchter, A., & Mack, J. 2012, The DrizzlePac Handbook +Goodman, J. & Weare, J. 2010, Communications in Applied Mathematics and +Computational Science, 5, 65–80 +Graham, M. T., Cappellari, M., Li, H., et al. 2018, Monthly Notices of the Royal +Astronomical Society, 477, 4711 +Greene, Z. S., Suyu, S. H., Treu, T., et al. 2013, ApJ, 768, 39 +Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90 +Jeans, J. H. 1922, Monthly Notices of the Royal Astronomical Society, 82, 122 +Jee, I., Suyu, S. H., Komatsu, E., et al. 2019, Science, 365, 1134 +Jones, E., Oliphant, T., Peterson, P., & Others. 2001, SciPy: Open source scien- +tific tools for Python +Kluyver, T., Ragan-Kelley, B., Pérez, F., et al. 2016, in Positioning and Power +in Academic Publishing: Players, Agents and Agendas, ed. F. Loizides & +B. Schmidt (IOS Press BV, Amsterdam, Netherlands), 87 – 90 +Knox, L. & Millea, M. 2020, Phys. Rev. D, 101, 043533 +Kochanek, C. S. 2020, MNRAS, 493, 1725–1735 +Kourkchi, E., Tully, R. B., Eftekharzadeh, S., et al. 2020, ApJ, 902, 145 +Krajnovi´c, D., Cappellari, M., de Zeeuw, P. T., & Copin, Y. 2006, Monthly No- +tices of the Royal Astronomical Society, 366, 787 +Li, H., Mao, S., Cappellari, M., et al. 2018, The Astrophysical Journal, 863, L19 +Millon, M., Courbin, F., Bonvin, V., et al. 2020a, Astronomy and Astrophysics, +640, A105 +Millon, M., Galan, A., Courbin, F., et al. 2020b, A&A, 639, A101 +More, A., Suyu, S. H., Oguri, M., More, S., & Lee, C.-H. 2017, The Astrophys- +ical Journal, 835, L25 +Article number, page 20 of 21 + +A. J. Shajib et al.: H0 from spatially resolved kinematics of RXJ1131−1231 +Morrissey, P., Matuszewski, M., Martin, C., et al. 2012, 8446, 844613 +Morrissey, P., Matuszewski, M., Martin, D. C., et al. 2018, The Astrophysical +Journal, 864, 93 +Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563 +Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493 +Oke, J. B., Cohen, J. G., Carr, M., et al. 1995, Publications of the Astronomical +Society of the Pacific, 107, 375 +Oliphant, T. E. 2015, Guide to NumPy, 2nd edn. (USA: CreateSpace Independent +Publishing Platform) +Pesce, D. W., Braatz, J. A., Reid, M. J., et al. 2020, ApJ, 891, L1 +Planck Collaboration. 2020, A&A, 641, A6 +Raftery, A. E. 1995, Sociological Methodology, 25, 111 +Refsdal, S. 1964, MNRAS, 128, 307 +Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, The Astrophysical Journal, +934, L7 +Robertson, J. G. 2013, Publications of the Astronomical Society of Australia, 30, +e048 +Rusu, C. E., Fassnacht, C. D., Sluse, D., et al. 2017, MNRAS, 467, 4220 +Rusu, C. E., Wong, K. C., Bonvin, V., et al. 2020, MNRAS, 498, 1440 +Schneider, P., Ehlers, J., & Falco, E. E. 1992, Gravitational Lenses +Schneider, P. & Sluse, D. 2013, A&A, 559, A37 +Schneider, P. & Sluse, D. 2014, A&A, 564, A103 +Shajib, A. J. 2019, MNRAS, 488, 1387–1400 +Shajib, A. J., Birrer, S., Treu, T., et al. 2020, MNRAS, 494, 6072 +Shajib, A. J., Birrer, S., Treu, T., et al. 2019, MNRAS, 483, 5649 +Shajib, A. J., Glazebrook, K., Barone, T., et al. 2022a, LensingETC: a tool to op- +timize multi-filter imaging campaigns of galaxy-scale strong lensing systems +Shajib, A. J., Treu, T., & Agnello, A. 2018, MNRAS, 473, 210 +Shajib, A. J., Vernardos, G., Collett, T. E., et al. 2022b, Strong Lensing by Galax- +ies +Sluse, D., Claeskens, J.-F., Hutsemékers, D., & Surdej, J. 2007, Astronomy and +Astrophysics, 468, 885 +Sluse, D., Surdej, J., Claeskens, J.-F., et al. 2003, A&A, 406, L43 +Sonnenfeld, A., Treu, T., Marshall, P. J., et al. 2015, ApJ, 800, 94 +Suyu, S. H., Auger, M. W., Hilbert, S., et al. 2013, ApJ, 766, 70 +Suyu, S. H., Marshall, P. J., Auger, M. W., et al. 2010, ApJ, 711, 201 +Suyu, S. H., Treu, T., Hilbert, S., et al. 2014, ApJ, 788, L35 +Sánchez-Blázquez, P., Peletier, R. F., Jiménez-Vicente, J., et al. 2006, Monthly +Notices of the Royal Astronomical Society, 371, 703 +Sérsic, J. L. 1968, Atlas de Galaxias Australes +Taubenberger, S., Suyu, S. H., Komatsu, E., et al. 2019, A&A, 628, L7 +Tewes, M., Courbin, F., Meylan, G., et al. 2013, A&A, 556, A22 +Tihhonova, O., Courbin, F., Harvey, D., et al. 2018, MNRAS, 477, 5657 +Treu, T., Agnello, A., Baumer, M. A., et al. 2018, MNRAS, 481, 1041 +Treu, T. & Koopmans, L. V. E. 2002, MNRAS, 337, L6 +Treu, T. & Marshall, P. J. 2016, A&A Rev., 24, 11 +Treu, T., Suyu, S. H., & Marshall, P. J. 2022, Strong lensing time-delay cosmog- +raphy in the 2020s +Valdes, F., Gupta, R., Rose, J. A., Singh, H. P., & Bell, D. J. 2004, The Astro- +physical Journal Supplement Series, 152, 251 +Van de Vyvere, L., Gomer, M. R., Sluse, D., et al. 2022a, Astronomy and Astro- +physics, 659, A127 +Van de Vyvere, L., Sluse, D., Gomer, M. R., & Mukherjee, S. 2022b, Astronomy +and Astrophysics, 663, A179 +Verde, L., Treu, T., & Riess, A. G. 2019, Nature Astronomy, 3, 891–895 +Waskom, M., Botvinnik, O., Hobson, P., et al. 2014, seaborn: v0.5.0 (November +2014) +Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, Astronomy and Astrophysics +Supplement Series, 143, 9 +Wong, K. C., Suyu, S. H., Chen, G. C. F., et al. 2020, MNRAS, 498, 1420 +Yahalomi, D. A., Schechter, P. L., & Wambsganss, J. 2017, A Quadruply Lensed +SN Ia: Gaining a Time-Delay...Losing a Standard Candle +Yıldırım, +A., +Suyu, +S. +H., +Chen, +G. +C.-F., +& +Komatsu, +E. +2021, +arXiv:2109.14615 [astro-ph] [arXiv:2109.14615] +Yıldırım, A., Suyu, S. H., & Halkola, A. 2020, Monthly Notices of the Royal +Astronomical Society, 493, 4783 +1 Department of Astronomy & Astrophysics, University of Chicago, +Chicago, IL 60637, USA; e-mail: ajshajib@uchicago.edu +2 Kavli Institute for Cosmological Physics, University of Chicago, +Chicago, IL 60637, USA +3 Department of Physics and Astronomy, University of California, +Davis, CA 95616, USA +4 Department of Physics and Astronomy, University of California, +Los Angeles, CA 90095, USA +5 Sub-Department of Astrophysics, Department of Physics, Univer- +sity of Oxford, Denys Wilkinson Building, Keble Road, Oxford, +OX1 3RH, UK +6 Technical University of Munich, TUM School of Natural Sciences, +Department of Physics, James-Franck-Str. 1, Garching, 85748, Ger- +many +7 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, +Garching, 85748, Germany +8 Institute of Astronomy and Astrophysics, Academia Sinica, 11F of +ASMAB, No.1, Section 4, Roosevelt Road, Taipei, 10617, Taiwan +9 Physics Department, California Polytechnic State University, San +Luis Obispo, CA 93407, USA +10 Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL +60510, USA +11 STAR Institute, Quartier Agora, Allée du Six Août, 19c, 4000 Liége, +Belgium +12 Kavli Institute for Particle Astrophysics and Cosmology and Depart- +ment of Physics, Stanford University, Stanford, CA 94305, USA +13 SLAC National Accelerator Laboratory, Menlo Park, CA, 94025 +14 Department of Physics and Astronomy, Stony Brook University, +Stony Brook, NY 11794, USA +15 Institute of Physics, Laboratory of Astrophysics, Ecole Polytech- +nique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, +1290 Versoix, Switzerland +Article number, page 21 of 21 + diff --git a/CdE0T4oBgHgl3EQfyQI7/content/tmp_files/load_file.txt b/CdE0T4oBgHgl3EQfyQI7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf802fd249ad0cf67b83d2dbdad4576ce4150124 --- /dev/null +++ b/CdE0T4oBgHgl3EQfyQI7/content/tmp_files/load_file.txt @@ -0,0 +1,2118 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf,len=2117 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms ©ESO 2023 January 9, 2023 TDCOSMO XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Improved Hubble constant measurement from lensing time delays using spatially resolved stellar kinematics of the lens galaxy Anowar J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib,1, 2,⋆ Pritom Mozumdar,3 Geoff C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Chen,4 Tommaso Treu,4 Michele Cappellari,5 Shawn Knabel,4 Sherry H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Suyu,6, 7, 8 Vardha N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Bennert,9 Joshua A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Frieman,1, 2, 10 Dominique Sluse,11 Simon Birrer,12, 13, 14 Frederic Courbin,15 Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Fassnacht,3 Lizvette Villafaña,4 Peter R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Williams4 (Affiliations can be found after the references) Received xxx, xxxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' accepted xxx, xxxx ABSTRACT Strong-lensing time delays enable measurement of the Hubble constant (H0) independently of other traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The main limitation to the precision of time-delay cosmography is mass-sheet degeneracy (MSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Some of the previous TDCOSMO analyses broke the MSD by making standard assumptions about the mass density profile of the lens galaxy, reaching 2% precision from seven lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, this approach could potentially bias the H0 measurement or underestimate the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In this work, for the first time, we break the MSD using spatially resolved kinematics of the lens galaxy in RXJ1131−1231 obtained from the Keck Cosmic Web Imager spectroscopy, in combination with previously published time delay and lens models derived from Hubble Space Telescope imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This approach allows us to robustly estimate H0, effectively implementing a maximally flexible mass model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Following a blind analysis, we estimate the angular diameter distance to the lens galaxy Dd = 865+85 −81 Mpc and the time-delay distance D∆t = 2180+472 −271 Mpc, giving H0 = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1 – for a flat Λ cold dark matter cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The error budget accounts for all uncertainties, including the MSD inherent to the lens mass profile and the line-of-sight effects, and those related to the mass–anisotropy degeneracy and projection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Our new measurement is in excellent agreement with those obtained in the past using standard simply parametrized mass profiles for this single system (H0 = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 km s−1 Mpc−1) and for seven lenses (H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 km s−1 Mpc−1), or for seven lenses using single-aperture kinematics and the same maximally flexible models used by us (H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 km s−1 Mpc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This agreement corroborates the methodology of time-delay cosmography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' cosmology: distance scale – gravitational lensing: strong – Galaxy: kinematics and dynamics – Galaxies: elliptical and lenticular, cD – Galaxies: individual: RXJ1131−1231 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Introduction The Hubble constant, H0, the current value of the Universe’s ex- pansion rate, is a crucial cosmological parameter that also sets the extragalactic distance scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Recently, tension has emerged between early- and late-Universe estimates of H0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Freed- man 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The temperature and polarisa- tion fluctuations in the cosmic microwave background (CMB) provide an estimate of the Hubble parameter at the last scatter- ing surface H(z ≈ 1100), which can be extrapolated to the cur- rent epoch using the Λ cold dark matter (ΛCDM) cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The CMB measurements from Planck give H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 km s−1 Mpc−1 (Planck Collaboration 2020) and H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1(Aiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the local Universe, H0 can be estimated using the cosmic distance ladder, which uses luminos- ity distances of type Ia supernovae (SNe Ia) with their absolute brightness calibrated using different classes of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Super- nova H0 for the Equation of State of the dark energy (SH0ES) team uses Cepheids and parallax distances for this calibration, and they find H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='04 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='04 km s−1 Mpc−1 (Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This value is in 5σ tension with the Planck CMB-based measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' If this difference is not due to systematic errors in either of these measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Efstathiou 2021), then this tension could point to new physics beyond the ΛCDM cosmo- logical model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Knox & Millea 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ⋆ NFHP Einstein Fellow To determine whether this “Hubble tension” is due to sys- tematics or new physics, multiple independent methods to mea- sure H0 are needed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Verde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Di Valentino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Freedman 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Carnegie–Chicago Hubble Project uses the tip of the red giant branch (TRGB) to calibrate the SNe Ia absolute brightness and measures H0 = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='9 km s−1 Mpc−1 (Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This TRGB-calibrated measurement is statistically consistent with both the SH0ES measurement and the CMB-based measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, sev- eral independent local probes strengthen the “H0 tension” by measuring values consistent with the SH0ES value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For exam- ple, the Megamaser Cosmology Project (MCP) estimates H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 km s−1 Mpc−1(Pesce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020), the surface bright- ness fluctuation (SBF) method measures H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 km s−1 Mpc−1 (Blakeslee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021), and the Tully–Fisher- relation-based method calibrated with Cepheids measures H0 = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 km s−1 Mpc−1 (Kourkchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Strong-lensing time delays provide an independent measure- ment of H0 (Refsdal 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' for an up-to-date review, see Bir- rer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022, for a historical perspective, see Treu & Marshall 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In strong lensing, a background source appears as multiple images due to the gravitational de- flection of photons by a massive foreground galaxy or galaxy cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The photons that were emitted at the same time from the background source arrive in different images with a rela- tive time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This time delay carries cosmological informa- Article number, page 1 of 21 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='02656v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='CO] 6 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms tion through a combination of angular diameter distances in- volved in the lensing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This combination is referred to as the “time-delay distance", which is inversely proportional to H0 (Refsdal 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Time-Delay COSMOgraphy (TDCOSMO) collaboration has an- alyzed seven time-delay lenses to measure H0 with 2% error, H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 km s−1 Mpc−1 assuming a power-law or compos- ite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', stars and Navarro–Frenk–White (NFW) halo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1996, 1997) mass profile for the lensing galaxies (Mil- lon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The TDCOSMO collaboration encompasses the COSmological MOnitoring of GRAvItational Lenses (COS- MOGRAIL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Courbin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020a), the H0 Lenses in COSMOGRAIL’s Wellspring (H0LiCOW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2010, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Bonvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020), the Strong-lensing High Angular Reso- lution Programme (SHARP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019), and the STRong- lensing Insights into the Dark Energy Survey (STRIDES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020) collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The simple parametric lens models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', the power law, adopted in the TDCOSMO analyses are “industry standard” con- sistent with non-lensing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The TDCOSMO collab- oration has performed various systematic checks on the adopted lens modeling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These checks find potential system- atic biases to be lower than the acceptable limit (∼1%) from the choice of mass model parametrization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', power law or composite, Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020b), from ignoring dark substruc- tures in the lens galaxy’s halo (Gilman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020), from ig- noring disky or boxy-ness in the baryonic distribution (Van de Vyvere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022a), from using different lens modeling soft- ware (Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022a), and from ignoring potential isoden- sity twists and ellipticity gradients in the lens galaxy (Van de Vyvere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, a significant source of potential systematics could arise due to the relatively simple parametriza- tion of the lens mass profile (Kochanek 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The well-known mass-sheet degeneracy (MSD) does not allow one to constrain the mass profile shape of the deflector galaxy from lens imaging observables alone (Falco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Schneider & Sluse 2013, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Non-lensing observables, such as the deflector galaxy’s velocity dispersion or the source’s unlensed intrinsic brightness, are required to break the mass-sheet degeneracy and simultane- ously constrain H0 and the mass profile shape (Treu & Koop- mans 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The TDCOSMO collaboration has redesigned the experi- ment to mitigate this systematic by relaxing the simple paramet- ric assumptions in the mass profile and constraining the profile shape solely from stellar velocity dispersion measurements of the lensing galaxies (Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Relaxing the assump- tion on the mass profile leads to an increase in the H0 uncer- tainty from 2 to 8% – which is dominated by the uncertainty of the measured velocity dispersion – giving H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' One approach to improving the precision is to incorpo- rate prior information on the mass profile shape from the mea- sured velocity dispersions of a larger sample of external lenses without measured time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Assuming that the Sloan Lens ACS (SLACS) survey’s lens galaxies are drawn from the same population as the TDCOSMO lens galaxies and using their ve- locity dispersions to constrain the mass profile shape, the un- certainty on H0 improves to 5%, giving H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 km s−1 Mpc−1(Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Note that this estimate is statistically consistent within 1σ with the larger 8% H0 measurement above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, the shift could also arise from systematic differences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', a difference between the parent populations of time-delay and non-time-delay lenses (Gomer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Such differ- ences could arise, for example, from evolutionary effects, as the SLACS sample is at lower redshift than the TDCOSMO lenses (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2015, for a discussion of the evolu- tion of mass density profiles of massive elliptical galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Spatially resolved velocity dispersion measurements of lens galaxies for systems with measured time delays are critical to drastically improving the H0 precision, given the limited sam- ple size of time-delay lenses (Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The spatially resolved nature of the measured veloc- ity dispersion is especially powerful in simultaneously break- ing the MSD and the mass-anisotropy degeneracy (Cappellari 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Barnabè et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2009, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Collett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Spatially resolved velocity dispersion measurements for ∼40 time-delay lens galaxies will yield an independent ≲2% H0 measurement without any mass profile assumption (Birrer & Treu 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Additional constraints from velocity dispersion measurements of non-time-delay lens galaxies or magnification information for standardizable lensed type Ia supernovae can further improve the uncertainty to ≲ 1% (Birrer & Treu 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In this paper, we measure the spatially resolved velocity dis- persion for the lens galaxy in the strongly lensed quasar system RXJ1131−1231using the Keck Cosmic Web Imager (KCWI) in- tegral field spectrograph on the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Keck Observatory (Mor- rissey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2012, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' and constrain H0 without any mass profile assumption from this single time-delay lens system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This is the first application of spatially resolved velocity dispersion from a time-delay lens to measure H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This lens system was pre- viously used to measure H0 by combining the observed imaging data, single-aperture velocity dispersion, time delays, and anal- ysis of the line-of-sight environment (Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, these previous studies assumed simple parametriza- tions for the mass profile, such as a power law or a combination of the NFW profile and the stellar profile with constant mass- to-light, which is the industry standard in modeling of galaxy- scale lenses (Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2016) marginal- ized over the MSD effect for the system RXJ1131−1231 to con- strain H0 using a single-aperture velocity dispersion measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Here, we allow the maximal freedom in the MSD by in- troducing one free parameter on top of the simply parametrized mass profile constrained by lens modeling, which is completely degenerate with H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In Section 2, we describe the observational strategy and data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In Section 3, we describe the procedures to extract the spatially resolved kine- matics map from the KCWI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In Section 4, we briefly review the lensing and dynamical formalisms and how we combine the two to mitigate the MSD in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Then in Section 5, we describe our dynamical models and present results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We infer the cosmological parameters from our analysis in the Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We discuss our results in Section 7 and conclude the paper in Sec- tion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We performed the cosmological inference blindly in this pa- per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The measurement of velocity dispersion was not blinded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, we blinded the cosmological and other model parame- ters directly related to cosmological parameters in the dynamical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Before unblinding, this analysis went through an in- ternal collaboration-wide review and code review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' After all the coauthors had agreed that the necessary systematic checks were satisfactorily performed, we froze the analysis and unblinded on 5 January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' All the sections in this paper except for the final discussion in Section 7 and summary in Section 8 were written before unblinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' After unblinding, we only made minor edits Article number, page 2 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 N E 1" A B C D G S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' HST/ACS image of RXJ1131−1231 in the F814W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The four quasar images are labeled with A, B, C, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The central deflec- tor is marked with G, of which we are measuring the spatially resolved velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' An arrow points to the nearby satellite S, which we mask out in the velocity dispersion measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='. The North and East directions and 1¨scale are also illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' for clarity and grammatical corrections in the previous sections and added the unblinded numbers where relevant in the abstract, main text, and plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Observations and data reduction In this section, we provide a brief description of the lens sys- tem RXJ1131−1231 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1), the spectroscopic observation with KCWI (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2), and the data reduction procedure (Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Description of lens system The quadruply imaged quasar lens system RXJ1131−1231 was discovered by Sluse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The deflector in this system is an elliptical galaxy with redshift zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='295, and the source redshift is zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='657 (Sluse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Due to its low red- shifts, the system is relatively bright and large in angular size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Einstein ring in this system contains intricate features, pro- viding a wealth of information to constrain the lens mass model (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Due to its early discovery and information-rich features, this system is one of the most studied lensed quasar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The time delays for this system were measured by Tewes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013, 2014) performed cosmo- graphic analyses of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These authors combined simply parametrized lens models based on the high-resolution imaging from the HST’s Advanced Camera for Surveys (ACS) instru- ment (HST-GO 9744;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' PI: Kochanek), the measured time delays, single-aperture velocity dispersion, and external convergence es- timate to infer H0 = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, such sim- ply parametrized lens models implicitly break the MSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2016) performed an independent mass modeling of this system while marginalizing the MSD with a prior on the source size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These authors found the prior choice on the anisotropy in the dynamical modeling to be the dominant systematic in infer- ring H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' KCWI Spectroscopy We obtained integral field unit (IFU) spectroscopy of RXJ1131−1231 on 16 May and 7 June 2021 with the KCWI in- strument on the Keck Observatory (Morrissey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2012, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We chose KCWI with the small IFU slicer and the low-resolution blue grating (BL) with a field-of-view (FoV) of 8′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 × 20′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The spectral resolution is R ≈ 3600, corresponding to an instrumental dispersion σinst ∼ 35 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The reciprocal dispersion is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 Å per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The observed wavelength range 3600–5600 Å covers the Ca H&K lines with wavelengths λλ3933, 3968 Å at the red- shift of the lens galaxy (zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='295).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We primarily use these lines to determine the stellar velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The redshifted 4304 Å G-band is beyond the observed range, so it is not accessible with the KCWI for the RXJ1131−1231 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We aligned the FoV’s longer side with the North direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', PA = 0◦) and dithered the individual exposures by 9′′ along the North-South direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' As the extent of the RXJ1131−1231 system is smaller than the FoV, each exposure contained the en- tire lens system within the FoV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In different exposures, the lens system occupied the upper or lower portion of the FoV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, the sky in an exposure with the system occupying the upper por- tion can be subtracted using another exposure with the system occupying the lower portion, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We obtained six ex- posures with a total integration time of 10,560 s on 16 May and three with a total integration time of 5,400 s on 7 June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' There- fore, the total exposure time is texp = 15, 960 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The airmass ranged from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='48 over the integrating period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Data Reduction We use the official Python-based data reduction pipeline1 (DRP) to reduce our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The DRP converts the 2D raw data captured on the detector into a 3D datacube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' It performs geometry cor- rection, differential atmospheric refraction correction, and wave- length calibration and produces a final standard-star-calibrated 3D datacube for each exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The calibration with the stan- dard star corrects for instrumental response and scales the data to flux units (Morrissey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We use the final output file with the suffix “_icubes” for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We stack the dithered datacubes through drizzling (Fruchter & Hook 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Since the exposures are obtained on different dates, the world coordinate system information is not accurate enough to determine the relative positions of the dithered expo- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We follow Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2021b) to determine the relative po- sitions by simultaneously fitting the point spread function (PSF) to the four quasar image positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' To perform the drizzling on the datacubes, we repurpose the drizzling routine of the DRP for OSIRIS, another IFU spectrograph on the Keck Observatory2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For the drizzling process, we set pixfrac = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 as recom- mended to reduce correlated uncertainties between the drizzled pixels (Avila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We calculate the drizzled weight image and ensure that the ratio of RMS/median < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 in the region of interest so that the trade-off is balanced between improving the 1 developed by Luca Rizzi, Don Neill, Max Brodheim;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' https:// kcwi-drp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='io/ 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='com/Keck-DataReductionPipelines/ OsirisDRP Article number, page 3 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms image resolution and increasing the background noise (Gonzaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The rectangular pixel size 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1457′′ × 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3395 of the KCWI is kept the same in the drizzled output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We transform the datacube to have square pixels of size 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1457 × 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1457 through resampling while conserving the total flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We converted the pix- els into square sizes for the convenience of Voronoi binning the spectra using the software vorbin as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We directly estimate the PSF from the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We produce a 2D image from the datacube by summing along the wavelength axis (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We create a model for this KCWI image using a high-resolution template from the HST imaging (Figure 1) that has a pixel size 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 and PSF full width at half maximum (FWHM) 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the model, the template is con- volved with a Gaussian PSF with a free FWHM parameter, and the positioning of the template on the KCWI image grid is fitted with two additional free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' By optimizing the model, we estimate that the PSF FWHM is 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Kinematics maps This section describes our procedure to obtain the final kine- matics map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We use the pPXF package3 to fit the spectra with a library of stellar templates and extract the velocity dispersion (Cappellari 2017, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1, we describe the stel- lar templates used for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2, we present the measurement of the spatially-resolved kinematics map of the lens galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3, we test the systematics of the velocity dispersion measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Library of Stellar Templates The popularly used template libraries Medium-resolution Isaac Newton Telescope library of empirical spectra (MILES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Sánchez-Blázquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2006) and INDO-US templates (Valdes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2004) are both too low resolution to fit our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The KCWI’s instrumental resolution of R ≈ 3600 leads to σinst ∼ 35 km s−1 for a Gaussian line spread function (LSF)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' MILES has a resolution of σtemplate ∼ 64 km s−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', R ∼ 2000), and the INDO-US templates have an approximately constant- wavelength resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 Å, which corresponds to σtemplate = 39 km s−1 over the Ca H&K wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Therefore, we choose the X-shooter Spectral Library (XSL), which contains 628 stars covering three segments, including UVB, Vis, and NIR bands (Gonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' As our data cover the rest-frame blue/UV range, we only use the UVB segment to fit the data, where its resolution is R ∼ 9700 and σtemplate ∼ 13 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Measuring the velocity dispersion We choose a cutout centred on the lens system with 6′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='235 × 6′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='235 (43 pixels × 43 pixels) to initiate the analysis (see Fig- ure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We estimate the lens galaxy light’s signal-to-noise ra- tio (S/N) in each spatial pixel (hereafter, spaxel) within this ini- tial cutout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We then select a region with sufficient S/N from the lens galaxy and relatively low quasar contamination for measur- ing the velocity dispersion (the yellow contour in Figure 2’s left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We perform Voronoi binning within this selected region 3 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='org/project/ppxf/ 4 We quantitatively verified that the shape of the instrumental LSF is Gaussian (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Figure 28 of Morrissey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, the treatment of the instrumental LSF in pPXF is self-consistent and avoids any system- atic bias due to inconsistent definitions of the LSF’s FWHM (Robertson 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' to preserve the maximal spatial resolution and reduce the bias in the lower-S/N region (Cappellari & Copin 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We elaborate on these steps below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' To estimate the lens galaxy’s S/N in each spaxel, we first si- multaneously fit the quasar and the lens galaxy in each spaxel to calculate the signal from each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We perform this fitting within the wavelength range 3400–4300 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' As the four quasar images surround the lens galaxy, each spaxel receives a differ- ent contribution from the quasar light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take spectra at the central spaxel of image A as the quasar template, ignoring the lens galaxy’s small contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Later in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3, we also choose the quasar template from images B and C to account for the associated systematic uncertainty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', the potential impact of chromatic microlensing that may change the contrast between the line and the continuum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Sluse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We determine a single optimal template spectrum for the lens galaxy template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For this purpose, we binned the spectra from spaxels within a circular region of radius 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 centered on the lens galaxy and fit it with pPXF using the 628 stellar templates from the XSL and the quasar template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We also include a Leg- endre polynomial of degree 3 as a component in the fitting to ac- count for any residual gradient in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' pPXF chooses 39 of the stellar templates and builds the optimal template by taking a weighted linear combination of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' See Figure 3 for the weighted distribution of spectral types of the full template li- brary and that of the 39 stars selected by pPXF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Among those stars in the XSL with stellar classes specified by the Simbad database (Wenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2000), G-type stars are selected with the highest total weight, consistent with the fact that massive elliptical galaxy spectra are dominated by G and K-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the pPXF fitting procedure, the stellar templates are broad- ened, corresponding to a freely varying velocity dispersion, but the velocity dispersion does not broaden the quasar template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Once the optimal galaxy template is constructed, we use this template and the quasar template to fit the spectrum of each spaxel individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We use this optimal template to fit the galaxy spectra in individual spaxels instead of the full template library to avoid large spurious fluctuations in the measured velocity dis- persion from spaxel to spaxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We show the decomposition of the spectra from one example spaxel into different components after fitting with pPXF in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We calculate the signal of the lens galaxy’s spectrum in each spaxel by subtracting the mod- eled quasar component from the observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The noise is estimated by adding in quadrature the Poisson noise of the total signal and the background noise estimated from an empty patch of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The noise values are multiplied by √ 2 to account for the fact that the square pixels are created from the rectan- gular pixels about double the size through resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We esti- mate the S/N using the restframe wavelength range 3985–4085 Å, slightly above the Ca H&K absorption lines in wavelength (see the purple shaded region in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' To perform Voronoi binning before the velocity dispersion measurement, we select the spaxels within a radius of 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='55 from the lens galaxy center that avoid the brightest spaxels contain- ing images A, B, and C and the lensed arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We also exclude a circular region around image D with radius 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' To avoid any po- tential bias due to contamination from the satellite galaxy S, we exclude the spaxel at its position (∆RA = 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='09, ∆Dec = 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='54 from the galaxy center, Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We also exclude pixels with S/N < 1 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the end, the spaxels within the selected re- gion have S/N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 Å−1 (see Figure 2 for the selected region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5 For reference, 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 corresponds to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 kpc at zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='295 for a fiducial flat ΛCDM cosmology with H0 = 70 km s−1 Mpc−1 and Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Article number, page 4 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 N E 1" KCWI data of RXJ1131 1231 3400 3600 3800 4000 4200 Restframe wavelength (�) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 Data Best fit model (galaxy + quasar) Best fit quasar model Data quasar Residual (= data model) 0 200 400 600 800 1000 1200 Flux (arbitrary unit) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Left: 2D representation (median-collapsed) of the 3D KCWI datacube for RXJ1131−1231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The yellow contour traces the region with 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 radial extent from the center selected for stellar kinematic measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A circular region with 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 radius around image D and the spaxel containing the satellite S are excluded from this selected region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' All the individual spaxels within this region have continuum S/N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 Å−1 for the lens galaxy’s light within 3985–4085 Å (the purple shaded range in the right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Right: The spectra (grey) from an example pixel (grey box in the left panel) and the estimate of the signal from the lens galaxy’s spectra (orange) after removing the contribution from the quasar light (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The full model of the spectra is presented with the red line, and the model’s residual is plotted in emerald color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The vertical purple shaded region marks where we compute the continuum S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' O B A F G K M L S C ~ Stellar type 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 Normalized density All Set 1 Set 2 Set 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Distribution of the stellar spectral types in the XSL according to the Simbad database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Unspecified stars are grouped in the ‘∼’ class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The dark grey color represents the full library of 628 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Set 1 (or- ange) refers to the 39 stars selected by pPXF out of the full library to construct an optimal template 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Set 2 (blue) refers to 32 stars selected from a random half of the full library and set 3 (emerald) refers to 33 stars selected from the other half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Sets 2 and 3 have 15 and 17 stars, re- spectively, in common with Set 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The alternating light grey and white vertical regions divide the spectral classes for easier visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We perform Voronoi binning using vorbin6 given the estimated S/N values for each spaxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In Figure 4, we show the 41 Voronoi bins obtained by setting the target S/N ≈ 23 Å−1 for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This target S/N was chosen so that the resultant S/N ≳ 20 Å−1 for each bin, which is standard practice (Figure 4, only bin 16 has S/N ≈ 18 Å−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For each Voronoi bin, we measure the velocity dispersion by fitting the binned spectra using pPXF using the optimal galaxy template described above, the quasar template, and the additive Legendre polynomial to model any slight gradient in the popula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A few examples of pPXF fit of the binned spectra are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 6 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='org/project/vorbin/ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Estimation of systematic uncertainty To estimate the systematic uncertainties in the velocity disper- sion measurement, we consider a range of plausible choices in the extraction procedure: the degrees of the additive Legendre polynomial used to correct the template continuum shape be- tween 2 to 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' the quasar template obtained from images A, B, and C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' the fitted wavelength range chosen from 3300–4200 Å, 3350– 4250 Å, and 3400–4300 Å;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' and three sets of template spectra used in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The first set of template spectra contains the complete XSL of 628 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The second set contains half of the entire sample that is randomly selected, and the third set con- tains the other half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The numbers of stars selected by pPXF in the three sets are 39, 32, and 33, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Sets 2 and 3 have 15 and 17 stars, respectively, in common with Set 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Figure 3 shows the distribution of spectral types in all three sets and the entire library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We do not take the quasar template from image D as it is much fainter than the other images, and thus the galaxy contribution in the brightest spaxel on image D is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Taking a combination of all of these choices yields 81 different setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We illustrate the shift in the extracted velocity dispersion maps for one change of setting at a time in Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We estimate the variance-covariance matrix of the binned ve- locity dispersions from these 81 setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' To do this, we generate 1,000 random realizations of the measured velocity dispersion map for each of the 81 setups using the corresponding statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We create the variance-covariance matrix from the 81,000 realizations combined from all the setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In this way, the diagonal terms of the variance-covariance matrix encode the total variance from systematic and statistical uncertainties, and the off-diagonal terms encode the systematic covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For example, if all 81 setups hypothetically provided the same velocity dispersion map and uncertainty, then the off-diagonal terms would be zero, and the diagonal terms would reflect only the statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We show the systematic variance- covariance relative to the statistical variance in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The systematic variance is subdominant relative to the statistical vari- ance (with a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='47 of the ratio between systematic and statistical covariances along the diagonal) except for bins 29 and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These two bins are closest to quasar images A and C, and Article number, page 5 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 Map of Voronoi bins 0 5 10 15 20 25 30 35 40 Bin number 16 18 20 22 24 26 28 30 S/N (� 1) Voronoi bins Target S/N Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Left: Voronoi binning of the selected spaxels within 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 from the galaxy center that avoid lensed arcs, quasar images, and the satellite galaxy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The different colors illustrate the regions for each Voronoi bin in a cartographic manner for easier visualization, with the bin number specified within each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We perform the binning with a target S/N ≈ 23 Å−1 for each bin, which results in 41 bins in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Right: Resultant S/N for each Voronoi bin (red points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 3400 3600 3800 4000 4200 Restframe wavelength (˚A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='06 Flux (arbitrary unit) bin 1: σlos = 282 ± 7 km s−1 3400 3600 3800 4000 4200 Restframe wavelength (˚A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='150 Flux (arbitrary unit) bin 6: σlos = 284 ± 11 km s−1 3400 3600 3800 4000 4200 Restframe wavelength (˚A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 Flux (arbitrary unit) bin 20: σlos = 274 ± 13 km s−1 3400 3600 3800 4000 4200 Restframe wavelength (˚A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 Flux (arbitrary unit) bin 37: σlos = 263 ± 16 km s−1 Data Best fit model (galaxy + quasar) Best fit quasar model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' pPXF fitting to the spectra from four examples of Voronoi bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The bin number and the measured velocity dispersion for the corresponding bin are specified in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The grey line presents the full spectra, the red line traces the best-fit model, and the blue line shows the quasar component in the best-fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' thus largely susceptible to the choice of quasar template (see Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=') We show the velocity dispersion and mean velocity maps av- eraged over the 81 setups in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We estimate a systematic velocity of 182 km s−1 using the pafit7 software program (Kra- jnovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2006) and subtract it from the mean velocity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The systematic velocity is the result of a slight deviation in the true redshift from the fiducial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The mean velocity map does 7 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='org/project/pafit/ Article number, page 6 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 not show any significant evidence of ordered rotation above the systematic and statistical noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus it is consistent with the lens galaxy being a slow rotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We use this systematic- averaged velocity dispersion map and the variance-covariance matrix estimated above when computing the likelihood function for dynamical modeling in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' To test the impact of our choice for the Voronoi binning scheme, we adopt an alternative target S/N ≈ 28 Å−1 for each bin, which results in 27 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We similarly produce another set of 81 model setups in this binning scheme and produce the variance-covariance matrix for these binned velocity disper- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We test the systematic impact of this different binning scheme on the cosmological measurement later in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We show the difference in the extracted kinematics between the two binning schemes in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Overview of lens and dynamical modeling This section reviews the theoretical formalism of lens and dy- namical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Lensing observables and modeling We briefly review the strong lensing formalism in the context of time-delay cosmography in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1, describe the mass- sheet transform (MST) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2, and explain the internal and external components of the MST in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Strong lensing formalism In the thin lens approximation applicable in this case, lensing observables are described using the surface mass density Σ(R) projected from the 3D mass density distribution ρ(r) in the lens galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Formally, the lensing observables depend on the dimen- sionless convergence defined as κ(θ) ≡ Σ(θ) Σcr , (1) which is the surface mass density normalized by the critical den- sity Σcr ≡ c2Ds 4πGDdDds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2) Here, c is the speed of light, G is the gravitational constant, Ds is the angular diameter distance between the observer and the source, Dd is the angular diameter distance between the observer and the lens galaxy, and Dds is the angular diameter distance between the lens galaxy and the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The on-sky deflection angle α(θ) relates to the convergence as κ(θ) = 1 2∇ · α(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (3) The time delay between two quasar images labeled A and B is given by ∆tAB = D∆t c �(θA − ς)2 2 − (θB − ς)2 2 − ψ(θA) + ψ(θB) � , (4) where θA is the angular position of image A, ς is the source’s angular position, ψ(θ) is the lensing potential, and the time-delay distance D∆t is defined as D∆t ≡ (1 + zd)DdDs Dds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Description of the MST The MST is a mathematical transform of the convergence pro- file that leaves invariant all the imaging observables, such as the image positions and the flux ratios (Falco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Schneider & Sluse 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This transform scales the convergence and the unknown source position as κ → κ′ = λMSTκ + (1 − λMST), ς → ς′ = λMSTς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (6) where λMST is the transformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The predicted time delay ∆t scales under the transform as ∆t → ∆t′ = λMST∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (7) Then, the inferred time-delay distance D∆t and the Hubble con- stant H0 based on the observed time delays will change as D′ ∆t = D∆t λMST , H′ 0 = λMSTH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (8) However, the MST changes the predicted velocity dispersion, thus measuring it breaks the MSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Notably, the MST also rescales the lensing magnifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, standardizable candles can also be used to break the MSD (Bertin & Lombardi 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022a) provided that microlensing and millilensing can be mitigated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Yahalomi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Foxley-Marrable et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Internal and external MST We can express the “true” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', physically present) lensing mass distribution as κtrue = κgal + κext, (9) where κgal is the mass distribution of the central lens galaxy (or galaxies) that is (are) considered in the lens modeling, and κext is called the external convergence, which approximates the projected mass distribution of line-of-sight structures as a mass sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Since limθ→∞ κgal = 0 has to be satisfied, we find that limθ→∞ κtrue = κext, hence the interpretation of κext as the lensing mass far from (or, “external” to) the central deflector(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' All the lensing observables including imaging observables result from κtrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, since only the central galaxies are usually considered in lens modeling with imaging observables, the lens model provides κ′ model with limθ→∞ κ′ model = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This κ′ model is an MST of κtrue for λMST = 1/(1 − κext) as κ′ model = κgal + κext 1 − κext + 1 − 1 1 + κext = κgal 1 − κext .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (10) Lens mass models are usually described with simply parametrized models, such as the power law or a combi- nation of the NFW profile and the observed stellar distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In that case, the assumption of a simple parametric form implicitly breaks the MSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Therefore, the simply parametrized model κmodel can be expressed as another approximate MST of the κ′ model as κ′ model ≈ λintκmodel + (1 − λint)κs(θ), (11) where λint is called the internal MST parameter, and κs is a “vari- able” mass sheet with limθ→∞ κs(θ) = 0 to ensure that both Article number, page 7 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms Range: 3350 4250 � Range: 3300 4200 � Polynomial degree 2 Polynomial degree 4 Stellar template set 2 Stellar template set 3 Quasar B Quasar C 20 0 20 20 0 20 20 0 20 20 0 20 20 0 20 20 0 20 20 0 20 20 0 20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Absolute difference in km s−1 between the extracted velocity dispersion from two setups that differ by one setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The baseline setup has the range: 3400–4300 Å, polynomial degree: 3, stellar template set 1, and quasar template from image A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The different setting for each case is specified at the top of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Range: 3350 4250 � Range: 3300 4200 � Polynomial degree 2 Polynomial degree 4 Stellar template set 2 Stellar template set 3 Quasar B Quasar C 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Same as Figure 6, but the difference is normalized by the statistical uncertainty of the baseline setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' limθ→∞ κ′ model = 0 and limθ→∞ κmodel = 0 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' How- ever, for Equation (11) to be an approximate MST, the variable mass-sheet needs to satisfy κs(θ) ≃ 1 within the central region that lensing observables are sensitive to (θ ≲ 2θE, Schneider & Sluse 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This can be achieved with the formulation (Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020) κs(θ) = θ2 s θ2 + θ2s , (12) where θs ≫ θE is a scale radius where the variable mass-sheet smoothly transitions from 1 − λint to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This approximate MST converges to the pure MST in the limit θs → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, the actual mass distribution of the central deflector(s) relates to the mod- eled mass distribution as κgal ≈ (1 − κext) [λintκmodel + (1 − λint)κs(θ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (13) The external convergence κext can be estimated by using rel- ative number counts of line-of-sight galaxies near the central de- Article number, page 8 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 10 20 30 40 Bin number 5 10 15 20 25 30 35 40 Bin number Fractional systematic covariance: xy h diag( 2 stat) i xy 2 stat,x 10 0 0 10 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Illustration of the systematic covariance relative to the statistical covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Σ is the variance-covariance matrix of the Voronoi-binned velocity dispersions (with target S/N ≈ 23 Å−1 for each bin), σstat,x is the statistical uncertainty in bin number x from our fiducial setup, and diag(σstat) is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Note that we assume no covariance in the statistical uncertainty from each setup for kinematic measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus the off-diagonal terms in the variance-covariance matrix purely represent the systematic covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Most diagonal terms are < 1 (with a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='47), showing that the systematic variances are subdom- inant to the statistical variances except for bins 29 and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These bins are close to images A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, they are largely susceptible to the choice of the quasar template, as seen in Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Velocity dispersion Velocity dispersion uncertainty Mean velocity Mean velocity uncertainty 225 250 275 300 los (km s 1) 20 40 los (km s 1) 50 0 50 vmean (km s 1) 20 30 vmean (km s 1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Maps of extracted velocity dispersion (top row) and mean ve- locity (bottom row) in Voronoi bins along with the corresponding un- certainties (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Voronoi binning was tuned to achieve S/N ≈ 23 Å−1 for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The illustrated maps (left column) corre- spond to the average values after combining 81 model setups, and the uncertainty maps correspond to the square root of the diagonal of the variance-covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A systematic velocity of 182 km s−1 was subtracted from the mean velocity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' los (km s 1) los/ los 20 0 20 2 0 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Absolute (left) and uncertainty-normalized (right) difference in the extracted velocity dispersion between two Voronoi binning schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The two binning schemes are obtained by setting the target S/N to 23 Å−1 and 28 Å−1 for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take the case with target S/N ≈ 23 Å−1 for each bin as the baseline in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' flector(s) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Buckley-Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020), or by using weak lensing of distant galaxies by the line-of-sight mass distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Ti- hhonova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The measured velocity dispersion then constrains the internal MST parameter λint (Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Dynamical modeling In this section, we describe the Jeans anisotropic multi- Gaussian-expansion (JAM) framework to model our dynamical observable, which is the spatially resolved stellar velocity dis- persion measured in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The orbital motions of the stars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', the distribution function f(x, v) of position x and velocity v, in the galactic potential Φ is described by the steady-state col- lisionless Boltzmann equation (Binney & Tremaine 1987, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4-13b) 3 � i=1 � vi ∂f ∂xi − ∂Φ ∂xi ∂f ∂vi � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (14) We assume an axisymmetric case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', ∂Φ/∂φ = ∂f/∂φ = 0 with φ being the polar angle in the spherical coordinate sys- tem), a spherically aligned velocity ellipsoid, and the anisotropy for each Gaussian component in the multi-Gaussian expan- sion (MGE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Cappellari 2002) to be spa- tially constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Slow rotators such as the deflector galaxy in RXJ1131−1231 are in general expected to be weakly triaxial or oblate but never flat and instead quite close to spherical in their central parts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Cappellari 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For this reason, we expect the spherical alignment of the velocity ellipsoid of jamsph (Cap- pellari 2020) to provide a better approximation to the galaxy dy- namics than the cylindrical alignment jamcyl solution (Cappellari 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Then, the above equation gives two Jeans equations in spherical coordinates (Jeans 1922;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' de Zeeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Cappellari 2020) ∂ � ζ⟨v2 r⟩ � ∂r + (1 + β)ζ⟨v2 r⟩ − ζ⟨v2 φ⟩ r = −ζ ∂Φ ∂r , (1 − β) ∂ � ζ⟨v2 r⟩ � ∂θ + (1 − β)ζ⟨v2 r⟩ − ζ⟨v2 φ⟩ tan θ = −ζ ∂Φ ∂θ , (15) Article number, page 9 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms where the following notations are used ζ⟨vpvq⟩ ≡ � vpvq fd3v, β ≡ 1 − ⟨v2 θ⟩ ⟨v2r⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (16) Here, β is the anisotropy parameter, and the velocity dispersion ellipsoid is assumed to be spherically aligned, giving ⟨vrvθ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The line-of-sight second moment ⟨v2 los⟩ is the integral given by S ⟨v2 los⟩(x, y) = � ∞ −∞ dz ζ⟨v2 z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (17) where S (x, y) is the surface density of the dynamical tracer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Given that there is no evidence of significant ordered rotation and the only significantly nonzero velocities are likely due to systematic errors (see Figure 9), we assume ⟨vlos⟩ = 0 and define ⟨v2 los⟩ = σ2 los.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The observed line-of-sight velocity dispersion is given a luminosity-weighted integral as � σ2 los � obs = � ⟨v2 los⟩ � obs = � ap dxdy I⟨v2 los⟩ ⊗ PSF � ap dxdy I ⊗ PSF , (18) where the symbol “⊗ PSF” denotes a convolution with the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the equation above, we have chosen the surface brightness profile I(x, y) as a substitute for the surface density S (x, y) of the dynamical tracer since the constant factor between surface brightness and surface number density cancels out in this ex- pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We use the dynamical modeling software jampy8 to compute the observed velocity dispersion by solving the Jeans equation from Equation (15) for a given 3D potential Φ(r) and anisotropy profile β(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Specifically, we use the jam_axi_proj() routine with the keyword align=‘sph’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' See Cappellari (2008, 2020) for a detailed formalism in computing Equation (18) by jampy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Cosmological inference from combining dynamical and lensing observables We parametrize the 3D potential Φ(r) using the lens model pa- rameters ξmass and the internal MST parameter λint to conve- niently use the lens model posterior from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013) as a mass model prior in the dynamical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus from Equa- tion (13), the surface mass density for our dynamical model is given by Σ(θ) = Σcr(1 − κext) [λint κmodel(θ) + (1 − λint)κs(θ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (19) We include D∆t and Dd as free parameters in our model, which give the critical density Σcr as Σcr = c2 4πG D∆t (1 + zd)D2 d = c2 4πG Dmodel ∆t (1 + zd)(1 − κext)λintD2 d , (20) where Dmodel ∆t is the time-delay distance predicted by the lens mass model κmodel(θ) for the time delays observed by Tewes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We approximate the surface mass density Σ(θ) with an MGE (Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Cappellari 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib 2019) using the 8 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='org/project/jampy/ software program mgefit9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' jampy deprojects the MGE compo- nents into an oblate or prolate spheroid with an inclination angle i (Cappellari 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The deprojected 3D mass density provides the 3D potential Φ for the kinematic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We also take the MGE of the surface brightness I(x, y) for deprojection to 3D with the inclination angle i for the kinematic computation by jampy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The combination of lens imaging observables and the stellar kinematics is sensitive to λint(1 − κext)Ds/Dds (Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We apply a prior on κext using the estimated κext distribution from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2014) to help break the de- generacy in distributing the total MSD into external and internal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Bayesian framework According to Bayes’ theorem, the posterior of the model param- eters Ξ = {ξmass, ξlight, Dmodel ∆t , i, κext, λint, Dd, β} as p(Ξ | D) ∝ p(D | Ξ) p(Ξ), (21) where p(D | Ξ) is the likelihood given data D and p(Ξ) is the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In this study, the data D is the measured velocity disper- sions in Voronoi bins (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The observational information from the published time delays, lens models using HST imag- ing, and the line-of-sight effects (Tewes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, 2014) is incorporated by adopting those previous posteri- ors as the prior on our model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The likelihood of the observed velocity dispersion vector σlos ≡ [σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' , σNbin], with Nbin being the number of Voronoi bins, is given by L(σlos | Ξ) ∝ exp � −1 2σT losΣ−1σlos � , (22) where Σ is the variance-covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Specific priors used in this Bayesian framework are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We ob- tain the posterior probability distribution function (PDF) of the model parameters using the Markov-chain Monte Carlo (MCMC) method using the affine-invariant ensemble sampler emcee (Goodman & Weare 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We ensure the MCMC chains’ convergence by running the chains for ≳20 times the autocorrelation length after the chains have stabilized (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Dynamical models We first describe our baseline dynamical model in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 and then perform various checks on systematics in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Baseline dynamical model This subsection describes the baseline settings in our dynamical model, namely the specific parametrization of the mass model (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1), the dynamical tracer profile (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2), the probability of oblate or prolate axisymmetry (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3), the inclination angle (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4), and the choice of anisotropy profile (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Parametrization of the mass model We adopt the power-law mass model as our baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In this model, the mass profile is defined with Einstein radius θE, 9 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='org/project/mgefit/ Article number, page 10 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 logarithmic slope γ, projected axis ratio qm, and position angle ϕmass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The convergence profile κmodel in Equation (19) for the power-law model is given by κpl model(θ1, θ2) = 3 − γ 2 ������������ θE � qmθ2 1 + θ2 2/qm ������������ γ−1 (23) Here, the coordinates (θ1, θ2) are rotated by ϕmass from the (RA, Dec) coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We adopt the lens model posterior from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013) as a prior in our dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For sim- plicity, we set the position angle ϕmass the same as the observed position angle of light ϕlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We use the estimated κext distribu- tion for the power-law model as the prior (see Figure 3 of Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We set θs = 12′′ (≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5θE) in the approximate mass-sheet κs (Equation 12) so that the imaging constraints alone cannot dif- ferentiate the power-law mass profile and its approximate MST from Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We obtain this lower limit by running the jupyter notebook that produces Figure 3 of (Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='10 However, we adjusted the fiducial lens model parameters in the notebook to match with those for RXJ1131−1231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take a uni- form prior for the internal MST parameter λint ∼ U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The upper limit of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='13 is set by the requirement that the trans- formed mass profile under the approximate MST must be mono- tonic so that the MGE can approximate the transformed pro- file sufficiently well (Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Previous studies also found similar or more restrictive upper limits for λint to satisfy the physical requirement of non-negative density (Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The appropriate number of MGE components for the mass or light profile is automatically chosen by jampy with a maxi- mum of 20 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We check that the MGE approximates the input mass or light profile very well (with a maximum 1% deviation at < 10′′ and maximum 10% deviation between 10′′– 50′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These deviations from the density profile have an oscilla- tory pattern due to the MGE approximation’s nature, except near the end of the fitted ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus the deviation in the integrated mass profile often averages out in the line-of-sight integration up to a very large radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We perform the MGE fitting up to 100′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, the large mismatch between the MGE approximation and the original profile occurs largely outside the integration limit ∼ 70′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The chosen number of maximum Gaussian components is not a dominant source of numerical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Setting this maxi- mum number to a very high value, such as 100, shifts the com- puted velocity dispersion by only < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5% within the observed region, which is insignificant compared to the 1% numerical sta- bility targeted by jampy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Dynamical tracer profile We update the light profile fitting for the lens galaxy from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013) using a larger HST image cutout than that therein, which did not contain the full extent of the lens galaxy’s light profile (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The lensed arcs and quasar images are first subtracted from the cutout using the prediction of the best- fit lens model from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We use the software pack- age lenstronomy11 to fit the residual light distribution attributed to the lens galaxy (Birrer & Amara 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 10 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='com/TDCOSMO/hierarchy_analysis_2020_ public/blob/6c293af582c398a5c9de60a51cb0c44432a3c598/ MST_impact/MST_pl_cored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='ipynb 11 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='com/lenstronomy/lenstronomy Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Values of the light model parameters for the double Sérsic model in our fitting of a large cutout and those from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The position angle ϕlight is defined as East of North.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Parameter This analysis Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013) Sérsic profile 1 I0 (e−1 s−1 pixel−1) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 θeff (′′) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='437 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='005 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='49 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='01 ns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='10 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='93 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='03 ql 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='865 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='878 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='004 Sérsic profile 2 I0 (e−1 s−1 pixel−1) 441 ±7 356 ±12 θeff (′′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='300 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='362 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='009 ns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='60 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='59 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='03 ql 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='847 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='849 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='004 ϕlight (◦) 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 Following Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013), we use the double Sérsic model to fit the light profile, which is a superposition of two concentric Sérsic profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Sérsic profile is defined as I(θ1, θ2) = I0 exp ������������� −bn ������������ � qlθ2 1 + θ2 2/ql θeff ������������ 1/ns + bn ������������� , (24) where I0 the amplitude, ql is the axis ratio, θeff is the effective radius, ns is the Sérsic index, and bn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='999n − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='327 is a nor- malizing factor so that θeff becomes the half-light radius (Sérsic 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The coordinates (θ1, θ2) are rotated by ϕlight from the (Ra, Dec) coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We first mask circular regions at the quasar image positions due to slightly saturated pixels producing significant residuals in the subtracted cutout (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We then iteratively mask the other pixels with significant residuals above statistical expec- tations to effectively perform an outlier rejection while preserv- ing the shape of a Gaussian tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For each iteration of this pro- cess, we take a discrepancy threshold, which we decrease from 5σ to 2σ with step size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5σ across these iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We then randomly mask a subset of the pixels with residuals more than the discrepancy level at the given iteration such that the number of remaining pixels with such high residuals is statistically ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The final masked area after the iterations is illustrated in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We tabulate the best-fit light model parameters in Ta- ble 1 and compare them with those from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The circularized half-light radius for our best-fit model is θeff = 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='91, which is slightly larger than the value θeff = 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='85 from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013) based on the same imaging data but from a smaller cutout (illustrated in 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We then take the MGE of the fitted double Sérsic profile as the light distribution I(x, y) in our dy- namical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We propagate the uncertainties and covari- ances from the light profile fitting into the dynamical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' To do that, we sample from the multivariate normal distribution corresponding to all the light model parameters for each call of the likelihood function within the MCMC process and then take the MGE of the light profile given the sampled parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Oblate or prolate shape of the axisymmetry The oblateness, prolateness, or triaxiality of a slow rotator galaxy can, in principle, be constrained from the kinematic mis- alignment angle ∆ϕkin ≡ ���ϕkin − ϕlight ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, we do not Article number, page 11 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms N E 1" Data 1" Reconstructed E N 1" Normalized Residuals E N 4 3 2 1 0 1 log10 flux 4 3 2 1 0 1 log10 flux 3 2 1 0 1 2 3 (data model) / noise Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Fit of the lens galaxy’s surface brightness profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Left: the HST/ACS imaging in the F814W filter of the lens system RXJ1131−1231 with the quasar images and the lensed arcs subtracted using the prediction from the best-fit lens model from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013), thus leaving only the lens galaxy’s light to be fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The orange circle shows the large circular region considered for fitting in our analysis, and the yellow square shows the smaller cutout used for lens modeling by Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The cyan annulus contains the region where pixels were fitted to reconstruct the source by Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus the lensed arcs from the quasar host galaxy were subtracted only within this annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The red contours mark quasar image positions with significant residuals due to saturated pixels, which we mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Middle: The fitted light profile with a double Sérsic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The black pixels correspond to masked pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The additional masked pixels within the orange circle not described above are randomly selected through an iterative process that performs outlier rejection while preserving the Gaussian tail (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Right: Normalized residual of the best-fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' detect any significant rotational pattern in the vmean map (Fig- ure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, the uncertainty for the constrained kinematic major axes is too large to be meaningful, and we cannot directly con- strain this galaxy’s oblateness from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Instead, we obtain the probability of oblateness from a population prior based on 189 slow rotator elliptical galaxies that are in the Sloan Digi- tal Sky Survey’s (SDSS’s) Mapping Nearby Galaxies at APO (MaNGA) sample (Abolfathi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take the distribution of ∆ϕkin for this sample of slow rota- tors (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018), where ∆ϕkin = 0◦ corresponds to a purely oblate shape, and ∆ϕkin = 90◦ corresponds to a purely prolate shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2018) find two distinct peaks in the distribution at ∆ϕkin = 0◦ and ∆ϕkin = 90◦ (see Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We, therefore, fit the data points with a double Gaussian profile with the means set at ∆ϕkin = 0◦ and ∆ϕkin = 90◦ (see the fit in Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Although the slow rotators with 0◦ < ∆ϕkin < 90◦ have triaxial shapes, we choose only oblate or prolate axisymmetric shapes in our dynamical modeling for computational simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' There- fore, we take ∆ϕkin < 45◦ as the oblate case and ∆ϕkin > 45◦ as the prolate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We obtain the prior probability p(oblate)pop of the galaxy being oblate as p(oblate)pop = � 45◦ 0◦ d(∆ϕkin) p(∆ϕkin)pop ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='65, (25) and thus p(prolate)pop = 1 − p(oblate)pop ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The jampy software package, by default, adopts the oblate case for deprojection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We implement the prolate case in jampy by setting qprolate = 1/q > 1 and switching the x and y axes in the input coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Due to the switching of x and y axes, σ parameters of the MGEs for mass and light models need to be scaled as σprolate = qσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Inclination The observed axis ratio of light ql,obs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='850 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='002 relates to ql,int through the inclination angle i as q2 l,obs = q2 l,int sin2 i + cos2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (26) We impose a prior on the intrinsic axis ratio ql,int from a sam- ple of massive elliptical galaxies in the SDSS with stellar mass 0 20 40 60 80 Kinematic misalignment, kin ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 Probability density (�90) oblate prolate Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=" (2018)'s population double Gaussian fit Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Population prior on kinematic misalignment angle ∆ϕkin ≡ ���ϕkin − ϕlight ��� for a sample of slow rotator elliptical galaxies from the SDSS’s MaNGA dataset (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Here, ∆ϕkin = 0◦ corresponds to a purely oblate shape, and ∆ϕkin = 90◦ corresponds to a purely prolate shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The vertical dashed grey lines mark ∆ϕkin = 0◦, 45◦, and 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The red points with error bars show the measurements from Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We fit this distribution with a double Gaussian model (blue line) with the means fixed to ∆ϕkin = 0◦ and ∆ϕkin = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take ∆ϕkin < 45◦ as the oblate case and ∆ϕkin > 45◦ as the prolate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Integrating the double Gaussian model from 0◦ to 45◦ gives the prior probability of oblateness p(oblate)pop ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 < log10(M⋆/M⊙) < 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='04 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='08 (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The distribution of ql,int by Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013) is differ- ent for oblate and prolate assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Therefore, we adopt the specific prior corresponding to the oblate or the prolate case (see Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Anisotropy profile We investigate two choices to parametrize the anisotropy pro- file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The first choice is a single spatially constant β = 1 − σ2 θ/σ2 r value for all the light MGE components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Numerically, we sam- Article number, page 12 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 Intrinsic axis ratio of light, ql,int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 Probablity density oblate case prolate case Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Prior on the intrinsic axis ratio ql,int of light for oblate (solid line) and prolate (dashed line) cases from Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The pri- ors correspond to massive elliptical galaxies from the SDSS survey at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='04 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='08 with 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 < log10(M⋆/M⊙) < 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ple σθ/σr with a uniform prior (σθ/σr) ∼ U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='78, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This range of σθ/σr allows −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='31 < β < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We adopt this range using the β values of eight slow rotator galaxies measured by Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2007, see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These measurements of β by Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2007) are from Schwarzschild modeling of data with one of the highest S/N values in the literature, al- lowing to constrain the Gauss–Hermite moments up to order six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Applying the student’s t-distribution on the sample mean of this small sample, we find the 95% confidence interval of the pop- ulation mean for β to be [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='17] and the standard devia- tion to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These values infer that 95% of the population is contained within β ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='31, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='38], which we take as the bound- aries of our prior range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The second choice of the anisotropy profile has two free parameters: the inner light MGE compo- nents with σ < rbreak = θeff = 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='91 are assigned one value for (σθ/σr)inner and the outer light MGE components with σ ≥ rbreak are assigned another independent value of (σθ/σr)outer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, this parametrization with two free parameters allows radial vari- ability in the anisotropy profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Both the inner and outer ratios have uncorrelated uniform priors (σθ/σr) ∼ U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='78, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For these two choices of parametrization, we compute the Bayesian information criterion (BIC) given by BIC ≡ k log(Nbin) − 2 log ˆL, (27) where k is the number of free model parameters, Nbin is the num- ber of data points, and ˆL is the maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We approx- imate ˆL from the highest likelihood value sampled in the MCMC chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The single-parameter β model provides the lowest BIC value excluding the two-parameter β model with ∆BIC ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', positively excluded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Raftery 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We check that the dif- ference between the highest and the second highest likelihood values among the MCMC samples is ≪ ∆BIC, thus this ∆BIC value is robust against our approximation of ˆL from the high- est likelihood value in the sampled chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The non-detection of varying anisotropy in our data is consistent with that ob- served in nearby elliptical galaxies, as even high-S/N SAURON data for a large sample of galaxies are accurately described by JAM models with constant anisotropy, as used here, within the noise of the kinematics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We com- pare the posterior distributions of the model parameters for the two anisotropy models in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' An example of a best-fit kinematic model and the corresponding residual with the single- parameter β model and oblate axisymmetry is illustrated in Fig- ure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The reduced χ2 ν value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='83 with ν = 41 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The distribution of residuals is similar to a normal dis- tribution expected from a perfect model for data with Gaussian noise, illustrating that our model is appropriate for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We show the range of velocity dispersion radial profiles sampled by our model in Figure 16 and compare it with the radially averaged measurements of the velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This illustration shows that our model reproduces the uncertainty range of the measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Checking potential systematics due to modeling choices In this section, we perform several checks on potential system- atics for different choices in the dynamical model setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison between power-law and composite mass models In addition to the power-law mass model, Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2014) also adopted a composite mass model individually describing the lens galaxy’s dark matter and baryonic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The dark matter distribution was modeled with an elliptical NFW profile in the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The parameters in this profile are the normalization of the NFW component κs, the NFW scale radius rscale, and the mass axis ratio qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The baryonic component was modeled with a mass-follow-light profile with a free mass-to-light ratio (M/L) parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, this mass model parametrization has one more free parameter than the power-law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' See Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2014) for parametric definitions of these profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We implement this composite mass profile as κcomp model in Equation (19) and adopt the model posterior from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2014) as a prior in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We appropriately convert the ellipticity defined in the potential by Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2014) to an ellipticity defined in the convergence in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take the MGE of this composite surface density model as done for the power-law surface density model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' How- ever, since the dark matter and baryonic components have differ- ent ellipticities, we take the MGE of each component separately to preserve the ellipticity information in deprojection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Specifi- cally, We take the MGE of the approximate MST with λint of the dark matter profile and the MGE of an accordingly rescaled baryonic profile, which effectively results in the total mass pro- file being transformed as the approximate MST with λint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This mass model with one more free parameter than the power-law model has a higher BIC score with ∆BIC = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, the BIC excludes the composite model with positive evidence (Raftery 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The median values of Dd from the power-law and composite mass models differ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='9% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='07σ, Figure 17), and the median D∆t values differ by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='26% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='06σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Therefore, we conclude that our power-law mass model with an additional degree of freedom to scale with the MST robustly describes the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison between prolate and oblate axisymmetry We compare the inferred Dd between the purely oblate and purely prolate cases in the deprojected 3D spheroidal shape of the mass and light models (Figure 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The median Dd values from these two cases differ by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3σ), and the median D∆t values differ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='94% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='04σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Our final distance poste- rior is the combination of oblate and prolate cases, with weights p(oblate)pop = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='65 and 1 − p(oblate)pop = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='35, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Article number, page 13 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='64 E ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 blinded Dd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='00 ( / r)outer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='00 / r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 ext 60 80 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='77 qm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='76 qm 60 80 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='23 ext 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 / r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 ( / r)outer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 blinded Dd different parameters for inner and outer MGE components same for all MGE components Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Constraints from axisymmetric JAM modeling on the power-law mass model parameters (θE, γ, and qm), internal MST parameter λint, external convergence κext, anisotropy profile parameter(s), and the cosmological distances D∆t and Dd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' assuming two anisotropy parametrizations: (i) one single constant β ≡ 1 − (σθ/σr)2 for all light MGE components (orange contours), and (ii) one free (σθ/σr)inner ≡ (σθ/σr) for light MGE components with σ < rbreak = θeff = 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='91 and another free (σθ/σr)outer for light MGE components with σ > rbreak(blue contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The blinded parameters are blinded as pblinded ≡ p/⟨p⟩ − 1 so that the distributions only reveal fractional uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The darker and lighter shaded regions in the 2D plots trace 68% and 95% credible regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The mass model parameters Einstein radius θE, power-law slope γ, axis ratio q, and position angle PA are additionally constrained through a prior from the imaging data from Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The two anisotropy parametrizations provide equally good fits to the kinematics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, the BIC selects the constant-β anisotropy model over the other one with one additional free parameter (∆BIC value is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus, this difference between the oblate and prolate cases is marginalized in our final cosmological distance posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We also compare the predictions from axisymmetric and spherical mass models in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The median Dd from the spherical model matches very well with the axisymmetric pro- late model, but the median D∆t differs by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='08σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The galaxy is only mildly elliptical in projection (ql ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='85), and the resulting axisymmetric models are not very flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For this rea- son, the relatively small difference between the axisymmetric and spherical models is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Article number, page 14 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 Data Model Residual 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 (data model)/noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 density Residual distribution 300 250 los (km s 1) 300 250 los (km s 1) 2 0 2 (data model)/noise Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Observed velocity dispersion map in Voronoi bins (first panel), the best-fit dynamical model with a power-law mass model, constant β anisotropy profile, and oblate shape (second panel), the normalized residual for the best-fit dynamical model (third panel), and the distribution of the normalized residual (orange, fourth panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The reduced χ2 quantity is χ2 ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='83 with degrees of freedom ν = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The grey dashed line in the fourth panel shows a normal distribution expected for residuals from a perfect model to the data with Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The residual distribution for 41 points is similar to this Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='50 ( ) 240 260 280 300 Velocity dispersion (km s 1) oblate prolate Radially binned measurements Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Radial profile of the line-of-sight velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The red points are radially binned values from the 2D maps, with the horizontal error bars illustrating the widths of the annuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The lines show the radial profiles for random samples from the dynamical model posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The radial profile of the model is averaged over the major, minor, and inter- mediate axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The solid purple lines correspond to 65 random samples for the oblate case, and the dashed green lines correspond to 35 ran- dom samples for the prolate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Note that the model was fit to the 2D kinematics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, we illustrate the 1D radial profile only for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison between Voronoi binning schemes Here, we compare the Voronoi binning schemes with two choices for the target S/N in each bin: ≈ 23 Å−1 and ≈ 28 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The two cases match very well with only a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='21% differ- ence (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='02σ) in the median values of Dd (Figure 19) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='28 % difference (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='01σ) in the median D∆t values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' As a result, we conclude that our choice of the Voronoi binning scheme is not a significant source of systematic error in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Based on the systematics tests performed above, we adopt a robust final distance posterior from the model with the power- law parametrization for the mass profile that the approximate in- ternal MST is applied to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We marginalize the oblate and prolate axisymmetrical cases by combining the posteriors from these two choices with weights of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='65 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='35, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the next section, we present the unblinded values from the distance posterior and infer the value of H0 from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 / r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 blinded Dd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 blinded Dd Power-law mass model Composite mass model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison of the constrained Dd from power-law (blue con- tours) and composite (orange contours) mass models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The blinded pa- rameters are blinded as pblinded ≡ p/⟨p⟩−1 so that the distributions only reveal fractional uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The darker and lighter shaded regions in the 2D plots trace 68% and 95% credible regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Cosmological inference In this section, we infer cosmological parameters from the joint distribution of Dd and D∆t, accounting for their covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The unblinded point estimates of these distances are Dd = 865+85 −81 Mpc (a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6% measurement) at zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='295, and D∆t = 2180+472 −271 Mpc (a 17% measurement) for zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We infer H0 and Ωm from our distance posterior for a flat ΛCDM cosmology (see Figure 20, left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We leave the ex- ploration of more exotic cosmologies based on our distance pos- terior for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We approximate the likelihood function L(H0, Ωm | Dd, D∆t) of the cosmological parameters using a 2D Gaussian kernel density estimate (KDE) from the 2D distance posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We adopt two choices of prior for Ωm: one is a uni- form prior Ωm ∼ U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5), and the other is a Gaussian prior Ωm ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='334, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='018) from the Pantheon+ analysis of type Ia Article number, page 15 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 / r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 blinded Dd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0 blinded Dd spherical axisymmetric prolate axisymmetric oblate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison of the constrained Dd between oblate (blue) and prolate (blue) cases of the deprojected spheroidal shape in the dynam- ical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The blinded parameters are blinded as pblinded ≡ p/⟨p⟩ − 1 so that the distributions only reveal fractional uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The darker and lighter shaded regions in the 2D plots trace 68% and 95% credible regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' supernovae relative distances (Brout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We infer the posterior joint PDF of H0 and Ωm by performing MCMC sam- pling using emcee, given the likelihood function and prior choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We infer H0 = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1(a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4% measurement) with the uniform Ωm-prior, and H0 = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 km s−1 Mpc−1 (a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1% measurement) with the Pantheon+ Ωm-prior (solid con- tours in the right panel of Figure 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We show the D∆t–Dd re- gion allowed by our priors in the left panel of Figure 20, which also shows the region allowed by our distance posterior that provides information for the cosmological inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Other cos- mological models beyond flat ΛCDM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Bonvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020) or combining other cosmological probes in a cosmology-independent manner (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Taubenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019) can utilize the additional cosmological information contained by our full 2D posterior outside the regions probed by our cosmo- logical priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For comparison, we also perform cosmological inference us- ing only the 1D posterior of Dd (dashed contours in right panel of Figure 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This gives H0 = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 km s−1 Mpc−1 (a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3% measurement) for the uniform Ωm-prior, and H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 km s−1 Mpc−1 (a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6% measurement) for the Pantheon+ Ωm-prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Dd-only constraints are lower by ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='15σ) than that from the full 2D distance posterior (for the uniform Ωm-prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This slight difference arises from the projection difference of the 2D posterior along the Dd direction and along the narrow track allowed by our choice of cosmological priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Discussion We now compare our results with previous works (Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1), discuss the improvement of the constraint in this paper over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 / r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 blinded Dd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 blinded int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 blinded D t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 blinded Dd target S/N 23 � 1 target S/N 28 � 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison of the constrained Dd between two choices of the target S/N for each bin in the Voronoi binning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The blinded pa- rameters are blinded as pblinded ≡ p/⟨p⟩−1 so that the distributions only reveal fractional uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The darker and lighter shaded regions in the 2D plots trace 68% and 95% credible regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' single-aperture stellar kinematics (Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2), and describe the limitations of this work (Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison with previous time-delay H0 measurements Our measured value H0 = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1 is consistent with previous measurements from lensing time delays with dif- ferent treatments of the MSD (see Figure 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These previous studies can be divided into two approaches: the first breaks the MSD by assuming simple parametric mass profiles such as the power law or composite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', NFW halo and stars with constant mass-to-light ratio), and the second breaks the MSD based solely on stellar kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Our study belongs to the second approach by allowing the freedom in the model to be maximally degener- ate with H0 and constraining it solely from the spatially resolved stellar kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, it is illustrative to compare our re- sult with the first approach to discuss the validity of their mass model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Following the first approach, Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013, 2014) mea- sured H0 = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 km s−1 Mpc−1 from this same system RXJ1131−1231 with simple parametric mass profiles using HST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2019) combined the HST imaging and adaptive-optics-assisted imaging from the Keck Telescope to measure H0 = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 km s−1 Mpc−1 ˙Although these studies used single-aperture stellar kinematics, the MSD was already broken by the assumption of parametric mass profiles, and the single-aperture velocity dispersion helped tighten the constraint and made the inferred H0 values from the power-law and com- posite models more consistent(Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Our measured value – albeit with a larger uncertainty due to the maximal free- dom allowed in the mass model – has a median value very close to these previous measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Such a good agreement in the medians suggests that these previous studies’ simple parametric Article number, page 16 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 2000 3000 D t (Mpc) 600 800 1000 Dd (Mpc) 800 1000 Dd (Mpc) Our measurement 60 80 100 H0 (km s 1 Mpc 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 m m ∼U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5) m ∼N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='334,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='018) from Dd only from Dd only Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Left: Final 2D posterior of the time-delay distance D∆t and the angular diameter distance Dd (emerald contour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The darker and lighter shaded regions in the 2D plots trace 68% and 95% credible regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We infer H0 and Ω from this distance posterior accounting for the covariance in a flat ΛCDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take a wide uniform prior on H0 ∼ U(0, 150) km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The blue-shaded region corresponds to a uniform prior Ωm ∼ U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5) and the orange-shaded region corresponds to a Gaussian prior Ωm ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='334, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='018) from the Pantheon+ analysis of type Ia supernovae relative distances (Brout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Right: Posterior PDF of H0 and Ωm in flat ΛCDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We constrain H0 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1% precision for the uniform and Pantheon+ Ωm-priors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We show the cosmological parameter posterior from only the 1D Dd posterior with dashed contours with colors matching the associated Ω prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In this case, the H0 precision is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6% for the uniform and Gaussian priors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Dd-only constraint on the H0 is lower by ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='15σ) than the constraint from the full 2D posterior, for the uniform Ωm-prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' mass models are close to the ground truth, and no bias is de- tected within the precision afforded by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Future spatially resolved velocity dispersion measurements for more time-delay lens systems or better quality data for this system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', from the James Webb Space Telescope) will allow us to make a more definitive statement on the validity of the parametric mass model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Following the second approach, Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2016) ana- lyzed this same system RXJ1131−1231 using HST imaging and single-aperture velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These authors marginalized the effect of MSD by incorporating a source on the prior but found that the H0 posterior strongly depends on the shape of the anisotropy prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' These authors use two different choices for this prior to find H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 km s−1 Mpc−1 and H0 = 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='9 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This large difference illustrates that single aper- ture velocity dispersion imposes only a weak constraint on the anisotropy profile and, thus, on the MSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This result highlights the need for spatially resolved velocity dispersion, such as the one presented in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Our measured H0 has a precision of 9% while allowing the data to constrain the MSD effect that is maximally degenerate with H0, illustrating the power of spa- tially resolved kinematics in constraining the anisotropy profile and the MSD, despite the seeing-limited nature of our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the future, exquisite data from the James Webb Space Telescope (JWST) will provide an even more dramatic improvement (4% H0 precision forecasted, Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We also compare our result with the measured values of H0 from the current TDCOSMO sample of seven time-delay lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' With the power-law mass model assumptions, the combination of seven time-delay lenses gives a 2% measurement with H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 km s−1 Mpc−1 (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, relaxing this mass profile assumption and constrain- ing the MSD solely from the single-aperture stellar kinematics of the TDCOSMO sample leads to a 9% uncertainty on the re- sultant H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In this study, we achieve the same 9% precision from a single system, highlighting the superb constraining power of spatially resolved kinematics over single-aperture ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' It is also worth comparing with the result obtained by Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2020) when combining the seven TDCOSMO lenses with information obtained from the external SLACS sample of non- time-delay lenses, H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Given the un- certainties, our new measurement is not statistically inconsistent with that result, although the difference is clearly important from a cosmological standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' With the data in hand, we cannot con- clude whether (a) the difference is real and the SLACS sam- ple cannot, therefore, be combined with the TDCOSMO sam- ple, or whether (b) it is due to a statistical fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This study demonstrates that, as we gather more and better data for spatially resolved kinematics and external samples of non-lenses, we will soon be able to conclude whether the difference is real or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the context of the “Hubble tension”, our new measurement strengthens the tension by reaffirming the previously obtained time-delay H0 measurements that agreed with other local mea- surement values, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', from SH0ES (Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Although the 9% uncertainty in H0 from our measurement alone is not suf- ficient to resolve the tension, it demonstrates that time-delay cos- mography can provide a powerful independent perspective with the help of future data from telescopes such as Keck, JWST, and the extremely large telescopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', see forecasts from Shajib Article number, page 17 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer & Treu 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We cannot help noticing that the median of our measurement is somewhat higher than the mean of the local values (∼73 km s−1 Mpc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, the difference is not significant, given the uncertain- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Therefore our likely explanation is that the difference origi- nates from the inevitable statistical fluctuation pertaining to this system, as the initial H0 measurements using simple parametric assumption all provided such higher values (Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We conclude by stressing that some dispersion around the mean is, of course, ex- pected, and indeed Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2020b) shows that the seven TDCOSMO lenses scatter around the mean by an amount con- sistent with the estimated errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Improvement from the spatial resolution of the stellar kinematics We investigate the improvement in constraints provided by the spatially resolved nature of the stellar kinematics presented in this paper over the unresolved or single-aperture case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2013) presents a single-aperture measurement of the line- of-sight velocity dispersion σlos = 323 ± 20 km s−1 obtained within a 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='81 × 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 aperture with a 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This mea- surement was from the Low-Resolution Imaging Spectrometer (LRIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Oke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1995) on the Keck Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The probed wavelength range was ∼3900–4700 Å, which probes mostly the redward range of the Ca H&K lines with a little overlap with the range probed by our data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', 3300–4200 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' If we take a luminosity-weighted-sum of the spatially resolved velocity dis- persion map within the same 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='81×0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 aperture, we get 288±5 km s−1, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7σ (11%) lower than the previous single- aperture measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Although the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7σ difference is not sta- tistically significant, some parts of it can be due to potential sys- tematics in the kinematic extraction procedure or due to differ- ent wavelength ranges probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' It is generally considered that the minimum error, considering systematics, on velocity dispersion measurements is 5%, even for very high-S/N data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, to illustrate the improvement in precision from the spatially resolved nature of the velocity dispersion presented in this study, we take a fiducial single-aperture measurement value of 288 ± 18 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This mean value is from the luminosity- weighted sum within the single aperture mentioned above, and the 18 km s−1 uncertainty comes from applying the 6% uncer- tainty of the 323 ± 20 km s−1 measurement on the fiducial mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We take the galaxy’s major axis to align with the rectangular aperture’s longer side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Rotating the aperture by 90◦ only changes the predicted velocity dispersion integrated within the aperture by ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1%, which is unsurprising given the mild ellipticity (ql ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='85) of the galaxy and the 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='96 seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We compare the key dynamical model parameters between the spatially resolved and single-aperture cases in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' As expected, the internal MST parameter λint and the anisotropy profile parameter σθ/σr are almost completely unconstrained in the case of the single- aperture stellar kinematics due to the mass-anisotropy degener- acy (Treu & Koopmans 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Courteau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, the angular diameter distance Dd can be constrained to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7% precision, largely by the anisotropy prior (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6% constraint on Dd from the spatially resolved data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This single-aperture pre- cision level on Dd agrees very well with the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='9% precision on Dd (= 810+160 −130 Mpc) obtained by Jee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2019) from the same system RXJ1131−1231 based on the previously available single-aperture stellar kinematics mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Hubble constant H0 can be inferred to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5% precision with the uniform Ωm prior from the full 2D posterior of the fiducial single-aperture case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Although the improvement in H0 precision (by ∼3%) from the spatially resolved kinematics does not appear to be dramatic, this is due to the fact that the projection of D∆t–Dd posterior along the narrow track allowed by our chosen prior happens to give a small difference between the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The improvement could have appeared more drastic if the full 2D posterior had a different orientation from the prior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In reality, the full cos- mological information (illustrated by the area enclosed within the 95% contour) contained by the single-aperture data is much less than that from the spatially resolved data presented in this study (see the D∆t–Dd contours in Figure 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Limitations of this study One limitation of our study is the data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Although our data are the first of their kind from a cutting-edge ground-based facil- ity such as the Keck Observatory, there are opportunities to ob- tain better-quality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The KCWI instrument is seeing-limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus the S/N on the lensing galaxy is degraded by contamina- tion from the nearby quasars, and the spatial resolution of the ve- locity dispersion map is limited by the seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Adaptive-optics- assisted IFU spectroscopy from the ground or observations from space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', with the JWST, can deliver exquisite spatially re- solved data for improved H0 precision in the future (Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Future data with higher spatial resolution will be particu- larly powerful in constraining the anisotropy profile better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Our measurement has only weak constraints on the anisotropy pro- file, which is largely bounded by the adopted uniform prior (see Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This prior is obtained from a sample of eight local massive ellipticals with one of the highest quality spatially re- solved kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, this is a small sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A tighter anisotropy prior from larger samples of massive ellipticals, even better if they are from a redshift range that matches with the one for our system, will be helpful to mitigate further the degener- acy induced by the anisotropy profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', the mass-anisotropy degeneracy (Treu & Koopmans 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Courteau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Conclusion We measured the spatially resolved stellar velocity dispersion of the lens galaxy in RXJ1131−1231 using the KCWI IFU spectro- graph on the Keck Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We combined the new spatially resolved stellar kinematics with previously obtained lens mod- els derived from HST imaging data, observed time delays, and estimated line-of-sight lensing effects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', the external conver- gence) to infer H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Combining the spatially resolved velocity dispersion with lens imaging and time delays simultaneously al- leviates the MSD in the measured D∆t and additionally measures the angular diameter distance Dd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In order to prevent conscious or unconscious experimenter bias, we blindly performed the dynamical modeling and the cos- mographic inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We unblinded the H0 value after all the co- authors had agreed on the modeling choices after various checks on systematics, and the analysis was frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The main conclu- sions from our study are as follows: – The 2D distance posterior of Dd and D∆t gives H0 = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1 for a uniform prior on Ωm ∼ U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5), and H0 = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 km s−1 Mpc−1for a Gaussian prior on Ωm from the Pantheon+ analysis (Brout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' – Our 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4% measurement from a single system with spatially resolved kinematics provides a similar precision as, and is in Article number, page 18 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 60 70 80 90 H0 (km s 1 Mpc 1) Probability density RXJ1131 1231 with the MSD broken assuming simple parametric mass models (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019) Seven lenses with the MSD broken assuming simple parametric mass models (Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, TDCOSMO-I) Seven lenses with the MSD broken using single- aperture kinematics (Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, TDCOSMO-IV) RXJ1131 1231 with the MSD broken using spatially resolved kinematics (this work) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison of our 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4% H0 measurement (red, 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1) from the single system RXJ1131−1231 with previous mea- surements from Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2019, blue), Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2020b, grey), and Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2020, emerald).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The distributions show the H0 posteriors as described in the figure legend, and the points with error bars mark the mean and 68% credible intervals of the corresponding posterior with matching color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' For the same flexible mass models, our analysis on a single system provides a similar precision on H0 with that from seven lenses with only single-aperture stellar kinematics (emerald, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8km s−1 Mpc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Moreover, the median value of our measurement falls very close to those from previous analyses on the same system but with simple parametric assumption on the mass model breaking the MSD (blue, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3km s−1 Mpc−1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' also 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 km s−1 Mpc−1by Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' excellent agreement with, the current TDCOSMO sample of seven time-delay lenses based only on single-aperture stellar kinematics (H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='1 km s−1 Mpc−1, Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Note that the system RXJ1131−1231 analyzed here is part of that sample of seven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' – The median value of H0 from our analysis is very close to the previously inferred values assuming simple parametric mass models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', H0 = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='3 km s−1 Mpc−1, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Thus we do not detect any potential bias in those mass profile assumptions within the precision afforded by our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' – Our measurement is in excellent agreement with that ob- tained by Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (2020a), based on the standard as- sumption of simply parametrized forms for the mass density profile of the lens to break the MSD (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 km s−1 Mpc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In conclusion, our study provides an important validation of previous work by our collaboration on the determination of H0 from time-delay cosmography (summarized in Figure 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This analysis also showcases the power of spatially resolved kinemat- ics in breaking the degeneracies that limit the H0 precision when mass profile assumptions on the galaxy density profile are re- laxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' As the first application of such methodology performed on real data, this study stands as an important proof of concept to pioneer future studies on many more time-delay lens systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' In the broader context of the "Hubble tension", the mea- surement presented here is on the high end of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' However, the precision is not yet sufficient to rule out the val- ues below 70km s−1 Mpc−1, generally favored by early universe probes (Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Larger samples of time-delay lenses with spatially resolved kinematics are needed to reach a conclu- sive answer without making assumptions about the mass den- sity profile of the deflectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' With JWST data already scheduled and ground-based data similar to those presented here for 7 sys- tems, ∼ 3% precision should be attainable in a relatively short time scale (Birrer & Treu 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Beyond that, a sample of ∼40 lensed quasars or supernovae with spatially resolved kinemat- ics can provide the ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='2% precision on H0 that is necessary to resolve or confirm the “Hubble tension” at 5σ confidence level, with maximally flexible models, thanks to spatially resolved stel- lar kinematics (Birrer & Treu 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We thank Elizabeth Buckley-Geer, Thomas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Collett, Philip J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Marshall, and Chiara Spiniello for useful discussions and comments that improved this study and the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Support for this work was provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51492 awarded to AJS by the Space Telescope Science Institute, which is operated by the Associ- ation of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', for NASA, under contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' TT and GCFC acknowledge support by NSF through grants NSF- AST-1906976 and NSF-AST-1836016, and from the Moore Foundation through grant 8548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' PM and CDF acknowledge support for this work from the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' AST-1907396 SHS thanks the Max Planck Society for support through the Max Planck Research Group and the Max Planck Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' SHS is supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This project has received funding from SNSF and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (COSMICLENS : grant agreement No 787886).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' VNB gratefully acknowledges assistance from National Science Foun- dation (NSF) Research at Undergraduate Institutions (RUI) grant AST-1909297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Note that findings and conclusions do not necessarily represent views of the NSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This work used computational and storage services associated with the Hoffman2 Shared Cluster provided by UCLA Institute for Digital Research and Education’s Research Technology Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The data presented herein were obtained at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Adminis- tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The Observatory was made possible by the generous financial support of the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' We are most fortunate to have the opportunity to conduct observations from this mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' This research made use of jampy (Cappellari 2008, 2020), pPXF (Cappellari 2017, 2022), pafit (Krajnovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2006), vorbin (Cappellari & Copin 2003), mgefit (Cappellari 2002) , lenstronomy (Birrer & Amara 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Article number, page 19 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='05 / r 600 1000 Dd (Mpc) 2000 4000 D t (Mpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 int 3000 D t (Mpc) 900 Dd (Mpc) spatially resolved single aperture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Comparison of the distance constraints between spatially resolved velocity dispersion and single-aperture velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Here, the integrated velocity dispersion is taken as the fiducial value of 287 ± 18 km s−1 to match the mean of our spatially resolved mea- surement, but the uncertainty of a single-aperture velocity dispersion measurement (Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The darker and lighter shaded regions in the 2D plots trace 68% and 95% credible regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' The single-aperture velocity dispersion cannot constrain the anisotropy pro- file parameter σθ/σr and the internal MST parameter λint, with both limited by the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' As a result, the D∆t–Dd posterior is constrained much more weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021), numpy (Oliphant 2015), scipy (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2001), astropy (Astropy Col- laboration 2013, 2018), jupyter (Kluyver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2016), matplotlib (Hunter 2007), seaborn (Waskom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014), emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013), and getdist (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='com/cmbant/getdist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' References Abdalla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Abellán, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Aboubrahim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022, Journal of High Energy Astrophysics, 34, 49 Abolfathi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Aguado, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Aguilar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, The Astrophysical Journal Supplement Series, 235, 42 Aiola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Calabrese, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Maurin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', 2020, 047–047 Astropy Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, A&A, 558, A33 Astropy Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, AJ, 156, 123 Avila, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Koekemoer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Mack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Fruchter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2015, Optimizing pixfrac in Astrodrizzle: An example from the Hubble Frontier Fields, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Simien, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Monnet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1983, Astronomy and Astrophysics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 128, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 405-410 (1983), 128, 405 Barnabè, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Czoske, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Koopmans, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2009, Monthly Notices of the Royal Astronomical Society, 399, 21 Barnabè, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Dutton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Marshall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2012, Monthly Notices of the Royal Astronomical Society, 423, 1073 Bertin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Lombardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2006, ApJ, 648, L17 Binney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Tremaine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1987, Galactic dynamics Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Amara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, Physics of the Dark Universe, 22, 189 Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Amara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Refregier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2016, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', 8, 020 Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Dhawan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022a, ApJ, 924, 2 Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Millon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022b, Time-Delay Cosmography: Mea- suring the Hubble Constant and other cosmological parameters with strong gravitational lensing Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Galan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, A&A, 643, A165 Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Gilman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021, JOSS, 6, 3283 Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021, A&A, 649, A61 Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Rusu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, MNRAS, 484, 4726 Blakeslee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Jensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Milne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Greene, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021, ApJ, 911, 65 Blum, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Castorina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Simonovi´c, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, arXiv e-prints, arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='07182 Bonvin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017, MNRAS, 465, 4914 Brout, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Scolnic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Popovic, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 938, 110 Buckley-Geer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Rusu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, MNRAS, 498, 3241 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2002, MNRAS, 333, 400 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2008, MNRAS, 390, 71 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2016, ARA&A, 54, 597 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017, MNRAS, 466, 798 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, Monthly Notices of the Royal Astronomical Society, 494, 4819 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022, Full spectrum fitting with photometry in ppxf: non- parametric star formation history, metallicity and the quenching boundary from 3200 LEGA-C galaxies at redshift z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='8 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Copin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2003, Monthly Notices of the Royal Astronomical Society, 342, 345 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Emsellem, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2007, MNRAS, 379, 418 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Scott, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Alatalo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, MNRAS, 432, 1709 Chang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', van der Wel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Rix, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, The Astrophysical Journal, 773, 149 Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Fassnacht, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, MNRAS, 490, 1743 Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Fassnacht, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021a, A&A, 652, A7 Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Fassnacht, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021b, Monthly Notices of the Royal Astronomical Society, 508, 755 Collett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Oldham, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, Science, 360, 1342 Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Eigenbrod, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Vuissoz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Meylan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Magain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2005, 225, 297 Courteau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', de Jong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014, Reviews of Modern Physics, 86, 47 de Zeeuw, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Evans, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Schwarzschild, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1996, Monthly Notices of the Royal Astronomical Society, 280, 903 Di Valentino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Mena, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Pan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021, Classical and Quantum Gravity, 38, 153001 Efstathiou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021, MNRAS, 505, 3866 Emsellem, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Monnet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Nieto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1994, A&A, 285, 739 Falco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Gorenstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Shapiro, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1985, ApJ, 289, L1 Foreman-Mackey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hogg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Lang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Goodman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, PASP, 125, 306 Foxley-Marrable, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Collett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Vernardos, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Goldstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Bacon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, Monthly Notices of the Royal Astronomical Society, 478, 5081 Freedman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021, ApJ, 919, 16 Freedman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Madore, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hatt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, ApJ, 882, 34 Freedman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Madore, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hoyt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, ApJ, 891, 57 Fruchter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Hook, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2002, Publications of the Astronomical Society of the Pacific, 114, 144 Gilman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, A&A, 642, A194 Gomer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Van de Vyvere, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022, A&A, 667, A86 Gonneau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Lyubenova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Lançon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, A&A, 634, A133 Gonzaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hack, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Fruchter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Mack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2012, The DrizzlePac Handbook Goodman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Weare, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2010, Communications in Applied Mathematics and Computational Science, 5, 65–80 Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, Monthly Notices of the Royal Astronomical Society, 477, 4711 Greene, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, ApJ, 768, 39 Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2007, Computing in Science and Engineering, 9, 90 Jeans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1922, Monthly Notices of the Royal Astronomical Society, 82, 122 Jee, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Komatsu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, Science, 365, 1134 Jones, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Peterson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2001, SciPy: Open source scien- tific tools for Python Kluyver, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Ragan-Kelley, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Pérez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2016, in Positioning and Power in Academic Publishing: Players, Agents and Agendas, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Loizides & B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Schmidt (IOS Press BV, Amsterdam, Netherlands), 87 – 90 Knox, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Millea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D, 101, 043533 Kochanek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, MNRAS, 493, 1725–1735 Kourkchi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Tully, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Eftekharzadeh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, ApJ, 902, 145 Krajnovi´c, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', de Zeeuw, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Copin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2006, Monthly No- tices of the Royal Astronomical Society, 366, 787 Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Mao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, The Astrophysical Journal, 863, L19 Millon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Bonvin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020a, Astronomy and Astrophysics, 640, A105 Millon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Galan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020b, A&A, 639, A101 More, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Oguri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', More, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017, The Astrophys- ical Journal, 835, L25 Article number, page 20 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' : H0 from spatially resolved kinematics of RXJ1131−1231 Morrissey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Matuszewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Martin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2012, 8446, 844613 Morrissey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Matuszewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Martin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, The Astrophysical Journal, 864, 93 Navarro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Frenk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1996, ApJ, 462, 563 Navarro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Frenk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1997, ApJ, 490, 493 Oke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Cohen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Carr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1995, Publications of the Astronomical Society of the Pacific, 107, 375 Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2015, Guide to NumPy, 2nd edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' (USA: CreateSpace Independent Publishing Platform) Pesce, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Braatz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Reid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, ApJ, 891, L1 Planck Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, A&A, 641, A6 Raftery, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1995, Sociological Methodology, 25, 111 Refsdal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1964, MNRAS, 128, 307 Riess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Yuan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Macri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 934, L7 Robertson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, Publications of the Astronomical Society of Australia, 30, e048 Rusu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Fassnacht, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017, MNRAS, 467, 4220 Rusu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Wong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Bonvin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, MNRAS, 498, 1440 Schneider, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Ehlers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Falco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1992, Gravitational Lenses Schneider, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, A&A, 559, A37 Schneider, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014, A&A, 564, A103 Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, MNRAS, 488, 1387–1400 Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, MNRAS, 494, 6072 Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Birrer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, MNRAS, 483, 5649 Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Glazebrook, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Barone, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022a, LensingETC: a tool to op- timize multi-filter imaging campaigns of galaxy-scale strong lensing systems Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Agnello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, MNRAS, 473, 210 Shajib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Vernardos, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Collett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022b, Strong Lensing by Galax- ies Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Claeskens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hutsemékers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Surdej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2007, Astronomy and Astrophysics, 468, 885 Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Surdej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Claeskens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2003, A&A, 406, L43 Sonnenfeld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Marshall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2015, ApJ, 800, 94 Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Auger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hilbert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, ApJ, 766, 70 Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Marshall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Auger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2010, ApJ, 711, 201 Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hilbert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014, ApJ, 788, L35 Sánchez-Blázquez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Peletier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Jiménez-Vicente, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2006, Monthly Notices of the Royal Astronomical Society, 371, 703 Sérsic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 1968, Atlas de Galaxias Australes Taubenberger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Komatsu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, A&A, 628, L7 Tewes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Meylan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2013, A&A, 556, A22 Tihhonova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Courbin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Harvey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, MNRAS, 477, 5657 Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Agnello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Baumer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2018, MNRAS, 481, 1041 Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Koopmans, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2002, MNRAS, 337, L6 Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' & Marshall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2016, A&A Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', 24, 11 Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Marshall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022, Strong lensing time-delay cosmog- raphy in the 2020s Valdes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Gupta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Rose, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Singh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Bell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2004, The Astro- physical Journal Supplement Series, 152, 251 Van de Vyvere, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Gomer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022a, Astronomy and Astro- physics, 659, A127 Van de Vyvere, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Sluse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Gomer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Mukherjee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2022b, Astronomy and Astrophysics, 663, A179 Verde, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Riess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2019, Nature Astronomy, 3, 891–895 Waskom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Botvinnik, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Hobson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2014, seaborn: v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='0 (November 2014) Wenger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Ochsenbein, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Egret, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2000, Astronomy and Astrophysics Supplement Series, 143, 9 Wong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2020, MNRAS, 498, 1420 Yahalomi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Schechter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Wambsganss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2017, A Quadruply Lensed SN Ia: Gaining a Time-Delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='Losing a Standard Candle Yıldırım, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', & Komatsu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' 2021, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='14615 [astro-ph] [arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content='14615] Yıldırım, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfyQI7/content/2301.02656v1.pdf'} +page_content=' H.' metadata={'source': 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SINGLE-PARTICLE DIFFUSION IN +MULTI-LAYERED MEDIA +PAUL C. BRESSLOFF∗ +Abstract. Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide +range of applications in the physical and life sciences, ranging from thermal conduction in composite +media, gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, and +intercellular gap junctions. Many of these systems involve three-dimensional (3D) diffusion in an +array of parallel planes with homogeneity in the lateral directions, so that they can be reduced to +effective one-dimensional (1D) models. +In this paper we develop a probabilistic model of single- +particle diffusion in 1D multi-layered media by constructing a multi-layered version of so-called +snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM, +each of which is restricted to a single layer. Each round of reflected BM is killed when the local time +at one end of the layer exceeds an independent, exponentially distributed random variable. (The +local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of +a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer +on the other side of the boundary with equal probability, and the process is iterated We proceed +by constructing a last renewal equation for multi-layered snapping out BM that relates the full +probability density to the probability densities of partially reflected BM in each layer. We then show +how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that +the renewal equation and corresponding multi-layer diffusion equation are equivalent. We illustrate +the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries +of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out +BM based on the encounter-based method for surface absorption; each round of reflected BM is now +killed according to a non-exponential distribution for each local time threshold. This is achieved +by considering a corresponding first renewal equation that relates the full probability density to +the FPT densities for killing each round of reflected BM. We show that for certain configurations, +non-exponential killing leads to an effective time-dependent permeability that is normalizable but +heavy-tailed. +1. Introduction. Diffusion in heterogeneous media partitioned by semi per- +meable barriers has a wide range of applications in natural and artificial systems. +Examples include multilayer electrodes and semi-conductors [27, 18, 24, 34], thermal +conduction in composite media [3, 33, 17, 44], waste disposal and gas permeation +in soils [55, 43, 52], diffusion magnetic resonance imaging (dMRI) [53, 13, 16], drug +delivery [49, 54], and intercellular gap junctions [20, 15, 26]. Many of these systems +involve three-dimensional (3D) diffusion in an array of parallel planes with homo- +geneity in the lateral directions, which means that they can be reduced to effective +one-dimensional (1D) models. Consequently, there have been a variety of analytical +and numerical studies of 1D multilayer diffusion that incorporate methods such as +spectral decompositions, Greens functions, and Laplace transforms [11, 51, 50, 19, 36, +37, 29, 35, 30, 14, 6, 46]. +Almost all studies of multilayer diffusion have focused on macroscopic models +in which the relevant field is the concentration of diffusing particles. Many of the +analytical challenges concern the derivation of time-dependent solutions that charac- +terize short-time transients or threshold crossing events. This requires either carrying +out a non-trivial spectral decomposition of the solution and/or inverting a highly +complicated Laplace transform. In general, it is necessary to develop some form of +approximation scheme or to supplement a semi-analytical solution with numerical +computations. As far as we are aware, single-particle diffusion or Brownian motion +∗Department +of +Mathematics, +University +of +Utah, +Salt +Lake +City, +UT +84112 +USA +(bressloff@math.utah.edu) +1 + +(BM) in multilayer media has not been investigated to anything like the same ex- +tent, with the possible exception of spatially discrete random walks [40, 47, 39, 2]. +On the other hand, a rigorous probabilistic formulation of 1D diffusion through a +single semi-permeable barrier has recently been introduced by Lejay [41] in terms of +so-called snapping out BM, see also Refs. [1, 42, 12]. Snapping out BM sews together +successive rounds of reflected BM that are restricted to either x < 0 or x > 0 with a +semi-permeable barrier at x = 0. Each round of reflected BM is killed when its local +time at x = 0± exceeds an exponentially distributed random variable with constant +rate κ0. (Roughly speaking, the local time at x = 0+ (x = 0−) specifies the amount +of time a positively (negatively) reflected Brownian particle spends in a neighborhood +of the right-hand (left-hand) side of the barrier [38].) It then immediately resumes +either negatively or positively reflected BM with equal probability, and so on. +We recently reformulated 1D snapping out BM in terms of a renewal equation +that relates the full probability density to the probability densities of partially re- +flected BMs on either side of the barrier [9]. (The theory of semigroups and resolvent +operators were used in Ref. [41] to derive a corresponding backward equation.) We +established the equivalence of the renewal equation with the corresponding single- +particle diffusion equation, and showed how to solve the former using a combination +of Laplace transforms and Green’s function methods. We subsequently extended the +theory to bounded domains and higher spatial dimensions [10]. Formulating interfa- +cial diffusion in terms of snapping out BM has at least two useful features. First, it +provides a more general probabilistic framework for modeling semi-permeable mem- +branes. For example, each round of partially reflected BM on either side of an in- +terface could be killed according to a non-Markovian process, along analogous lines +to encounter-based models of surface absorption [31, 32, 7, 8]. +That is, partially +reflected BM is terminated when its local time at the interface exceeds a random +threshold that is not exponentially distributed. As we have shown elsewhere, this +leads to a time-dependent permeability that tends to be heavy-tailed [9, 10]. Sec- +ond, numerical simulations of snapping out BM generate sample paths that can be +used to obtain approximate solutions of boundary value problems in the presence of +semi-permeable interfaces [41].1 +In this paper we develop a multi-layered version of snapping out BM and its as- +sociated renewal equations for both exponential and non-Markovian killing processes. +In particular, we consider a single particle diffusing in a finite interval [0, L] that is +partitioned into m subintervals (or layers) (aj, aj+1), j = 0, . . . , m − 1, with a0 = 0, +am = L, see Fig. 1.1. The interior interfaces at x = a1, . . . , am−1 are taken to be +semi-permeable barriers with constant permeabilities κj, j = 1, . . . , m − 1, whereas +partially reflecting or Robin boundary conditions are imposed at the ends x = 0, L +with absorption rates 2κ0 and 2κl, respectively. +(The factors of 2 are convenient +when formulating snapping out BM.) The diffusion coefficient is also heterogeneous +with D(x) = Dj for all x ∈ (aj−1, aj). We begin in section 2 by writing down the +multi-layered diffusion equation, which we formally solve using Laplace transforms +and an iterative method based on transfer matrices, following along analogous lines +to Refs. [51, 46]. In section 3, we construct the multi-layered version of snapping out +BM and write down the corresponding last renewal equation, which relates the full +1An efficient computational schemes for finding solutions to the single-particle diffusion equation +in the presence of one or more semi-permeable interfaces has also been developed in terms of under- +damped Langevin equations [21, 22]. However, this is distinct from snapping out BM, which is an +exact single-particle realization of diffusion through an interface in the overdamped limit. +2 + +x = a1 +x = 0 +x = L +x = a m-2 +x = a m-1 +x = a 2 +layer 1 +layer 2 +layer m-1 +layer m +Fig. 1.1. A 1D layered medium consisting of m layers x ∈ (aj, aj+1), j = 0, 1, . . . m − 1, with +a0 = 0 and am = L. The interior interfaces at x = aj, j = 1, . . . m − 1 act as semi-permeable +membranes, whereas partially absorbing boundary conditions are imposed on the exterior boundaries +at x = 0, L. +probability density to the probability densities of partially reflected BM in each layer. +We then show how transfer matrices can be used to solve the Laplace transformed +renewal equation, although the details differ significantly from the iterative solution +of the diffusion equation. We also prove that the renewal equation and diffusion equa- +tion are equivalent. This exploits a subtle feature of partially reflected BM, namely, +the Robin boundary condition is modified when the initial position of the particle is +on the boundary itself. In section 4 we illustrate the theory by analyzing the first +passage time (FPT) problem for the particle to escape from one of the ends of the +domain. The FPT statistics can be analyzed in terms of the small-s behavior of the +Laplace transformed probability fluxes at the ends x = 0, L, where s is the Laplace +variable. This means that it is sufficient to solve the multi-layer renewal equation in +Laplace space, without having to invert the Laplace transformed solution using some +form of spectral decomposition, for example. Finally, in section 5, we use the renewal +approach to incorporate a generalization of snapping out BM based on the encounter- +based method for surface absorption. This is achieved by considering a corresponding +first renewal equation that relates the full probability density to the FPT densities +for killing each round of reflected BM. +2. Single-particle diffusion equation in a 1D layered medium. Before +developing the more general renewal approach for single-particle diffusion in the multi- +layer domain of Fig. 1.1, it is useful to briefly consider the classical formulation in +terms of the diffusion equation with constant permeabilities. Let ρj(x, t) denote the +probability density of the particle position in the j-th layer. For concreteness, we +assume that the particle starts in the first layer, that is, x0 ∈ [0, a1], although it +is straightforward to adapt the analysis to include more general initial conditions, +see section 3. (For notational convenience, we drop the explicit dependence of ρj on +x0.) Single-particle diffusion can be represented by the following piecewise system of +partial differential equations (PDEs): +∂ρj +∂t = Dj +∂2ρj +∂x2 , +x ∈ (aj−1, aj), j = 1, . . . m, +(2.1a) +Dj +∂ρj(x, t) +∂x +���� +x=a− +j += Dj+1 +∂ρj+1(x, t) +∂x +���� +x=a+ +j += κj[ρj+1(a+ +j , t) − ρj(a− +j , t)], +j = 1, . . . m − 1, +(2.1b) +D1 +∂ρ1(x, t) +∂x +���� +x=0 += 2κ0ρ1(0, t), +Dm +∂ρm(x, t) +∂x +���� +x=L += −2κmρm(L, t), +(2.1c) +together with the initial condition ρj(x, t) = δ(x − x0)δj,1. Finally, we denote the +composite solution on the domain G = ∪m +j=1[a+ +j−1, a− +j ] by ρ(x, t). Laplace transforming +3 + +equations (2.1a)–(2.1c) gives +Dj +∂2�ρj +∂x2 − s�ρj = −δ(x − x0)δj,1, +x ∈ (aj−1, aj), j = 1, . . . m, +(2.2a) +Dj +∂�ρj(x, s) +∂x +���� +x=a− +j += Dj+1 +∂�ρj+1(x, s) +∂x +���� +x=a+ +j += κj[�ρj+1(a+ +j , s) − �ρj(a− +j , s)], +j = 1, . . . m − 1, +(2.2b) +D1 +∂�ρ1(x, s) +∂x +���� +x=0 += 2κ0�ρ1(0, s), +Dm +∂�ρm(x, s) +∂x +���� +x=L += −2κm�ρm(L, s). +(2.2c) +Equations (2.2a)–(2.2b) can be solved using transfer matrices along similar lines to +Refs. [51, 46]. We sketch the steps here. +First, note that for all 1 ≤ j ≤ m, equation (2.2a) has the general solution +�ρj(x, s) = Al +j(s) cosh( +� +s/Dj[x − aj−1]) + Bl +j(s) sinh( +� +s/Dj[x − aj−1]) +(2.3) +or, equivalently +�ρj(x, s) = Ar +j(s) cosh( +� +s/Dj[x − aj]) + Br +j (s) sinh( +� +s/Dj[x − aj]). +(2.4) +For 1 < j ≤ m, the coefficients Al +j, Bl +j are related to Ar +j, Br +j according to +� Ar +j +Br +j +� += Uj(s) +� Al +j +Bl +j +� +, +Uj(s) = +� +cosh( +� +s/DjLj) +sinh( +� +s/DjLj) +sinh( +� +s/DjLj) +cosh( +� +s/DjLj) +� +, +(2.5) +where Lj = aj − aj−1 is the length of the j-th layer. The presence of the Dirac delta +function for j = 1 means that the relationship between the coefficients (Ar +1(s), Br +1(s)) +and (Al +1(s), Bl +1(s)) is determined by imposing the continuity condition �ρ1(x+ +0 , s) = +�ρ1(x− +0 , s) and the flux discontinuity condition ∂x�ρ1(x+ +0 , s) − ∂x�ρ1(x− +0 , s) = −1/D1. +This yields the result +� Ar +1 +Br +1 +� += U1(s) +� Al +1 +Bl +1 +� ++ +1 +√sD1 +� +sinh( +� +s/D1[x0 − a1]) +− cosh( +� +s/D1[x0 − a1]) +� +. +(2.6) +Given the relationships �ρj(aj, s) = Ar +j(s), �ρj(aj−1, s) = Al +j(s), Dj∂x�ρj(aj, s) = +�sDjBr +j (s) and Dj∂x�ρj(aj−1, s) = �sDjBl +j(s), the boundary conditions (2.2b) can +be written in the form +� +sDjBr +j (s) = +� +sDj+1Bl +j+1(s) = κj[Al +j+1(s) − Ar +j(s)]. +(2.7) +That is, for 1 ≤ j < m, +� Al +j+1 +Bl +j+1 +� += Vj(s) +� Ar +j +Br +j +� +, +Vj(s) = + + +1 +� +sDj/κj +0 +� +Dj/Dj+1 + + . +(2.8) +Iterating equations (2.5) and (2.8) for m ≥ 2, we have +� Ar +m +Br +m +� += Mm(s) +� Ar +1 +Br +1 +� +, +(2.9) +4 + +with +M2(s) = U2(s)V1(s), +Mm(s) = Um(s) + + +m−1 +� +j=2 +Vj(s)Uj(s) + + V1(s) for m ≥ 3. +(2.10) +Hence, we have shown how the solution in any layer can be expressed in terms of +the two unknown coefficients Al +1(s) and Bl +1(s). The latter are then determined by +imposing the Robin boundary conditions at x = 0, L: +� +sD1Bl +1(s) = 2κ0Al +1(s), +� +sDmBr +m(s) = −2κmAr +m(s). +(2.11) +3. Snapping out BM in a 1D layered medium. We now develop an al- +ternative formulation of multi-layer diffusion, which is based on a generalization of +1D snapping out BM for a single semi-permeable interface [41, 7]. In particular, we +construct a renewal equation that relates ρ(x, t) on G to the probability densities of +partially reflected BM in each of the layers [aj−1, aj], j = 1, . . . , m. +3.1. Single layer with partially reflecting boundaries. Consider BM in the +interval [aj−1, aj] with both ends totally reflecting. Let X(t) ∈ [aj−1, aj] denote the +position of the Brownian particle at time t and introduce the pair of Brownian local +times +ℓ+ +j−1(t) = lim +h→0 +Dj +h +ˆ t +0 +H(aj−1 + h − X(τ))dτ, +(3.1a) +ℓ− +j (t) = lim +h→0 +Dj +h +ˆ t +0 +H(aj − h − X(τ))dτ, +(3.1b) +where H is the Heaviside function. Note that ℓ+ +j−1(t) determines the amount of time +that the Brownian particle spends in a neighborhood to the right of x = aj−1 over the +interval [0, t]. Similarly, ℓ− +j (t) determines the amount of time spent in a neighborhood +to the left of x = aj. (The inclusion of the factor Dj means that the local times have +units of length.) It can be shown that the local times exist and are nondecreasing, +continuous function of t [38]. The corresponding stochastic differential equation (SDE) +for X(t) is given by the Skorokhod equation +dX(t) = +� +2DjdW(t) + dℓ+ +j−1(t) − dℓ− +j (t). +(3.2) +Roughly speaking, each time the particle hits one of the ends it is given an impul- +sive kick back into the bulk domain. It can be proven that the probability density +for particle position evolves according to the single-particle diffusion equation with +Neumann boundary conditions at both ends. +Partially reflected BM can now be defined by introducing a pair of exponentially +distributed independent random local time thresholds �ℓ+ +j−1 and �ℓ− +j such that +P[�ℓ+ +j−1 > ℓ] = e−2κj−1ℓ/Dj, +P[�ℓ− +j > ℓ] = e−2κjℓ/Dj. +(3.3) +The stochastic process is then killed as soon as one of the local times exceeds its +corresponding threshold, which occurs at the stopping time Tj = min{τ − +j , τ + +j } with +τ + +j = inf{t > 0 : ℓ+ +j−1(t) > �ℓ+ +j−1}, +τ − +j = inf{t > 0 : ℓ− +j (t) > �ℓ− +j }. +(3.4) +5 + +local time lj-1 +Tj +x +time t +x0 +interface +aj-1 +aj +totally reflecting +* +Fig. 3.1. Sketch of a course-grained trajectory of a Brownian particle in the interval [aj−1, aj] +with a partially reflecting boundary at x = aj−1 and a totally reflecting boundary at x = aj. The +particle is absorbed as soon as the time ℓj−1(t) spent in a boundary layer around x = aj−1 exceeds +an exponentially distribution threshold �ℓj−1, which occurs at the stopping time Tj. +In Fig. 3.1 we illustrate the basic construction using a simplified version of partially +reflected BM in which x = aj−1 is partially reflecting (0 < κj−1 < ∞) but x = aj is +totally reflecting (κj = 0). +It can be shown that the probability density for particle position prior to absorp- +tion at one of the ends (see also section 5), +pj(x, t|x0)dx = P[x ≤ X(t) < x + dx, t < Tj|X0 = x0], x ∈ [aj−1, aj], +(3.5) +satisfies the single-particle diffusion equation (Fokker-Planck equation) with Robin +boundary conditions at x = aj−1, aj [25, 48, 45, 5, 28]: +∂pj(x, t|x0) +∂t += Dj +∂2pj(x, t|x0) +∂x2 +, +aj−1 < x0, x < aj, +(3.6a) +Dj∂xpj(aj−1, t|x0) = 2κj−1p(aj−1, t|x0), +(3.6b) +Dj∂xpj(aj, t|x0) = −2κjp(aj, t|x0), +(3.6c) +and pj(x, 0|x0) = δ(x − x0). +It is convenient to Laplace transform with respect to t, which gives +Dj +∂2�pj(x, s|x0) +∂x2 +− s�pj(x, s|x0) = −δ(x − x0), +aj−1 < x0, x < aj +(3.7a) +Dj∂x�pj(aj−1, s|x0) = 2κj−1�pj(aj−1, s|x0), +(3.7b) +Dj∂x�p(aj, s|x0) = −2κj�pj(aj, s|x0). +(3.7c) +We can identify �pj(x, s|x0) as the Green’s function of the modified Helmholtz equation +with Robin boundary conditions at x = aj−1, aj: +�pj(x, s|x0) = + + + +AjFj(x, s)F j(x0, s), +aj−1 ≤ x ≤ x0 +AjFj(x0, s)Fj(x, s), +x0 ≤ x ≤ aj +(3.8) +6 + +where +Fj(x, s) = +� +sDj cosh( +� +s/Dj[x − aj−1]) + 2κj−1 sinh( +� +s/Dj[x − aj−1]), +(3.9a) +Fj(x, s) = +� +sDj cosh( +� +s/Dj[aj − x]) + 2κj sinh( +� +s/Dj[aj − x]), +(3.9b) +Aj = +1 +�sDj +1 +2(κj−1 + κj) +� +sDj cosh( +� +s/DjLj) + [sDj + 4κj−1κj] sinh( +� +s/DjLj) +, +(3.9c) +and Lj = aj − aj−1 is the width of the layer. It can be checked that the Robin +boundary conditions are satisfied at x = aj−1, aj for all aj−1 < x0 < aj. However, for +x0 = aj−1, aj, we have +Dj∂x�pj(aj−1, s|aj−1) = 2κj−1�p(aj−1, s|aj−1) − 1, +(3.10a) +Dj∂x�pj(aj, s|aj) = −2κj�pj(aj, s|aj) + 1. +(3.10b) +In other words, +lim +ǫ→0 +� +∂ +∂x +���� +x=aj +�pj(x, s|aj − ǫ) +� +̸= +∂ +∂x +���� +x=aj +� +lim +ǫ→0 �pj(x, s|aj − ǫ) +� +(3.11) +etc. The modification of the Robin boundary condition when the particle starts at +the barrier plays a significant role in establishing the equivalence of snapping out BM +with single particle diffusion in a multi-layered medium (see section 3.3). +3.2. Last renewal equation. We now construct snapping out BM in the multi- +layered domain shown in Fig. 1.1 by sewing together multiple rounds of reflected BM. +For the moment, assume that the exterior boundaries are totally reflecting. For each +interface we introduce a pair of local time ℓ± +j and a corresponding pair of independent +exponentially distributed thresholds �ℓ± +j with rates 2κj, j = 1, . . . , m − 1. Suppose +that the particle starts at x = x0 in the first layer. It realizes positively reflected +BM until its local time ℓ− +1 (t) at x = a1 exceeds the random threshold �ℓ− +1 with rate +2κ1. The process immediately restarts as a new reflected BM with probability 1/2 in +either [0, a1] or [a1, a2]. If the particle is in layer 2, then the reflected BM is stopped +as soon as one of the local times (ℓ+ +1 (t), ℓ− +2 (t)) exceeds its corresponding threshold. +Each time the BM is restarted all local times are reset to zero. Finally, taking the +exterior boundaries to be partially reflecting, we introduce an additional pair of local +times, ℓ0(t), ℓm(t) for the external boundaries at x = 0, L, and a corresponding pair of +exponentially distributed random thresholds �ℓ0, �ℓm with rates 2κ0, 2κm, respectively. +The stochastic process is then permanently terminated at the stopping time +T = min{T0, Tm}, +Tk = inf{t > 0 : ℓk(t) > �ℓk}, k = 0, m. +(3.12) +We illustrate the basic construction in Fig. +3.2 in the simplified case of a single +semi-permeable interface at x = aj and totally reflecting boundaries x = aj−1 and +x = aj+1. The statistics of diffusion across the interface can be captured by sewing +together successive rounds of partially reflected BM in the intervals [aj−1, a− +j ] and +[a+ +j , aj+1] with each round killed according to an exponentially distributed local time +threshold, and the new domain selected with probability 1/2. +7 + +x = aj +x = aj-1 +reflecting +x0 +(a) +Robin +x = aj+1 +Robin +reflecting +reflecting +reflecting +x = aj +x = aj-1 +x = aj+1 +x0 +reflecting +reflecting +x = aj +x = aj-1 +x = aj+1 +(b) +(a) +(c) +Fig. 3.2. Decomposition of snapping out BM on the interval [aj−1, aj+1] with reflecting bound- +ary conditions at the ends and a semi-permeable barrier at x = aj. (a) Diffusion across the interface. +(b) Partially reflected BM in [a+ +j , aj+1]. (c) Partially reflected BM in [aj−1, a− +j ]. +Consider a general initial probability density φ(x0) with x0 ∈ G and set +ρj(x, t) = +ˆ +G +ρj(x, t|x0)φ(x0)dx0, +pj(x, t) = +ˆ +G +pj(x, t|x0)φ(x0)dx0. +(3.13) +Following our previous work on snapping out BM for single semi-permeable interfaces +[9, 10], the renewal equation for the j-th interior layer, j = 2, . . . , m − 1, takes the +form +ρj(x, t) = pj(x, t) + κj−1 +ˆ t +0 +pj(x, τ|aj−1)[ρj−1(a− +j−1, t − τ) + ρj(a+ +j−1, t − τ)]dτ ++ κj +ˆ t +0 +pj(x, τ|aj)[ρj(a− +j , t − τ) + ρj+1(a+ +j , t − τ)]dτ +(3.14a) +for all x ∈ (a+ +j−1, a− +j ), with the probability density pj(x, τ|y) given by the solution +to equations (3.6). The first term pj(x, t) on the right-hand side of equation (3.14a) +represents all trajectories that reach x at time t without ever being absorbed by the +interfaces at x = a+ +j−1, a− +j . The first integral on the right-hand side sums over all +trajectories that were last absorbed (stopped) at time t − τ by hitting the interface +at x = aj−1 from either the left-hand or right-hand side and then switching with +probability 1/2 to BM in the j-th layer such that it is at position x ∈ (a+ +j−1, a− +j ) at +time t. Since the particle is not absorbed over the interval (t − τ, t], the probability of +reaching x is pj(x, τ|aj−1). In addition, the probability that the last stopping event +occurred in the interval (t−τ, t−τ+dτ) irrespective of previous events is 2κj−1dτ. (We +see that the inclusion of the factor 2 in the definition of the permeability cancels the +probability factor of 1/2.) The second integral has the corresponding interpretation +for trajectories that were last stopped by hitting the interface at x = aj. In the case +of the end layers, we have +ρ1(x, t) = p1(x, t) + κ1 +ˆ t +0 +p1(x, τ|a1)[ρ1(a− +1 , t − τ) + ρ2(a+ +1 , t − τ)]dτ, +(3.14b) +ρm(x, t) = pm(x, t) +(3.14c) ++κm−1 +ˆ t +0 +pm(x, τ|am−1)[ρm−1(a− +m−1, t − τ) + ρm(a+ +m−1, t − τ)]dτ. +8 + +Note that there is only a single integral contribution in the end layers since only one +of the boundaries is semi-permeable. One interesting difference between the renewal +equation formulation and the PDE analyzed in section 2 is that the exterior boundary +conditions are already incorporated into the solutions p1(x, t|x0) and pm(x, t|x0), so +that they do not have to be imposed separately. +Given the fact that the renewal equations (3.14a)–(3.14c) are convolutions in time, +it is convenient to Laplace transform them by setting �ρj(x, s) = +´ ∞ +0 +e−stρj(x, t)dt etc. +This gives +�ρ1(x, s) = �p1(x, s) + κ1�p1(x, s|a1)Σ1(s), x ∈ [0+, a− +1 ], +(3.15a) +�ρj(x, s) = �pj(x, s) + κj−1�pj(x, s|aj−1)Σj−1(s) + κj �pj(x, s|aj)Σj(s), x ∈ [a+ +j−1, a− +j ], +1 < j < m, +(3.15b) +�ρm(x, s) = �pm(x, s) + κm−1�pm(x, s|am−1)Σm−1(s), x ∈ [a+ +m−1, L−], +(3.15c) +where +Σj(s) = �ρj(a− +j , s) + �ρj+1(a+ +j , s). +(3.16) +The functions Σj(s) can be determined self-consistently by setting x = a± +k for k = +1, . . . , m−1 and performing various summations. More specifically, substituting equa- +tion (3.15b) into the right-hand side of (3.16) for 1 < j < m gives +Σj(s) = Σp +j(s) + κj−1�pj(aj, s|aj−1)Σj−1(s) + κj �pj(aj, s|aj)Σj(s) ++ κj �pj+1(aj, s|aj)Σj(s) + κj+1�pj+1(aj, s|aj+1)Σj+1(s) +(3.17a) +for 1 < j < m − 1 and Σp +j(s) ≡ �pj(aj, s) + �pj+1(aj, s). On the other hand, equations +(3.15b) and (3.15a) for j = 2 implies that +Σ1(s) = Σp +1(s) + κ1�p1(a1, s|a1)Σ1(s) ++ κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s), +(3.17b) +while equations (3.15c) and (3.15a) for j = m − 1 yields +Σm−1(s) = Σp +m−1(s) + κm−1�pm(am−1, s|am−1)Σm−1(s) +(3.17c) ++ κm−2�pm−1(am−1, s|am−2)Σm−2(s) + κm−1�pm−1(am−1, s|am−1)Σm−1(s). +Equations (3.17a)–(3.17c) can be rewritten in the more compact matrix form +m−1 +� +k=1 +Θjk(s)Σk(s) = −Σp +j(s), +(3.18) +where Θ(s) is a tridiagonal matrix with non-zero elements +Θj,j(s) = dj(s) ≡ κj[�pj+1(aj, s|aj) + �pj(aj, s|aj)] − 1, j = 1, . . . m − 1, +(3.19a) +Θj,j−1(s) = cj(s) ≡ κj−1�pj(aj, s|aj−1), +j = 2, . . . m − 1, +(3.19b) +Θj,j−1(s) = bj(s) ≡ κj+1�pj+1(aj, s|aj+1), +j = 1, . . . , m − 2. +(3.19c) +Assuming that the matrix Θ(s) is invertible, we obtain the formal solution +Σj(s) = − +m−1 +� +k=1 +Θ−1 +jk (s)Σp +k(s). +(3.20) +9 + +Substituting into equations (3.15a)–(3.15c) gives +�ρj(x, s) = �pj(x, s) − +m−1 +� +k=1 +� +κj−1�pj(x, s|aj−1)Θ−1 +j−1,k(s) + κj �pj(x, s|aj)Θ−1 +jk (s) +� +× [�pj(aj, s) + �pj+1(aj+1, s)]. +(3.21) +An alternative way to solve for Σj(s) is to use transfer matrices analogous to the +analysis of the PDE in section 2. For simplicity, suppose that the particle starts in +the first layer at a point x0 ∈ [0, a1] so that �pj(x, s) = �p1(x, s|x0)δj,1. It follows that +equations (3.17a)–(3.17c) can be rewritten in the iterative form +� +Σj +Σj+1 +� += Wj(s) +� Σj−1 +Σj +� +, +Wj(s) = + + +0 +1 +−cj(s) +bj(s) +−dj(s) +bj(s) + + +(3.22) +for 1 < j < m − 1. In particular, +� Σm−2 +Σm−1 +� += N(s) +� Σ1 +Σ2 +� +, +N(s) = +m−2 +� +k=2 +Wk(s), +(3.23) +with, see equation (3.17b), +Σ2(s) = − +1 +b1(s) (�p1(a1, s|x0) + d1(s)Σ1(s)) . +(3.24) +Finally, having determined Σ2, . . . , Σm−1 in terms of Σ1, we can calculate Σ1 by +imposing equation (3.17c), after rewriting it in the more compact form +Σm−2(s) = −dm−1(s) +cm−2(s) Σm−1(s). +(3.25) +We thus obtain the following self-consistency condition for Σ1: +� +1, dm−1(s) +cm−2(s) +� +N(s) +� +Σ1(s) +− +1 +b1(s) (�p1(a1, s|x0) + d1(s)Σ1(s)) +� += 0. +(3.26) +3.3. Equivalence of the renewal and diffusion equations. We now have +two alternative methods of solution in Laplace space, one based on the diffusion +equations (2.2a)–(2.2c) and the other based on the renewal equations (3.15a)–(3.15c). +Both methods involve transfer matrices that can be iterated to express the solution in +the final layer in terms of the solution in the first layer. It is useful to check that the +renewal equations (3.15a)–(3.15c) are indeed equivalent to the Laplace transformed +diffusion equations (2.1a)–(2.1c). +(This is simpler than showing that the iterative +solutions are equivalent.) Clearly, the composite density �ρ(x, s) satisfies the diffusion +equation in the bulk and the exterior boundary conditions, so we only have to check +the boundary conditions across the interior interfaces. First, differentiating equations +(3.15a) and (3.15b) for j = 2 with respect to x and setting x = a± +1 gives +∂x�ρ1(a− +1 , s) = ∂x�p1(a1, s|x0) + κ1∂x�p1(a1, s|a1)Σ1(s), +(3.27a) +∂x�ρ2(a+ +1 , s) = κ1∂x�p2(a1, s|a1)Σ1(s) + κ2∂x�p2(a1, s|a2)Σ2(s). +(3.27b) +10 + +Imposing the Robin boundary condition (3.6) implies that +D1∂x�p1(a1, s|x0) = −2κ1�p(a1, s|x0), +D2∂x�p2(a1, s|a2) = 2κ1�p(a1, s|a2). +On the other hand, equations (3.10a) and (3.10b) yield +D1∂x�p1(a1, s|a1) = −2κ1�p(a1, s|a1) + 1, +D2∂x�p2(a1, s|a1) = 2κ1�p2(a1, s|a1) − 1. +Substituting into equations (3.27a) and (3.27b), we have +D1∂x�ρ1(a− +1 , s) = −2κ1�p1(a1, s|x0) − κ1[2κ1�p1(a1, s|a1) − 1]Σ1(s), +(3.28a) +D2∂x�ρ2(a+ +1 , s) = κ1[2κ1�p2(a1, s|a1) − 1]Σ1(s) + 2κ2κ1�p2(a1, s|a2)Σ2(s). +(3.28b) +Subtracting equations (3.28a) and (3.28b), and using equation (3.17b) implies that +D2∂x�ρ2(a+ +1 , s) − D1∂x�ρ1(a− +1 , s) = 2κ1 +� +κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s) ++ �p1(a1, s|x0) + κ1�p1(a1, s|a1)Σ1(s) − Σ1(s) +� += 0. +(3.29) +Similarly, adding equations (3.28a) and (3.28b) gives +D2∂x�ρ2(a+ +1 , s) + D1∂x�ρ1(a− +1 , s)] = 2κ1 +� +κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s) +− �p1(a1, s|x0) − κ1�p1(a1, s|a1)Σ1(s) +� +. +(3.30) +On the other hand setting x = a± +1 in equations (3.15a) and (3.15b) for j = 2 shows +that +�ρ1(a− +1 , s) = �p1(a1, s|x0) + κ1�p1(a1, s|a1)Σ1(s), +(3.31a) +�ρ2(a+ +1 , s) = κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s). +(3.31b) +Hence, we obtain the expected semi-permeable boundary conditions at x = a1, +D2∂x�ρ2(a+ +1 , s) = D1∂x�ρ1(a− +1 , s) = κ1[�ρ2(a+ +1 , s) − �ρ1(a− +1 , s)]. +(3.32) +A similar analysis can be carried out at the other interfaces. +We have thus established the equivalence of the renewal equations (3.14a)–(3.14c) +and the Laplace transformed diffusion equations (2.2a)–(2.2c). Hence, snapping out +BM X(t) on G is the single-particle realization of the stochastic process whose prob- +ability density evolves according to the multi-layer diffusion equation. +4. First-passage time problem. One of the useful features of working in +Laplace space is that one can solve various first passage time problems without having +to calculate any inverse Laplace transforms. We will illustrate this by considering the +escape of the Brownian particle from one of the ends at x = 0, L. For simplicity, +we again assume that the particle starts in the first layer. Let Q(x0, t) denote the +survival probability that a particle starting at x0 ∈ (0, a1) has not been absorbed at +either end over the interval [0, t). It follows that +Q(x0, t) = +ˆ L +0 +ρ(x, t)dx = +m−1 +� +j=0 +ˆ aj+1 +aj +ρj(x, t)dx. +(4.1) +11 + +(We drop the explicit dependence of ρ and ρj on the initial position x0 for notational +convenience.) Differentiating both sides of equation (4.1) with respect to t and using +equations (2.1a)–(2.1c) shows that +dQ(x0, t) +dt += +m +� +j=1 +ˆ aj +aj−1 +∂ρj(x, t) +∂t +dx = +m +� +j=1 +ˆ aj +aj−1 +Dj +∂2ρj(x, t) +∂x2 +dx += +m +� +j=1 +Dj +�∂ρj(aj, t) +∂x +− ∂ρj(aj−1, t) +∂x +� += Dm +∂ρm(am, t) +∂x +− D1 +∂ρ1(a0, t) +∂t +≡ −Jm(x0, t) − J0(x0, t). +(4.2) +We have used flux continuity across each interior interface so that the survival proba- +bility decreases at a rate equal to the sum of the outward fluxes at the ends x = 0, L, +which are denoted by J0 and JL respectively. Laplace transforming equation (4.2) +and imposing the initial condition Q(x0, 0) = 1 gives +s �Q(x0, s) − 1 = − �J0(x0, s) − �JL(x0, s). +(4.3) +Assuming that κ0+κm > 0, the particle is eventually absorbed at one of the ends with +probability one, which means that limt→∞ Q(x0, t) = lims→0 s �Q(x0, s) = 0. Hence, +�J0(x0, 0)+ �Jm(x0, 0) = 1. Let π0(x0) and πL(x0) denote the splitting probabilities for +absorption at x = 0 and x = L, respectively, and denote the corresponding conditional +MFPTs by T0(x0) and TL(x0). It can then be shown that +π0(x0) = �J0(x0, 0), +πL(x0) = �JL(x0, 0), +(4.4) +and +π0(x0)T0(x0) = − ∂ +∂s +�J0(x0, s) +���� +s=0 +, +πL(x0)TL(x0) = − ∂ +∂s +�JL(x0, s) +���� +s=0 +. +(4.5) +Hence, analyzing the statistics of escape from the domain [0, L] reduces to determining +the small-s behavior of the solutions ∂x�ρ1(0, s) and ∂x�ρm(L, s). We will proceed using +the renewal equation approach of section 3. +4.1. Identical layers. A considerable simplification of the iterative equation +(3.22) occurs in the case of identical layers with Dj = D, κj = κ and aj = ja for all +j = 1, . . . , m. The solution (3.8) for partially reflected BM is now the same in each +layer. That is, �pj(x, s|x0) = �p(x − (j − 1)a, s|x0 − (j − 1)a) for x, x0 ∈ [aj−1, aj] with +�p(x, s|x0) = + + + +AF(x, s)F(x0, s), +a ≤ x ≤ x0 +AF(x0, s)F(x, s), +x0 ≤ x ≤ a +, +(4.6) +F(x, s) = +√ +sD cosh( +� +s/D[x − a]) + 2κ sinh( +� +s/D[x − a]), +(4.7a) +F(x, s) = +√ +sD cosh( +� +s/D[a − x]) + 2κ sinh( +� +s/D[a − x]), +(4.7b) +A = +1 +√ +sD +1 +4κ +√ +sD cosh( +� +s/Da) + [sD + 4κ2] sinh( +� +s/Da) +. +(4.7c) +12 + +In addition equations (3.22)–(3.26) for identical layers imply that +N(s) = W(s)m−3, +W(s) = +� +0 +1 +−1 +−g(a, s) +� +, +(4.8) +with +g(y, s) ≡ 2κ�p(a, s|y) − 1 +κ�p(a, s|0) += 2g0(y, s) − g1(s), +(4.9) +where +g0(y, s) ≡ �p(a, s|y) +�p(a, s|0) = +√ +sD cosh( +� +s/Dy) + 2κ sinh( +� +s/Dy) +√ +sD +, +(4.10a) +g1(s) ≡ +1 +κ�p(a, s|0) = 4κ +√ +sD cosh( +� +s/Da) + [sD + 4κ2] sinh( +� +s/Da) +κ +√ +sD +. +(4.10b) +The matrix W(s) can be diagonalized according to +W(s) = UWd(s)U†, +Wd(s) = diag(λ+(s), λ−(s)), +(4.11) +with +λ±(s) = −g(a, s) ± +� +g(a, s)2 − 4 +2 +, +λ+ + λ− = −g, +λ+λ− = 1, +(4.12) +and +U = +� +1 +1 +λ+ +λ− +� +, +U† = +� +1 +1−λ2 ++ +− +λ+ +1−λ2 ++ +1 +1−λ2 +− +− +λ− +1−λ2 +− +� +, +U†U = UU† = +� 1 +0 +0 +1 +� +. +(4.13) +Substituting (4.11) into (3.23) and (3.26) gives +� +1, g(a, s) +� +U(s)Wd(s)m−3U†(s) +� +Σ1(s) +Σ2(s) +� += 0, +(4.14) +and +Σm−1(s) = +� +0, 1 +� +U(s)Wd(s)m−3U†(s) +� Σ1(s) +Σ2(s) +� +, +(4.15) +with +Σ2(s) = −g0(x0, s) +κ +− g(a, s)Σ1(s). +(4.16) +In addition, from equations (3.15a) and (3.15c) we have +�J0(x0, s) = f(x0, s) + κf(a, s)Σ1(s), +�JL(x0, s) = κf(a, s)Σm−1(s), +(4.17) +where +D∂x�p(0, s|y) = f(y, s) ≡ 2κ[ +√ +sD cosh( +� +s/D[a − y]) + 2κ sinh( +� +s/D[a − y])] +4κ +√ +sD cosh( +� +s/Da) + [sD + 4κ2] sinh( +� +s/Da) +, +(4.18) +13 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +κ = 100 +κ = 10 +κ = 1 +κ = 0.1 +splitting probability +initial position x0 +0 +0.2 +0.5 +0.8 +1 +0.3 +0.7 +0.1 +0.4 +0.6 +0.9 +Fig. 4.1. Splitting probabilities for escape from a three-layer, homogeneous medium. Plots of +π0(x0) and πL(x0) as a function of x0 for various rates κ. Other parameters are D = 1 and a = 1. +and D∂x�p(L, s|L − a) = −f(a, s). +For the sake of illustration, consider three layers (m = 3). Equation (4.14) implies +that for κ > 0 +Σ1(s) = 1 +κ +g(a, s)g0(x0, s) +1 − g(a, s)2 +, +Σ2(s) = − 1 +κ +g0(x0, s) +1 − g(a, s)2 . +(4.19) +Using the limits +lim +s→0 g0(y, s) = 1 + 2κy/D, +lim +s→0 g1(s) = 4(1 + κa/D), +(4.20) +lim +s→0 g(y, s) = −2(1 + 2κ[a − y]/D), +lim +s→0 f(y, s) = (1 + 2κ[a − y]/D) +2(1 + κa/D) +, +(4.21) +we can thus determine the splitting probabilities π0(x0) and πL(x0). Example plots +of π0(x0) and πL(x0) as a function of x0 ∈ [0, a] are shown in Fig. 3.2 for a = D = 1. +It can be checked that π0(x0) + πL(x0) = 1 for all x0. Moreover, in the limit κ → ∞, +we see that π0(0) → 1 and πL(0) → 0 as expected. Also note that for x0 < 1/2 +(x0 > 1/2), π0(x0) is an increasing (a decreasing) function of κ. +4.2. Large number of layers (m → ∞). For a large number of layers (m ≫ 1) +we have +Wm−3 +d += +� +λm−3 ++ +0 +0 +λm−3 +− +� += λm−3 +− +� +ǫ +0 +0 +1 +� +, +ǫ = +�λ+ +λ− +�m−3 +(4.22) +with |ǫ| ≪ 1 since |λ−| > |λ+|. It follows that +N(s) = U(s)Wd(s)m−3U†(s) = λ−(s)m−3{M0(s) + ǫM1(s)}, +(4.23) +where +M0 = +1 +1 − λ2 +− +� +1 +−λ− +λ− +−λ2 +− +� +, +M1 = +1 +1 − λ2 ++ +� +1 +−λ+ +λ+ +−λ2 ++ +� +. +(4.24) +14 + +The next step is to introduce the series expansions +Σj(s) = Σ(0) +j (s) + ǫΣ(1) +j (s) + O(ǫ2), +j = 1, 2, +(4.25) +with +Σ(0) +2 (s) = −g0(x0, s) +κ +− g(a, s)Σ(0) +1 (s), +Σ(n) +2 (s) = −g(a, s)Σ(n) +1 (s) for n ≥ 1. (4.26) +Substituting equations (4.23) and (4.25) into (4.14) and collecting terms in powers of +ǫ gives the O(1) and O(ǫ) equations +� +1, g(a, s) +� +M0(s) +� +Σ(0) +1 (s) +Σ(0) +2 (s) +� += 0, +(4.27a) +� +1, g(a, s) +� � +M0(s) +� +Σ(1) +1 (s) +Σ(1) +2 (s) +� ++ M1(s) +� +Σ(0) +1 (s) +Σ(0) +2 (s) +�� += 0. +(4.27b) +Equation (4.27a) has the solution +Σ(0) +1 (s) = − +λ−(s)g0(x0, s) +κ(1 + g(a, s)λ−(s)) = g0(x0, s) +κλ−(s) , +(4.28) +so that +Σ(1) +1 (s) = λ+(s)4 +λ−(s)4 +1 − λ+(s)2 +1 − λ−(s)2 +� +Σ(0) +1 +− g0(x0, s) +κλ+(s) +� +. +(4.29) +Finally, +Σm−1(s) = (0, 1)N(s) +� Σ1(s) +Σ2(s) +� +(4.30) += λ−(s)m−3(0, 1){M0(s) + ǫM1(s)} +� +Σ(0) +1 (s) + ǫΣ(1) +1 (s) + O(ǫ2) +Σ(0) +2 (s) + ǫΣ(1) +1 (s) + O(ǫ2) +� += λ+(s)m−3(0, 1) +� +M0(s) +� +Σ(1) +1 (s) +Σ(1) +2 (s) +� ++ M1(s) +� +Σ(0) +1 (s) +Σ(0) +2 (s) +� ++ O(ǫ) +� +. +We have used the fact the O(1) solution (Σ(0) +1 , Σ(0) +2 )⊤ is actually a null-vector of the +matrix M0 so the leading contribution to Σm−1(s) is proportional to ǫλ−(s)m−3 = +λ+(s)m−3. Hence, Σm−1(s) → 0 as m → ∞ due to the fact that |λ+(s)| < 1 for all s. +Equations (4.4) and (4.17) then imply that πm(x0) → 0 as m → ∞, with the rate of +decay determined by λ+(0)m−3. +5. Generalized model of multi-layer diffusion. The analysis of the FPT +problem in section 4 could also have been carried out using the solution of the diffusion +equation constructed in section 2. However, one advantage of the renewal approach +is that it is based on snapping out BM, which can be used to generate sample paths +of single-particle diffusion in a multi-layer medium. Rather than exploring numerical +aspects here, we consider another advantage of the renewal approach, namely, it +supports a more general model of semi-permeable membranes. This is based on an +extension of snapping out BM that modifies the rule for killing each round of reflected +BM within a layer. We proceed by applying the encounter-based model of absorption +[31, 32, 7, 8] to reflected BM in each of the layers separately. +15 + +5.1. Local time propagator for a single layer. As we mentioned in sec- +tion 3.1, partially reflected BM in an interval can be implemented by introducing +exponentially distributed local time thresholds at either end of the interval, which +then determine when reflected BM is killed. Here we generalize the killing mecha- +nism. Given the local times (3.1a) and (3.1b) of the j-th layer with totally reflecting +boundaries, the local time propagator is defined according to [31] +Pj(x, ℓ, ℓ′, t|x0)dx dℓ ℓ′ += P[x < X(t) < x + dx, ℓ < ℓ+ +j−1 < ℓ + dℓ, ℓ′ < ℓ− +j < ℓ′ + dℓ′|X(0) = x0]. +(5.1) +Next, for each interface we introduce a pair of independent identically distributed +random local time thresholds �ℓ± +j such that P[�ℓ± +j > ℓ] ≡ Ψ± +j (ℓ). The special case of +exponential distributions is given by equations (3.3). The stochastic process in the +j-th layer is then killed as soon as one of the local times ℓ+ +j−1 and ℓ− +j exceeds its +corresponding threshold, which occurs at the FPT time Tj = min{τ + +j , τ − +j }, see equa- +tion (3.4). Since the corresponding local time thresholds �ℓ+ +j−1 and �ℓ− +j are statistically +independent, the relationship between the resulting probability density pj(x, t|x0) for +partially reflected BM in the j-th layer and Pj(x, ℓ1, ℓ2, t|x0) can be established as +follows: +pj(x, t|x0)dx = P[X(t) ∈ (x, x + dx), t < Tj|X0 = x0] += P[X(t) ∈ (x, x + dx), ℓ+ +j−1(t) < �ℓ+ +j−1, ℓ− +j (t) < �ℓ− +j |X0 = x0] += +ˆ ∞ +0 +dℓψ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′ψ− +j (ℓ′)P[X(t) ∈ (x, x + dx), ℓ+ +j−1 < ℓ, ℓ− +j < ℓ′|X0 = x0] += +ˆ ∞ +0 +dℓψ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′ψ− +j (ℓ′) +ˆ ℓ +0 +dˆℓ +ˆ ℓ′ +0 +dˆℓ′[Pj(x, ˆℓ, ˆℓ′, t|x0)dx]. +We have also introduced the probability densities ψ± +j (ℓ) = −∂ℓΨ± +j (ℓ). Reversing the +orders of integration yields the result +pj(x, t|x0) = +ˆ ∞ +0 +dℓΨ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′Ψ− +j (ℓ′)Pj(x, ℓ, ℓ′, t|x0). +(5.2) +An evolution equation for the local time propagator can be derived as follows +[7, 8]. Since the local times only change at the boundaries x = aj−1, aj, the propagator +satisfies the diffusion equation in the bulk of the domain +∂Pj +∂t = Dj +∂2Pj +∂x2 , x ∈ (aj−1, aj). +(5.3) +The nontrivial step is determining the boundary conditions at x = aj−1, aj. Here we +give a heuristic derivation based on a boundary layer construction. For concreteness, +consider the left-hand boundary layer [aj−1, aj−1 + h] and define +ℓh +j−1(t) = Dj +h +ˆ t +0 +�ˆ h +0 +δ(Xt′ − x)dx +� +dt′. +(5.4) +By definition, hℓh +j−1(t)/Dj is the residence or occupation time of the process X(t) +in the boundary layer up to time t. Although the width h and the residence time +16 + +in the boundary layer vanish in the limit h → 0, the rescaling by 1/h ensures that +limh→0 ℓh +j−1(t) = ℓ+ +j−1(t). Moreover, from conservation of probability, the flux into +the boundary layer over the residence time hδℓ/Dj generates a corresponding shift in +the probability Pj within the boundary layer from ℓ → ℓ + δℓ. That is, for ℓ > 0, +−Jj(aj−1 + h, ℓ, ℓ′, t|x0)hδℓ = [Pj(aj−1, ℓ + δℓ, ℓ′, t|x0) − Pj(aj−1, ℓ, ℓ′, t|x0)]h, +where Jj(x, ℓ, ℓ′, t|x0) = −D∂xPj(x, ℓ, ℓ′, t|x0). Dividing through by hδℓ and taking +the limits h → 0 and δℓ → 0 yields +−Jj(aj−1, ℓ, ℓ′, t|x0) = ∂ℓPj(aj−1, ℓ, ℓ′, t|x0), ℓ > 0. +Moreover, when ℓ = 0 the probability flux Jj(aj−1, 0, ℓ′, t|x0) is identical to that +of a Brownian particle with a totally absorbing boundary at x = aj−1, which we +denote by Jj,∞(aj−1, ℓ′, t|x0). In addition, it can be shown that Pj(aj−1, 0, ℓ′, t|x0) = +−Jj,∞(aj−1, ℓ′, t|x0). Applying a similar argument at the end x = aj, we obtain the +pair of boundary conditions +D∂xPj(aj−1, ℓ, ℓ′, t|x0) = −Pj(aj−1, 0, ℓ′, t|x0)δ(ℓ) + ∂Pj(aj−1, ℓ, ℓ′, t|x0) +∂ℓ +, +(5.5a) +−D∂xPj(aj, ℓ, ℓ′, t|x0) = −Pj(aj, ℓ, 0, t|x0)δ(ℓ′) + ∂Pj(aj, ℓ, ℓ′, t|x0) +∂ℓ′ +. +(5.5b) +The crucial step in the encounter-based approach is to note that for exponentially +distributed local time thresholds, see equation (3.3), the right-hand side of equation +(5.2) reduces to a double Laplace transform of the local time propagator: +pj(x, t|x0) = Pj(x, z+ +j−1, z− +j , t|x0), +z+ +j−1 = 2κj−1 +Dj +, +z− +j = 2κj +Dj +, +(5.6) +with +Pj(x, z, z′, t|x0) ≡ +ˆ ∞ +0 +dℓe−zℓ +ˆ ∞ +0 +dℓ′e−z′ℓ′Pj(x, ℓ, ℓ′, t|x0). +(5.7) +Laplace transforming the propagator boundary conditions (5.5a) and (5.5b) then +shows that the probability density pj of equation (5.6) is the solution to the Robin +BVP given by equations (3.6a) and (3.6b). Hence, the probability density of partially +reflected BM in the j-th layer is equivalent to the doubly Laplace transformed local +time propagator with the pair of Laplace variables z+ +j−1 and z− +j . Assuming that the +Laplace transforms can be inverted, we can then incorporate non-exponential proba- +bility distributions Ψ+ +j−1(ℓ) and Ψ− +j (ℓ′) such that the corresponding marginal density +is now +pj(x, t|x0) = +ˆ ∞ +0 +dℓΨ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′Ψ− +j (ℓ′)L−1 +ℓ L−1 +ℓ′ Pj(x, z, z′, t|x0), +(5.8) +where L−1 denotes the inverse Laplace transform. One major difference from the +exponential case is that the stochastic process X(t) is no longer Markovian. +5.2. Killing time densities. In order to sew together successive rounds of re- +flected BM in the case of general distributions Ψj we will need the conditional FPT +densities f + +j−1(x0, t) and f − +j (x0, t) for partially reflected BM in the j-th layer to be +killed at the ends x = aj−1 and x = aj, respectively. The corresponding conditional +17 + +killing times were defined in equation (3.4). +The FPT densities are given by the +outward probability fluxes at the two ends: +f + +j−1(x0, t) = Dj∂xpj(aj−1, t|x0), +f − +j (x0, t) = −Dj∂xpj(aj, t|x0). +(5.9) +As in previous sections, it is convenient to Laplace transform with respect to t. Laplace +transforming equation (5.8) and using the Green’s function (3.8) gives +�pj(x, s|x0) = +ˆ ∞ +0 +dℓΨ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′Ψ− +j (ℓ′)L−1 +ℓ L−1 +ℓ′ �Pj(x, z, z′, s|x0), +(5.10) +where +�Pj(x, z, z′, s|x0) = + + + +Aj(z, z′, s)Fj(x, z, s)Fj(x0, z′, s), +aj−1 ≤ x ≤ x0, +Aj(z, z′, s)Fj(x0, z, s)Fj(x, z′, s), +x0 ≤ x ≤ aj, +(5.11) +with +Fj(x, z, s) = +� +s/Dj cosh( +� +s/Dj[x − aj−1]) + z sinh( +� +s/Dj[x − aj−1]), +(5.12a) +Fj(x, z′, s) = +� +s/Dj cosh( +� +s/Dj[aj − x]) + z′ sinh( +� +s/Dj[aj − x]), +(5.12b) +Aj = +1 +� +sDj +1 +(z + z′) +� +s/Dj cosh( +� +s/DjLj) + [s/Dj + zz′] sinh( +� +s/DjLj) +. +(5.12c) +Since �Pj(x, z, z, s|x0) satisfies the Robin boundary conditions +Dj∂x �Pj(aj−1, z, z′, s|x0) = Djz �Pj(aj−1, z, z′, s|x0), +Dj∂x �Pj(aj, z, z′, s|x0) = −Djz′ �Pj(aj, z, z′, s|x0), +it follows that +�f + +j−1(x0, s) ≡ Dj∂x�pj(aj−1, s|x0) += Dj +ˆ ∞ +0 +dℓΨ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′Ψ− +j (ℓ′) +� +∂ℓ �Pj(aj−1, ℓ, ℓ′, s|x0) + �Pj(aj−1, 0, ℓ′, s|x0) +� += Dj +ˆ ∞ +0 +dℓψ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′Ψ− +j (ℓ′) �Pj(aj−1, ℓ, ℓ′, s|x0). +(5.13) +Similarly, +�f − +j (x0, s) ≡ −Dj∂x�pj(aj, s|x0) = Dj +ˆ ∞ +0 +dℓΨ+ +j−1(ℓ) +ˆ ∞ +0 +dℓ′ψ− +j (ℓ′) �Pj(aj, ℓ, ℓ′, s|x0). +(5.14) +Evaluation of the FPT densities reduces to the problem of calculating the prop- +agator �Pj(ak, ℓ, ℓ′, s|x0) by inverting the double Laplace transform �Pj(ak, z, z′, s|x0) +with respect to z and z′, k = j − 1, j, and then evaluating the double integrals in +equations (5.13) and (5.14). In general, this is a non-trivial calculation. However, a +18 + +major simplification occurs if we take one of the densities Ψ+ +j−1 or Ψ− +j to be an expo- +nential. First suppose that Ψ+ +j−1(ℓ) = e−2κj−1ℓ/Dj. We then have a Robin boundary +condition at x = aj−1, +�f + +j−1(x0, s) = 2κj−1�pj(aj−1, s|x0), +(5.15) +whereas +�f − +j (x0, s) = Dj +ˆ ∞ +0 +dℓ′ψ− +j (ℓ′) �Pj(aj, z+ +j−1, ℓ′, s|x0). +(5.16) +From equation (5.10) we find that +�Pj(aj, zj, z′, s|x0) = 1 +Dj +Λj(x0, s) +z′ + hj(s), +(5.17) +where +Λj(x0, s) = +� +s/Dj cosh( +� +s/Dj[x0 − aj−1]) + z+ +j−1 sinh( +� +s/Dj[x0 − aj−1]) +� +s/Dj cosh( +� +s/DjLj) + z+ +j−1 sinh( +� +s/DjLj +, +(5.18) +and +hj(s) = +� +s/Dj +� +s/Dj tanh( +� +s/DjLj) + z+ +j−1 +� +s/Dj + z+ +j−1 tanh( +� +s/DjLj) +. +(5.19) +Inverting the Laplace transform with respect to z′ then gives +�Pj(aj, z+ +j−1, ℓ′, s|x0) = D−1 +j Λj(x0, s)e−hj(s)ℓ′ +(5.20) +and, hence, +�f − +j (x0, s) = Λj(x0, s) �ψ− +j (hj(s)). +(5.21) +On the other hand, +�pj(aj, s|x0) = D−1 +j Λj(x0, s)�Ψ− +j (hj(s)). +(5.22) +We thus obtain the following boundary condition at x = aj: +�f − +j (x0, s) = �K− +j (s)�pj(aj, s|x0), +�K− +j (s) = +Dj �ψ− +j (hj(s)) +�Ψ− +j (hj(s)) +. +(5.23) +Finally, using the convolution theorem, the boundary condition at x = aj in the time +domain takes the form +Dj∂xpj(aj, t|x0) = − +ˆ t +0 +K− +j (τ)pj(aj, t − τ|x0)dτ. +(5.24) +That is, in the case of a non-Markovian density for killing partially reflected BM at +one end of an interval, the corresponding boundary condition involves an effective +time-dependent absorption rate K− +j (t), which acts as a memory kernel. +19 + +Now suppose that Ψ− +j (ℓ) = e−2κjℓ/Dj so that +�f − +j (x0, s) = 2κj�pj(aj, s|x0), �f + +j−1(x0, s) = Dj +ˆ ∞ +0 +dℓψ+ +j−1(ℓ) �Pj(aj−1, ℓ, z− +j , s|x0). +(5.25) +From equation (5.10) we have +�Pj(aj, z, z− +j , s|x0) = 1 +Dj +Λj(x0, s) +z + hj(s), +(5.26) +where +Λj(x0, s) = +� +s/Dj cosh( +� +s/Dj[aj − x0]) + z− +j sinh( +� +s/Dj[aj − x0]) +� +s/Dj cosh( +� +s/DjLj) + z− +j sinh( +� +s/DjLj +, +(5.27) +and +hj(s) = +� +s/Dj +� +s/Dj tanh( +� +s/DjLj) + z− +j +� +s/Dj + z− +j tanh( +� +s/DjLj) +. +(5.28) +Using identical arguments to the previous case we find that the boundary condition +at x = aj−1 is +�f + +j−1(x0, s) = �K+ +j−1(s)�pj(aj−1, s|x0), +�K+ +j−1(s) = +Dj �ψ+ +j−1(hj(s)) +�Ψ+ +j−1(hj(s)) +. +(5.29) +5.3. Generalized snapping out BM and the first renewal equation. We +now define a generalized snapping out BM by sewing together successive rounds of +reflected BM along identical lines to section 3.2, except that now each round is killed +according to the general process introduced in section 5.1. (For simplicity, we assume +that the exterior boundaries at x = 0, L are totally reflecting.) Although each round +of partially reflected Brownian motion is non-Markovian, all history is lost following +absorption and restart so that we can construct a renewal equation. However, it is +now more convenient to use a first rather than a last renewal equation. Again we +consider a general probability density φ(x0) of initial conditions x0 ∈ G. +Let f + +j−1(t) and f − +j (t) denote the conditional FPT densities for partially reflected +BM in the j-th layer to be killed at the end x = aj−1 and x = aj, respectively, in the +case of a general initial distribution φ(x0). It follows that +f + +j−1(t) = +ˆ aj +aj−1 +f + +j−1(x0, t)φ(x0)dx0 = Dj +ˆ aj +aj−1 +∂xpj(aj−1, t|x0)φ(x0)dx0, +(5.30a) +f − +j (t) = +ˆ aj +aj−1 +f − +j (x0, t)φ(x0)dx0 = −Dj +ˆ aj +aj−1 +∂xpj(aj, t|x0)φ(x0)dx0. +(5.30b) +with f + +j−1(x0, t) and f − +j (x0, t) defined in equations (5.9). We also set f + +1 (t) ≡ 0 and +f − +m(t) ≡ 0. Generalizing previous work [9, 10], the first renewal equation in the j-th +layer, 1 ≤ j ≤ m, takes the form +ρj(x, t) ≡ +ˆ +G +ρj(x, t|x0)φ(x0)dx0 +(5.31) += pj(x, t) + 1 +2 +m−1 +� +k=1 +ˆ t +0 +[ρj(x, t − τ|a− +k ) + ρj(x, t − τ|a+ +k )][f − +k (τ) + f + +k (τ)]dτ +20 + +for x ∈ (aj−1, aj) and +pj(x, t) = +ˆ aj +aj−1 +pj(x, t|x0)φ(x0)dx0. +(5.32) +The first term on the right-hand side of equation (5.31) represents all sample trajecto- +ries that start in the k-th layer and have not been absorbed at the ends x = ak−1, ak +up to time t. The integral term represents all trajectories that were first absorbed +(stopped) at a semi-permeable interface at time τ and then switched to either posi- +tively or negatively reflected BM state with probability 1/2, after which an arbitrary +number of switches can occur before reaching x ∈ (aj−1, aj) at time t. The probability +that the first stopping event occurred at the k-th interface in the interval (τ, τ + dτ) +is [f + +k (τ) + f − +k (τ)]dτ. Laplace transforming the renewal equation (5.31) with respect +to time t gives +�ρj(x, s) = �pj(x, s) + 1 +2 +m−1 +� +k=1 +[�ρj(x, s|a− +k ) + �ρj(x, s|a+ +k )][ �f − +k (s) + �f + +k (s)]. +(5.33) +In order to determine the factors +Σjk(x, s) = �ρj(x, s|a− +k ) + �ρj(x, s|a+ +k ), +1 ≤ k < m, +(5.34) +we substitute into equation (5.33) the initial density φ(x0) = 1 +2[δ(x0−a− +k )+δ(x0−a+ +k )]. +This gives +Σjk(x, s) = �pj(x, s|ak)[δj,k + δj,k+1] + 1 +2Σjk(x, s)[ �f − +k (a− +k , s) + �f + +k (a+ +k , s)] ++ 1 +2Σj,k−1(x, s) �f + +k−1(a− +k , s) + 1 +2Σj,k+1(x, s) �f − +k+1(a+ +k , s)]. +(5.35) +Comparison with equations (3.17a)–(3.17c) implies that the above equation can +be rewritten in the matrix form +m−1 +� +l=1 +Θkl(s)Σjl(s) = −�pj(x, s|ak)[δj,k + δj,k+1], +(5.36) +where Θ(s) is a tridiagonal matrix with non-zero elements +Θk,k(s) = dk(s) ≡ �f − +k (a− +k , s) + �f + +k (a+ +k , s) − 1, k = 1, . . . m − 1, +(5.37a) +Θk,k−1(s) = ck(s) ≡ �f + +k−1(a− +k , s), +k = 2, . . . m − 1, +(5.37b) +Θk,k+1(s) = bk(s) ≡ �f − +k+1(a+ +k , s), +k = 1, . . . , m − 2. +(5.37c) +Assuming that the matrix Θ(s) is invertible, we obtain the formal solution +Σjk(x, s) = −Θ +−1 +kj (s)�pj(x, s|aj) − Θ +−1 +k,j−1(s)�pj(x, s|aj−1). +(5.38) +Substituting into equation (5.33) yields the result +�ρj(x, s) = �pj(x, s) + 1 +2 +m−1 +� +k=1 +[Θ +−1 +kj (s)�pj(x, s|aj) + Θ +−1 +k,j−1(s)�pj(x, s|aj−1)] +(5.39) +× [ �f − +k (s) + �f + +k (s)]. +21 + +Equivalence of first and last renewal equations for exponential killing. An im- +portant check of our analysis is to show that the solution (5.39) of the first renewal +equation is equivalent to the solution (3.21) of the last renewal equation when each +round of reflecting BM is killed according to an independent exponential distribu- +tion for each local time threshold. Since �pj(x, s|x0) then satisfies Robin boundary +conditions at x = aj−1, aj we find that +Θk,k(s) = κk[�pk+1(ak, s|ak) + �pk(ak, s|ak)] − 1 = Θkk(s), +(5.40a) +Θk,k−1(s) = κk−1�pk(ak−1, s|ak) = κk−1�pk(ak, s|ak−1) = Θk,k−1(s), +(5.40b) +Θk,k+1(s) = κk+1�pk+1(ak+1, s|ak) = κk+1�pk+1(ak, s|ak+1) = Θk,k+1(s). +(5.40c) +We have used two important properties of partially reflected BM: +i) Symmetry of the Green’s function �p(x, s|x0) = �p(x0, s|x). +ii) The solution for the functions Σjk(x, s) is obtained by introducing the initial con- +ditions (5.34). The FPT densities are thus evaluated at the initial points a± +k . This +means that when we impose the Robin boundary conditions we do not pick up the +additional constant term in equations (3.10a) and (3.10b). +It follows that the solution (5.39) reduces to the form +�ρj(x, s) = �pj(x, s) + +m−1 +� +k=1 +� +�pj(x, s|aj−1)Θ−1 +k,j−1(s) + �pj(x, s|aj)Θ−1 +kj (s) +� +× κk[�pk(ak, s) + �pk+1(ak, s)]. +(5.41) +Finally, using the fact that κkΘ−1 +kj (s) = κjΘ−1 +jk (s), we recover the solution (3.21). +Non-exponential killing. The above analysis shows that the same solution struc- +ture holds for both exponential and non-exponential killing, provided that we ex- +press the tridiagonal matrix Θij in terms of the conditional FPT densities �f ± +k (a± +k , s), +�f + +k−1(a− +k , s) and �f − +k+1(a+ +k , s). The latter are themselves determined from equations +(5.13) and (5.14). One configuration that is analytically tractable is a 1D domain +with a sequence of semi-permeable barriers whose distributions Ψ± +j +alternate be- +tween exponential and non-exponential. For example, suppose Ψ− +j (ℓ) = e−2κj/Dj and +Ψ+ +j (ℓ) = e−2κj/Dj+1 for all odd layers j = 1, 3, . . ., whereas Ψ± +j (ℓ) are non-exponential +for even layers j = 2, 4, . . .. Combining the analysis of the FPT densities in section +5.2 with the analysis of the first renewal equation and its relationship with the last +renewal equation, we obtain the following generalization of the interfacial boundary +conditions (2.2b): +Dj∂xρj(a− +j , s) = Dj+1∂xρj+1(a+ +j , s) = 1 +2[ �K+ +j (s)ρj+1(a+ +j , s) − �K− +j (s)ρj(a− +j , s)], +(5.42) +with �K± +j (s) = 2κj for odd j and +�K− +j (s) = +Dj �ψ− +j (hj(s)) +�Ψ− +j (hj(s)) +, +�K+ +j (s) = +Dj+1 �ψ+ +j (hj+1(s)) +�Ψ+ +j (hj+1(s)) +(5.43) +for even j, with hj(s) and hj+1(s) given by equations (5.19) and (5.28), respectively. +We thus have the setup shown in Fig. 5.1. Note, in particular, that the time-dependent +permeability kernels of the even interfaces are asymmetric. +22 + +x = a1 +x = 0 +x = a4 +x = a2 +x = a3 +K2-(t) +κ1 +κ1 +K2+(t) +K4-(t) +κ2 +κ2 +K4+(t) +Fig. 5.1. A 1D layered medium partitioned by a sequence of semi-permeable interfaces that +alternate between symmetric constant permeabilities κj, j = 1, 3, . . . and asymmetric time-dependent +permeabilities K± +j (t), j = 2, 4, . . .. +Permeability kernels for the gamma distribution. For the sake of illustration, sup- +pose that ψ± +j (ℓ) for even j are given by the gamma distributions +ψ± +j (ℓ) = +z± +j (z± +j ℓ)µ−1e−z± +j ℓ +Γ(µ) +, µ > 0, +(5.44) +where Γ(µ) is the gamma function. The corresponding Laplace transforms are +�ψ± +j (z) = +� +z± +j +z± +j + z +�µ +, +�Ψ± +j (z) = 1 − �ψ± +j (z) +z +. +(5.45) +If µ = 1 then ψ± +j reduce to the exponential distributions with constant reactivity κj. +The parameter µ thus characterizes the deviation of ψ± +j (ℓ) from the exponential case. +If µ < 1 (µ > 1) then ψ± +j (ℓ) decreases more rapidly (slowly) as a function of the local +time ℓ. Substituting the gamma distributions into equations (5.43) yields +�K− +j (s) = +Djhj(s)(z− +j )µ +[z− +j + hj(s)]µ − (z− +j )µ , �K+ +j (s) = +Dj+1hj+1(s)(z+ +j )µ +[z+ +j + hj+1(s)]µ − (z+ +j )µ . +(5.46) +If µ = 1 then �K± +j (s) = 2κj as expected. On the other hand if µ = 2, say, then +�K− +j (s) = +2κj +2 + Djhj(s)/2κj +, +�K+ +j (s) = +2κj +2 + Dj+1hj+1(s)/2κj +. +(5.47) +The corresponding time-dependent kernels K± +j (t) are normalizable since +ˆ ∞ +0 +K− +j (t)dt = �K− +j (0) = +2κj +2 + κ−1 +j κj−1/[1 + 2κj−1Lj/Dj), +(5.48a) +ˆ ∞ +0 +K+ +j (t)dt = �K+ +j (0) = +2κj +2 + κ−1 +j κj+1/[1 + 2κj+1Lj/Dj+1). +(5.48b) +However, the kernels are heavy-tailed with infinite moments. For example, +⟨t⟩− ≡ +1 +�K− +j (0) +ˆ ∞ +0 +tK− +j (t)dt = − +1 +�K− +j (0) +lim +s→0 ∂s �K− +j (s) += +1 +�K− +j (0) +lim +s→0 +Djh′ +j(s)/2 +[2 + Djhj(s)/2κj]2 = Dj +4κj +�K− +j (0) +2κj +lim +s→0 h′ +j(s) = ∞. +(5.49) +That is, all moments are infinite since all derivatives of hj(s) are singular at s = 0. +An analogous result was previously found for a single interface in 1D and 3D [9, 10]. +23 + +6. Discussion. In this paper we developed a probabilistic framework for analyz- +ing single-particle diffusion in heterogeneous multi-layered media. Our approach was +based on a multi-layered version of snapping out BM. We showed that the distribution +of sample trajectories satisfied a last renewal equation that related the full probabil- +ity density to the probability densities of partially reflected BM in each layer. The +renewal equation was solved using a combination of Laplace transforms and transfer +matrices. +We also proved the equivalence of the renewal equation and the corre- +sponding multi-layered diffusion equation in the case of constant permeabilities. We +then used the renewal approach to incorporate a more general probabilistic model of +semipermeable interfaces. This involved killing each round of partially reflected BM +according to a non-Markovian encounter-based model of absorption at an interface. +We constructed a corresponding first renewal equation that related the full probability +density to the FPT densities for killing each round of reflected BM. In particular, we +showed that non-Markovian models of absorption can generate asymmetric, heavy- +tailed time-dependent permeabilities. +In developing the basic mathematical framework, we focused on relatively simple +examples such as identical layers with constant permeabilities or alternating Marko- +vian and non-Markovian interfaces. We also restricted our analysis to the Laplace +domain rather than the time domain. However, it is clear that in order to apply +the theory more widely, it will be necessary to develop efficient numerical schemes +for solving the last or first renewal equations in Laplace space, and then inverting +the Laplace transformed probability density to obtain the solution in the time do- +main. In the case of non-Markovian models of absorption at both ends of a layer, it +will also be necessary to compute the double inverse Laplace transform of the local +time propagator and evaluate the resulting double integral in equation (5.8). Another +computational issue is developing an efficient numerical scheme for simulating sample +trajectories of snapping out BM in heterogeneous multi-layer media. +Finally, from a modeling perspective, it would be interesting to identify plausible +biophysical mechanisms underlying non-Markovian models of semi-permeable mem- +branes. As previously highlighted within the context of encounter-based models of +absorption [31, 32, 7, 8], various surface-based reactions are better modeled in terms +of a reactivity that is a function of the local time. For example, the surface may +become progressively activated by repeated encounters with a diffusing particle, or an +initially highly reactive surface may become less active due to multiple interactions +with the particle (passivation) [4, 23]. +REFERENCES +[1] V. Aho, K. Mattila, T. K¨uhn, P. Kek¨al¨ainen, O. Pulkkine, R. B. Minussi, M. Vihinen- +Ranta and J. Timonen Diffusion through thin membranes: Modeling across scales. Phy. +Rev. E 93 (2016) 043309 +[2] I. Alemany, J. N. Rose, J. Garnier-Brun, A. D. Scott and D. J. Doorly Random walk dif- +fusion simulations in semi-permeable layered media with varying diffusivity Science Reports +12 (2022) 10759 +[3] S. Barbaro, C. Giaconia and A. Orioli A computer oriented method for the analysis of non +steady state thermal behaviour of buildings. Build. Environ. 23 (1988) 19-24 +[4] C. H. Bartholomew. Mechanisms of catalyst deactivation. Appl. Catal. A: Gen. 212 (2001) +17-60. +[5] A. N. Borodin and P. Salminen. Handbook of Brownian Motion: +Facts and Formulae +Birkhauser Verlag, Basel-Boston-Berlin (1996). +[6] P.C. Bressloff Diffusion in cells with stochastically-gated gap junctions. SIAM J. Appl. Math. +76 (2016) 1658-1682 +24 + +[7] P.C. Bressloff Diffusion-mediated absorption by partially reactive targets: Brownian function- +als and generalized propagators. J. Phys. A. 55 (2022) 205001 +[8] P.C. Bressloff Spectral theory of diffusion in partially absorbing media. Proc. R. Soc. A 478 +(2022) 20220319 +[9] P.C. Bressloff A probabilistic model of diffusion through a semipermeable barrier. Proc. Roy. +Soc. A 478 (2022) 20220615. +[10] P.C. Bressloff Renewal equation for single-particle diffusion through a semipermeable inter- +face. Phys. Rev. E. In press (2023) +[11] P. R. Brink and S. V. Ramanan A model for the diffusion of fluorescent probes in the sep- +tate giant axon of earthworm: axoplasmic diffusion and junctional membrane permeability. +Biophys. J. 48 (1985) 299-309 +[12] A. Bobrowski. Semigroup-theoretic approach to diffusion in thin layers separated by semi- +permeable membranes. J. Evol. Equ. 21 (2021) 1019-1057 +[13] P. T. Callaghan, A. Coy, T. P. J. Halpin, D. MacGowan, K. J. Packer and F. O. Zelaya. +Diffusion in porous systems and the influence of pore morphology in pulsed gradient spin- +echo nuclear magnetic resonance studies. J. Chem. Phys. 97 (1992) 651-662 +[14] E. Carr and I. Turner A semi-analytical solution for multilayer diffusion in a composite +medium consisting of a large number of layers. Appl. Math. Model. 40 (2016) 7034-7050 +[15] B. W. Connors and M. A. Long Electrical synapses in the mammalian brain. Ann. Rev. +Neurosci. 27 (2004) 393-418 +[16] A. Coy and P. T. Callaghan. Pulsed gradient spin echo nuclear magnetic resonance for +molecules diffusing between partially reflecting rectangular barriers. J. Chem. Phys. 101 +(1994) 4599-4609. +[17] F. deMonte. Transient heat conduction in one-dimensional composites lab. A natural analytic +approach. Int. J. Heat Mass Transf. 43 (2000) 3607-3619 +[18] J.-P. Diard, N. Glandut, C. Montella and J.-Y. Sanchez. One layer, two layers, etc. An +introduction to the EIS study of multilayer electrodes. Part 1: Theory. J. Electroanal. Chem. +578 (2005) 247-257 +[19] O. K. Dudko, A. M. Berezhkovskii and G. H. Weiss. Diffusion in the presence of periodically +spaced permeable membranes. J. Chem. Phys. 121 (2004) 11283 +[20] W. J. Evans and P. E. Martin Gap junctions: structure and function. Mol. Membr. Biol. 19 +(2002) 121-136 +[21] S. Regev and O. Farago. Application of underdamped Langevin dynamics simulations for the +study of diffusion from a drug-eluting stent. Phys. A, Stat. Mech. Appl. 507 (2018) 231-239 +[22] O. Farago Algorithms for Brownian dynamics across discontinuities. J. Comput. Phys. 423 +(2020) 109802. +[23] M. Filoche, D. S. Grebenkov, J. S. Andrade and B. Sapoval. Passivation of irregular +surfaces accessed by diffusion. Proc. Natl. Acad. Sci. 105 (2008) 7636-7640. +[24] V. Freger. Diffusion impedance and equivalent circuit of a multilayer film. Electrochem. Com- +mun. 7 (2005) 957-961 +[25] M. Freidlin. Functional Integration and Partial Differential Equations Annals of Mathematics +Studies, Princeton University Press, Princeton (1985) New Jersey +[26] D. A. Goodenough and D. L. Paul Gap junctions. Cold Spring Harb Perspect Biol 1 (2009) +a002576 +[27] G. L. Graff, R. E. Williford and P. E. Burrows. Mechanisms of vapor permeation through +multilayer barrier films: lag time versus equilibrium permeation. J. Appl. Phys. 96 (2004) +1840-1849 +[28] D. S. Grebenkov Partially Reflected Brownian Motion: A Stochastic Approach to Transport +Phenomena, in “Focus on Probability Theory”, Ed. Velle LR pp. 135-169. Hauppauge: Nova +Science Publishers (2006) +[29] D. S. Grebenkov Pulsed-gradient spin-echo monitoring of restricted diffusion in multilayered +structures. J. Magn. Reson. 205 (2010) 181-195 +[30] D. S. Grebenkov, D. V. Nguyen and J.-R. Li Exploring diffusion across permeable barriers +at high gradients. I. Narrow pulse approximation. J. Magn. Reson. 248 (2014) 153-163. +[31] D. S. Grebenkov Paradigm shift in diffusion-mediated surface phenomena. Phys. Rev. Lett. +(2020) 125, 078102. +[32] D. S. Grebenkov An encounter-based approach for restricted diffusion with a gradient drift. +J. Phys. A. (2022) 55 045203. +[33] P. Grossel and F. Depasse. Alternating heat diffusion in thermophysical depth profiles: mul- +tilayer and continuous descriptions. J. Phys. D: Appl. Phys. 31 (1998) 216. +[34] Y. Gurevich, I. Lashkevich and C. G. delaCruz. Effective thermal parameters of layered +films:an application to pulsed photothermal techniques. Int. J. Heat Mass Transf. 52 (2009) +25 + +4302-4307. +[35] D. W. Hahn and M. N. Ozisik One-Dimensional Composite Medium Ch. 10 pp. 393-432. +Wiley, Hoboken (2012) +[36] R. Hickson, S. Barry and G. Mercer. Critical times in multilayer diffusion. Part 1: Exact +solutions. Int. J. Heat Mass Transf. 52 (2009) 5776-5783. +[37] R. Hickson, S. Barry and G. Mercer. Critical times in multilayer. diffusion. Part. 2: Ap- +proximate solutions. Int. J. Heat Mass Transf. 52 (2009) 5784-5791. +[38] K. Ito and H. P. McKean. Diffusion Processes and Their Sample Paths Springer-Verlag, +Berlin (1965) +[39] T. Kay and Giuggioli. Diffusion through permeable interfaces: Fundamental equations and +their application to first-passage and local time statistics. Phys. Rev. Res. 4 (2022) L032039 +[40] V. M. Kenkre, L. Giuggiol and Z. Kalay. Molecular motion in cell membranes: analytic +study of fence-hindered random walks. Phys. Rev. E 77 (2008) 051907 +[41] A. Lejay The snapping out Brownian motion. The Annals of Applied Probability 26 (2016) +1727-1742. +[42] A. Lejay Monte Carlo estimation of the mean residence time in cells surrounded by thin layers. +Mathematics and Computers in Simulation 143 (2018) 65-77 +[43] G. Liu, L. Barbour and B. C. Si. Unified multilayer diffusion model and application to diffu- +sion experiment in porous media by method of chambers. Environ. Sci. Technol. 43 (2009) +2412-2416 +[44] X. Lu and P. Tervola. Transient heat conduction in the composites lab-analytical method. J. +Phys. A: Math. Gen. 38 (2005) 81 +[45] G. N. Milshtein. The solving of boundary value problems by numerical integration of stochastic +equations. Math. Comp. Sim. 38(1995) 77-85 +[46] N. Moutal and D. S. Grebenkov Diffusion across semi-permeable barriers: spectral proper- +ties, efficient computation, and applications J. Sci. Comput. 81 (2019) 1630-1654 +[47] D. Novikov, E. Fieremans, J. Jensen and J. A. Helpern. Random walks with barriers. Nat. +Phys. 7 (2011) 508-514 +[48] V. G. Papanicolaou. The probabilistic solution of the third boundary value problem for second +order elliptic equations Probab. Th. Rel. Fields 87 (1990) 27-77 +[49] G. Pontrelli and F. de Monte. Mass diffusion through two-layer porous media: an applica- +tion to the drug-eluting stent. Int. J. Heat Mass Transf. 50 (2007) 3658-3669. +[50] J. G. Powles, M. Mallett, G. Rickayzen and W. Evans. Exact analytic solutions for dif- +fusion impeded by an infinite array of partially permeable barriers. Proc. R. Soc. Lond. A +436 (1992) 391 +[51] S. V. Ramanan and P. R. Brink. Exact solution of a model of diffusion in an infinite chain +or monlolayer of cells coupled by gap junctions. Biophys. J. 58 (1990) 631-639 +[52] C. D. Shackelford and S. M. Moore. Fickian diffusion of radio nuclides for engineered +containment barriers: diffusion coefficients, porosities, and complicating issues. Eng. Geol. +152 (2013) 133-147. 123 +[53] J. E. Tanner. Transient diffusion in a system partitioned by permeable barriers. application to +NMR measurements with a pulsed field gradient. J. Chem. Phys. 69 (1978) 1748. +[54] H. Todo, T. Oshizaka, W. R. Kadhum and K. Sugibayashi. Mathematical model to predict +skin concentration after topical application of drugs. Pharmaceutics 5 (2013) 634-651. +[55] S. R. Yates, S. K. Papiernik, F. Gao and J. Gan. Analytical solutions for the transport of +volatile organic chemicals in unsaturated layered systems. Water Resour. Res. 36 (2000) +1993-2000. +26 + diff --git a/FtE1T4oBgHgl3EQfEwPV/content/tmp_files/load_file.txt b/FtE1T4oBgHgl3EQfEwPV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..71676423d31f2bafea01a3e3592d0c6c5afb2849 --- /dev/null +++ b/FtE1T4oBgHgl3EQfEwPV/content/tmp_files/load_file.txt @@ -0,0 +1,1205 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf,len=1204 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='02895v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='stat-mech] 7 Jan 2023 RENEWAL EQUATIONS FOR SINGLE-PARTICLE DIFFUSION IN MULTI-LAYERED MEDIA PAUL C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' BRESSLOFF∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide range of applications in the physical and life sciences, ranging from thermal conduction in composite media, gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, and intercellular gap junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Many of these systems involve three-dimensional (3D) diffusion in an array of parallel planes with homogeneity in the lateral directions, so that they can be reduced to effective one-dimensional (1D) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In this paper we develop a probabilistic model of single- particle diffusion in 1D multi-layered media by constructing a multi-layered version of so-called snapping out Brownian motion (BM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The latter sews together successive rounds of reflected BM, each of which is restricted to a single layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Each round of reflected BM is killed when the local time at one end of the layer exceeds an independent, exponentially distributed random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of a boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') The particle then immediately resumes reflected BM in the same layer or the layer on the other side of the boundary with equal probability, and the process is iterated We proceed by constructing a last renewal equation for multi-layered snapping out BM that relates the full probability density to the probability densities of partially reflected BM in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that the renewal equation and corresponding multi-layer diffusion equation are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We illustrate the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Finally, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter-based method for surface absorption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' each round of reflected BM is now killed according to a non-exponential distribution for each local time threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We show that for certain configurations, non-exponential killing leads to an effective time-dependent permeability that is normalizable but heavy-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diffusion in heterogeneous media partitioned by semi per- meable barriers has a wide range of applications in natural and artificial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Examples include multilayer electrodes and semi-conductors [27, 18, 24, 34], thermal conduction in composite media [3, 33, 17, 44], waste disposal and gas permeation in soils [55, 43, 52], diffusion magnetic resonance imaging (dMRI) [53, 13, 16], drug delivery [49, 54], and intercellular gap junctions [20, 15, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Many of these systems involve three-dimensional (3D) diffusion in an array of parallel planes with homo- geneity in the lateral directions, which means that they can be reduced to effective one-dimensional (1D) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Consequently, there have been a variety of analytical and numerical studies of 1D multilayer diffusion that incorporate methods such as spectral decompositions, Greens functions, and Laplace transforms [11, 51, 50, 19, 36, 37, 29, 35, 30, 14, 6, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Almost all studies of multilayer diffusion have focused on macroscopic models in which the relevant field is the concentration of diffusing particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Many of the analytical challenges concern the derivation of time-dependent solutions that charac- terize short-time transients or threshold crossing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This requires either carrying out a non-trivial spectral decomposition of the solution and/or inverting a highly complicated Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In general, it is necessary to develop some form of approximation scheme or to supplement a semi-analytical solution with numerical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' As far as we are aware, single-particle diffusion or Brownian motion ∗Department of Mathematics, University of Utah, Salt Lake City, UT 84112 USA (bressloff@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='edu) 1 (BM) in multilayer media has not been investigated to anything like the same ex- tent, with the possible exception of spatially discrete random walks [40, 47, 39, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' On the other hand, a rigorous probabilistic formulation of 1D diffusion through a single semi-permeable barrier has recently been introduced by Lejay [41] in terms of so-called snapping out BM, see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [1, 42, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Snapping out BM sews together successive rounds of reflected BM that are restricted to either x < 0 or x > 0 with a semi-permeable barrier at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Each round of reflected BM is killed when its local time at x = 0± exceeds an exponentially distributed random variable with constant rate κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (Roughly speaking, the local time at x = 0+ (x = 0−) specifies the amount of time a positively (negatively) reflected Brownian particle spends in a neighborhood of the right-hand (left-hand) side of the barrier [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') It then immediately resumes either negatively or positively reflected BM with equal probability, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We recently reformulated 1D snapping out BM in terms of a renewal equation that relates the full probability density to the probability densities of partially re- flected BMs on either side of the barrier [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (The theory of semigroups and resolvent operators were used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [41] to derive a corresponding backward equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') We established the equivalence of the renewal equation with the corresponding single- particle diffusion equation, and showed how to solve the former using a combination of Laplace transforms and Green’s function methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We subsequently extended the theory to bounded domains and higher spatial dimensions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Formulating interfa- cial diffusion in terms of snapping out BM has at least two useful features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' First, it provides a more general probabilistic framework for modeling semi-permeable mem- branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For example, each round of partially reflected BM on either side of an in- terface could be killed according to a non-Markovian process, along analogous lines to encounter-based models of surface absorption [31, 32, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' That is, partially reflected BM is terminated when its local time at the interface exceeds a random threshold that is not exponentially distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' As we have shown elsewhere, this leads to a time-dependent permeability that tends to be heavy-tailed [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sec- ond, numerical simulations of snapping out BM generate sample paths that can be used to obtain approximate solutions of boundary value problems in the presence of semi-permeable interfaces [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1 In this paper we develop a multi-layered version of snapping out BM and its as- sociated renewal equations for both exponential and non-Markovian killing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In particular, we consider a single particle diffusing in a finite interval [0, L] that is partitioned into m subintervals (or layers) (aj, aj+1), j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m − 1, with a0 = 0, am = L, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The interior interfaces at x = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , am−1 are taken to be semi-permeable barriers with constant permeabilities κj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m − 1, whereas partially reflecting or Robin boundary conditions are imposed at the ends x = 0, L with absorption rates 2κ0 and 2κl, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (The factors of 2 are convenient when formulating snapping out BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') The diffusion coefficient is also heterogeneous with D(x) = Dj for all x ∈ (aj−1, aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We begin in section 2 by writing down the multi-layered diffusion equation, which we formally solve using Laplace transforms and an iterative method based on transfer matrices, following along analogous lines to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [51, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In section 3, we construct the multi-layered version of snapping out BM and write down the corresponding last renewal equation, which relates the full 1An efficient computational schemes for finding solutions to the single-particle diffusion equation in the presence of one or more semi-permeable interfaces has also been developed in terms of under- damped Langevin equations [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' However, this is distinct from snapping out BM, which is an exact single-particle realization of diffusion through an interface in the overdamped limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 2 x = a1 x = 0 x = L x = a m-2 x = a m-1 x = a 2 layer 1 layer 2 layer m-1 layer m Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A 1D layered medium consisting of m layers x ∈ (aj, aj+1), j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1, with a0 = 0 and am = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The interior interfaces at x = aj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1 act as semi-permeable membranes, whereas partially absorbing boundary conditions are imposed on the exterior boundaries at x = 0, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' probability density to the probability densities of partially reflected BM in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, although the details differ significantly from the iterative solution of the diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We also prove that the renewal equation and diffusion equa- tion are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This exploits a subtle feature of partially reflected BM, namely, the Robin boundary condition is modified when the initial position of the particle is on the boundary itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In section 4 we illustrate the theory by analyzing the first passage time (FPT) problem for the particle to escape from one of the ends of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The FPT statistics can be analyzed in terms of the small-s behavior of the Laplace transformed probability fluxes at the ends x = 0, L, where s is the Laplace variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This means that it is sufficient to solve the multi-layer renewal equation in Laplace space, without having to invert the Laplace transformed solution using some form of spectral decomposition, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Finally, in section 5, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter- based method for surface absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Single-particle diffusion equation in a 1D layered medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Before developing the more general renewal approach for single-particle diffusion in the multi- layer domain of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1, it is useful to briefly consider the classical formulation in terms of the diffusion equation with constant permeabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Let ρj(x, t) denote the probability density of the particle position in the j-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For concreteness, we assume that the particle starts in the first layer, that is, x0 ∈ [0, a1], although it is straightforward to adapt the analysis to include more general initial conditions, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (For notational convenience, we drop the explicit dependence of ρj on x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') Single-particle diffusion can be represented by the following piecewise system of partial differential equations (PDEs): ∂ρj ∂t = Dj ∂2ρj ∂x2 , x ∈ (aj−1, aj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1a) Dj ∂ρj(x, t) ∂x ���� x=a− j = Dj+1 ∂ρj+1(x, t) ∂x ���� x=a+ j = κj[ρj+1(a+ j , t) − ρj(a− j , t)], j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1b) D1 ∂ρ1(x, t) ∂x ���� x=0 = 2κ0ρ1(0, t), Dm ∂ρm(x, t) ∂x ���� x=L = −2κmρm(L, t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1c) together with the initial condition ρj(x, t) = δ(x − x0)δj,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Finally, we denote the composite solution on the domain G = ∪m j=1[a+ j−1, a− j ] by ρ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Laplace transforming 3 equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1a)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1c) gives Dj ∂2�ρj ∂x2 − s�ρj = −δ(x − x0)δj,1, x ∈ (aj−1, aj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2a) Dj ∂�ρj(x, s) ∂x ���� x=a− j = Dj+1 ∂�ρj+1(x, s) ∂x ���� x=a+ j = κj[�ρj+1(a+ j , s) − �ρj(a− j , s)], j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2b) D1 ∂�ρ1(x, s) ∂x ���� x=0 = 2κ0�ρ1(0, s), Dm ∂�ρm(x, s) ∂x ���� x=L = −2κm�ρm(L, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2c) Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2a)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2b) can be solved using transfer matrices along similar lines to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [51, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We sketch the steps here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' First, note that for all 1 ≤ j ≤ m, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2a) has the general solution �ρj(x, s) = Al j(s) cosh( � s/Dj[x − aj−1]) + Bl j(s) sinh( � s/Dj[x − aj−1]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3) or, equivalently �ρj(x, s) = Ar j(s) cosh( � s/Dj[x − aj]) + Br j (s) sinh( � s/Dj[x − aj]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4) For 1 < j ≤ m, the coefficients Al j, Bl j are related to Ar j, Br j according to � Ar j Br j � = Uj(s) � Al j Bl j � , Uj(s) = � cosh( � s/DjLj) sinh( � s/DjLj) sinh( � s/DjLj) cosh( � s/DjLj) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5) where Lj = aj − aj−1 is the length of the j-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The presence of the Dirac delta function for j = 1 means that the relationship between the coefficients (Ar 1(s), Br 1(s)) and (Al 1(s), Bl 1(s)) is determined by imposing the continuity condition �ρ1(x+ 0 , s) = �ρ1(x− 0 , s) and the flux discontinuity condition ∂x�ρ1(x+ 0 , s) − ∂x�ρ1(x− 0 , s) = −1/D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This yields the result � Ar 1 Br 1 � = U1(s) � Al 1 Bl 1 � + 1 √sD1 � sinh( � s/D1[x0 − a1]) − cosh( � s/D1[x0 − a1]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6) Given the relationships �ρj(aj, s) = Ar j(s), �ρj(aj−1, s) = Al j(s), Dj∂x�ρj(aj, s) = �sDjBr j (s) and Dj∂x�ρj(aj−1, s) = �sDjBl j(s), the boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2b) can be written in the form � sDjBr j (s) = � sDj+1Bl j+1(s) = κj[Al j+1(s) − Ar j(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7) That is, for 1 ≤ j < m, � Al j+1 Bl j+1 � = Vj(s) � Ar j Br j � , Vj(s) = \uf8eb \uf8ed 1 � sDj/κj 0 � Dj/Dj+1 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) Iterating equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) for m ≥ 2, we have � Ar m Br m � = Mm(s) � Ar 1 Br 1 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9) 4 with M2(s) = U2(s)V1(s), Mm(s) = Um(s) \uf8ee \uf8f0 m−1 � j=2 Vj(s)Uj(s) \uf8f9 \uf8fb V1(s) for m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10) Hence, we have shown how the solution in any layer can be expressed in terms of the two unknown coefficients Al 1(s) and Bl 1(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The latter are then determined by imposing the Robin boundary conditions at x = 0, L: � sD1Bl 1(s) = 2κ0Al 1(s), � sDmBr m(s) = −2κmAr m(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='11) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Snapping out BM in a 1D layered medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We now develop an al- ternative formulation of multi-layer diffusion, which is based on a generalization of 1D snapping out BM for a single semi-permeable interface [41, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In particular, we construct a renewal equation that relates ρ(x, t) on G to the probability densities of partially reflected BM in each of the layers [aj−1, aj], j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Single layer with partially reflecting boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Consider BM in the interval [aj−1, aj] with both ends totally reflecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Let X(t) ∈ [aj−1, aj] denote the position of the Brownian particle at time t and introduce the pair of Brownian local times ℓ+ j−1(t) = lim h→0 Dj h ˆ t 0 H(aj−1 + h − X(τ))dτ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1a) ℓ− j (t) = lim h→0 Dj h ˆ t 0 H(aj − h − X(τ))dτ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1b) where H is the Heaviside function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Note that ℓ+ j−1(t) determines the amount of time that the Brownian particle spends in a neighborhood to the right of x = aj−1 over the interval [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Similarly, ℓ− j (t) determines the amount of time spent in a neighborhood to the left of x = aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (The inclusion of the factor Dj means that the local times have units of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') It can be shown that the local times exist and are nondecreasing, continuous function of t [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The corresponding stochastic differential equation (SDE) for X(t) is given by the Skorokhod equation dX(t) = � 2DjdW(t) + dℓ+ j−1(t) − dℓ− j (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2) Roughly speaking, each time the particle hits one of the ends it is given an impul- sive kick back into the bulk domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It can be proven that the probability density for particle position evolves according to the single-particle diffusion equation with Neumann boundary conditions at both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Partially reflected BM can now be defined by introducing a pair of exponentially distributed independent random local time thresholds �ℓ+ j−1 and �ℓ− j such that P[�ℓ+ j−1 > ℓ] = e−2κj−1ℓ/Dj, P[�ℓ− j > ℓ] = e−2κjℓ/Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3) The stochastic process is then killed as soon as one of the local times exceeds its corresponding threshold, which occurs at the stopping time Tj = min{τ − j , τ + j } with τ + j = inf{t > 0 : ℓ+ j−1(t) > �ℓ+ j−1}, τ − j = inf{t > 0 : ℓ− j (t) > �ℓ− j }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4) 5 local time lj-1 Tj x time t x0 interface aj-1 aj totally reflecting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sketch of a course-grained trajectory of a Brownian particle in the interval [aj−1, aj] with a partially reflecting boundary at x = aj−1 and a totally reflecting boundary at x = aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The particle is absorbed as soon as the time ℓj−1(t) spent in a boundary layer around x = aj−1 exceeds an exponentially distribution threshold �ℓj−1, which occurs at the stopping time Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1 we illustrate the basic construction using a simplified version of partially reflected BM in which x = aj−1 is partially reflecting (0 < κj−1 < ∞) but x = aj is totally reflecting (κj = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It can be shown that the probability density for particle position prior to absorp- tion at one of the ends (see also section 5), pj(x, t|x0)dx = P[x ≤ X(t) < x + dx, t < Tj|X0 = x0], x ∈ [aj−1, aj], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5) satisfies the single-particle diffusion equation (Fokker-Planck equation) with Robin boundary conditions at x = aj−1, aj [25, 48, 45, 5, 28]: ∂pj(x, t|x0) ∂t = Dj ∂2pj(x, t|x0) ∂x2 , aj−1 < x0, x < aj, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6a) Dj∂xpj(aj−1, t|x0) = 2κj−1p(aj−1, t|x0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6b) Dj∂xpj(aj, t|x0) = −2κjp(aj, t|x0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6c) and pj(x, 0|x0) = δ(x − x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It is convenient to Laplace transform with respect to t, which gives Dj ∂2�pj(x, s|x0) ∂x2 − s�pj(x, s|x0) = −δ(x − x0), aj−1 < x0, x < aj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7a) Dj∂x�pj(aj−1, s|x0) = 2κj−1�pj(aj−1, s|x0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7b) Dj∂x�p(aj, s|x0) = −2κj�pj(aj, s|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7c) We can identify �pj(x, s|x0) as the Green’s function of the modified Helmholtz equation with Robin boundary conditions at x = aj−1, aj: �pj(x, s|x0) = \uf8f1 \uf8f2 \uf8f3 AjFj(x, s)F j(x0, s), aj−1 ≤ x ≤ x0 AjFj(x0, s)Fj(x, s), x0 ≤ x ≤ aj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) 6 where Fj(x, s) = � sDj cosh( � s/Dj[x − aj−1]) + 2κj−1 sinh( � s/Dj[x − aj−1]), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9a) Fj(x, s) = � sDj cosh( � s/Dj[aj − x]) + 2κj sinh( � s/Dj[aj − x]), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9b) Aj = 1 �sDj 1 2(κj−1 + κj) � sDj cosh( � s/DjLj) + [sDj + 4κj−1κj] sinh( � s/DjLj) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9c) and Lj = aj − aj−1 is the width of the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It can be checked that the Robin boundary conditions are satisfied at x = aj−1, aj for all aj−1 < x0 < aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' However, for x0 = aj−1, aj, we have Dj∂x�pj(aj−1, s|aj−1) = 2κj−1�p(aj−1, s|aj−1) − 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10a) Dj∂x�pj(aj, s|aj) = −2κj�pj(aj, s|aj) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10b) In other words, lim ǫ→0 � ∂ ∂x ���� x=aj �pj(x, s|aj − ǫ) � ̸= ∂ ∂x ���� x=aj � lim ǫ→0 �pj(x, s|aj − ǫ) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='11) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The modification of the Robin boundary condition when the particle starts at the barrier plays a significant role in establishing the equivalence of snapping out BM with single particle diffusion in a multi-layered medium (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Last renewal equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We now construct snapping out BM in the multi- layered domain shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1 by sewing together multiple rounds of reflected BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For the moment, assume that the exterior boundaries are totally reflecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For each interface we introduce a pair of local time ℓ± j and a corresponding pair of independent exponentially distributed thresholds �ℓ± j with rates 2κj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Suppose that the particle starts at x = x0 in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It realizes positively reflected BM until its local time ℓ− 1 (t) at x = a1 exceeds the random threshold �ℓ− 1 with rate 2κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The process immediately restarts as a new reflected BM with probability 1/2 in either [0, a1] or [a1, a2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' If the particle is in layer 2, then the reflected BM is stopped as soon as one of the local times (ℓ+ 1 (t), ℓ− 2 (t)) exceeds its corresponding threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Each time the BM is restarted all local times are reset to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Finally, taking the exterior boundaries to be partially reflecting, we introduce an additional pair of local times, ℓ0(t), ℓm(t) for the external boundaries at x = 0, L, and a corresponding pair of exponentially distributed random thresholds �ℓ0, �ℓm with rates 2κ0, 2κm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The stochastic process is then permanently terminated at the stopping time T = min{T0, Tm}, Tk = inf{t > 0 : ℓk(t) > �ℓk}, k = 0, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='12) We illustrate the basic construction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2 in the simplified case of a single semi-permeable interface at x = aj and totally reflecting boundaries x = aj−1 and x = aj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The statistics of diffusion across the interface can be captured by sewing together successive rounds of partially reflected BM in the intervals [aj−1, a− j ] and [a+ j , aj+1] with each round killed according to an exponentially distributed local time threshold, and the new domain selected with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 7 x = aj x = aj-1 reflecting x0 (a) Robin x = aj+1 Robin reflecting reflecting reflecting x = aj x = aj-1 x = aj+1 x0 reflecting reflecting x = aj x = aj-1 x = aj+1 (b) (a) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Decomposition of snapping out BM on the interval [aj−1, aj+1] with reflecting bound- ary conditions at the ends and a semi-permeable barrier at x = aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (a) Diffusion across the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (b) Partially reflected BM in [a+ j , aj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (c) Partially reflected BM in [aj−1, a− j ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Consider a general initial probability density φ(x0) with x0 ∈ G and set ρj(x, t) = ˆ G ρj(x, t|x0)φ(x0)dx0, pj(x, t) = ˆ G pj(x, t|x0)φ(x0)dx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='13) Following our previous work on snapping out BM for single semi-permeable interfaces [9, 10], the renewal equation for the j-th interior layer, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m − 1, takes the form ρj(x, t) = pj(x, t) + κj−1 ˆ t 0 pj(x, τ|aj−1)[ρj−1(a− j−1, t − τ) + ρj(a+ j−1, t − τ)]dτ + κj ˆ t 0 pj(x, τ|aj)[ρj(a− j , t − τ) + ρj+1(a+ j , t − τ)]dτ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14a) for all x ∈ (a+ j−1, a− j ), with the probability density pj(x, τ|y) given by the solution to equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The first term pj(x, t) on the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14a) represents all trajectories that reach x at time t without ever being absorbed by the interfaces at x = a+ j−1, a− j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The first integral on the right-hand side sums over all trajectories that were last absorbed (stopped) at time t − τ by hitting the interface at x = aj−1 from either the left-hand or right-hand side and then switching with probability 1/2 to BM in the j-th layer such that it is at position x ∈ (a+ j−1, a− j ) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Since the particle is not absorbed over the interval (t − τ, t], the probability of reaching x is pj(x, τ|aj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In addition, the probability that the last stopping event occurred in the interval (t−τ, t−τ+dτ) irrespective of previous events is 2κj−1dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (We see that the inclusion of the factor 2 in the definition of the permeability cancels the probability factor of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') The second integral has the corresponding interpretation for trajectories that were last stopped by hitting the interface at x = aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In the case of the end layers, we have ρ1(x, t) = p1(x, t) + κ1 ˆ t 0 p1(x, τ|a1)[ρ1(a− 1 , t − τ) + ρ2(a+ 1 , t − τ)]dτ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14b) ρm(x, t) = pm(x, t) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14c) +κm−1 ˆ t 0 pm(x, τ|am−1)[ρm−1(a− m−1, t − τ) + ρm(a+ m−1, t − τ)]dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 8 Note that there is only a single integral contribution in the end layers since only one of the boundaries is semi-permeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' One interesting difference between the renewal equation formulation and the PDE analyzed in section 2 is that the exterior boundary conditions are already incorporated into the solutions p1(x, t|x0) and pm(x, t|x0), so that they do not have to be imposed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Given the fact that the renewal equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14c) are convolutions in time, it is convenient to Laplace transform them by setting �ρj(x, s) = ´ ∞ 0 e−stρj(x, t)dt etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This gives �ρ1(x, s) = �p1(x, s) + κ1�p1(x, s|a1)Σ1(s), x ∈ [0+, a− 1 ], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a) �ρj(x, s) = �pj(x, s) + κj−1�pj(x, s|aj−1)Σj−1(s) + κj �pj(x, s|aj)Σj(s), x ∈ [a+ j−1, a− j ], 1 < j < m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15b) �ρm(x, s) = �pm(x, s) + κm−1�pm(x, s|am−1)Σm−1(s), x ∈ [a+ m−1, L−], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15c) where Σj(s) = �ρj(a− j , s) + �ρj+1(a+ j , s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='16) The functions Σj(s) can be determined self-consistently by setting x = a± k for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m−1 and performing various summations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' More specifically, substituting equa- tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15b) into the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='16) for 1 < j < m gives Σj(s) = Σp j(s) + κj−1�pj(aj, s|aj−1)Σj−1(s) + κj �pj(aj, s|aj)Σj(s) + κj �pj+1(aj, s|aj)Σj(s) + κj+1�pj+1(aj, s|aj+1)Σj+1(s) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17a) for 1 < j < m − 1 and Σp j(s) ≡ �pj(aj, s) + �pj+1(aj, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' On the other hand, equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15b) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a) for j = 2 implies that Σ1(s) = Σp 1(s) + κ1�p1(a1, s|a1)Σ1(s) + κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17b) while equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15c) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a) for j = m − 1 yields Σm−1(s) = Σp m−1(s) + κm−1�pm(am−1, s|am−1)Σm−1(s) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17c) + κm−2�pm−1(am−1, s|am−2)Σm−2(s) + κm−1�pm−1(am−1, s|am−1)Σm−1(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17c) can be rewritten in the more compact matrix form m−1 � k=1 Θjk(s)Σk(s) = −Σp j(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='18) where Θ(s) is a tridiagonal matrix with non-zero elements Θj,j(s) = dj(s) ≡ κj[�pj+1(aj, s|aj) + �pj(aj, s|aj)] − 1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='19a) Θj,j−1(s) = cj(s) ≡ κj−1�pj(aj, s|aj−1), j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='19b) Θj,j−1(s) = bj(s) ≡ κj+1�pj+1(aj, s|aj+1), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='19c) Assuming that the matrix Θ(s) is invertible, we obtain the formal solution Σj(s) = − m−1 � k=1 Θ−1 jk (s)Σp k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='20) 9 Substituting into equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15c) gives �ρj(x, s) = �pj(x, s) − m−1 � k=1 � κj−1�pj(x, s|aj−1)Θ−1 j−1,k(s) + κj �pj(x, s|aj)Θ−1 jk (s) � × [�pj(aj, s) + �pj+1(aj+1, s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='21) An alternative way to solve for Σj(s) is to use transfer matrices analogous to the analysis of the PDE in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For simplicity, suppose that the particle starts in the first layer at a point x0 ∈ [0, a1] so that �pj(x, s) = �p1(x, s|x0)δj,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It follows that equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17c) can be rewritten in the iterative form � Σj Σj+1 � = Wj(s) � Σj−1 Σj � , Wj(s) = \uf8eb \uf8ed 0 1 −cj(s) bj(s) −dj(s) bj(s) \uf8f6 \uf8f8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='22) for 1 < j < m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In particular, � Σm−2 Σm−1 � = N(s) � Σ1 Σ2 � , N(s) = m−2 � k=2 Wk(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='23) with, see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17b), Σ2(s) = − 1 b1(s) (�p1(a1, s|x0) + d1(s)Σ1(s)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='24) Finally, having determined Σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , Σm−1 in terms of Σ1, we can calculate Σ1 by imposing equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17c), after rewriting it in the more compact form Σm−2(s) = −dm−1(s) cm−2(s) Σm−1(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='25) We thus obtain the following self-consistency condition for Σ1: � 1, dm−1(s) cm−2(s) � N(s) � Σ1(s) − 1 b1(s) (�p1(a1, s|x0) + d1(s)Σ1(s)) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='26) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Equivalence of the renewal and diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We now have two alternative methods of solution in Laplace space, one based on the diffusion equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2a)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2c) and the other based on the renewal equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Both methods involve transfer matrices that can be iterated to express the solution in the final layer in terms of the solution in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It is useful to check that the renewal equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15c) are indeed equivalent to the Laplace transformed diffusion equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1a)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (This is simpler than showing that the iterative solutions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') Clearly, the composite density �ρ(x, s) satisfies the diffusion equation in the bulk and the exterior boundary conditions, so we only have to check the boundary conditions across the interior interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' First, differentiating equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15b) for j = 2 with respect to x and setting x = a± 1 gives ∂x�ρ1(a− 1 , s) = ∂x�p1(a1, s|x0) + κ1∂x�p1(a1, s|a1)Σ1(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27a) ∂x�ρ2(a+ 1 , s) = κ1∂x�p2(a1, s|a1)Σ1(s) + κ2∂x�p2(a1, s|a2)Σ2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27b) 10 Imposing the Robin boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6) implies that D1∂x�p1(a1, s|x0) = −2κ1�p(a1, s|x0), D2∂x�p2(a1, s|a2) = 2κ1�p(a1, s|a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' On the other hand, equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10b) yield D1∂x�p1(a1, s|a1) = −2κ1�p(a1, s|a1) + 1, D2∂x�p2(a1, s|a1) = 2κ1�p2(a1, s|a1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Substituting into equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27b), we have D1∂x�ρ1(a− 1 , s) = −2κ1�p1(a1, s|x0) − κ1[2κ1�p1(a1, s|a1) − 1]Σ1(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28a) D2∂x�ρ2(a+ 1 , s) = κ1[2κ1�p2(a1, s|a1) − 1]Σ1(s) + 2κ2κ1�p2(a1, s|a2)Σ2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28b) Subtracting equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28b), and using equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17b) implies that D2∂x�ρ2(a+ 1 , s) − D1∂x�ρ1(a− 1 , s) = 2κ1 � κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s) + �p1(a1, s|x0) + κ1�p1(a1, s|a1)Σ1(s) − Σ1(s) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='29) Similarly, adding equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28b) gives D2∂x�ρ2(a+ 1 , s) + D1∂x�ρ1(a− 1 , s)] = 2κ1 � κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s) − �p1(a1, s|x0) − κ1�p1(a1, s|a1)Σ1(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='30) On the other hand setting x = a± 1 in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15b) for j = 2 shows that �ρ1(a− 1 , s) = �p1(a1, s|x0) + κ1�p1(a1, s|a1)Σ1(s), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='31a) �ρ2(a+ 1 , s) = κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='31b) Hence, we obtain the expected semi-permeable boundary conditions at x = a1, D2∂x�ρ2(a+ 1 , s) = D1∂x�ρ1(a− 1 , s) = κ1[�ρ2(a+ 1 , s) − �ρ1(a− 1 , s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='32) A similar analysis can be carried out at the other interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We have thus established the equivalence of the renewal equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14c) and the Laplace transformed diffusion equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2a)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hence, snapping out BM X(t) on G is the single-particle realization of the stochastic process whose prob- ability density evolves according to the multi-layer diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' First-passage time problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' One of the useful features of working in Laplace space is that one can solve various first passage time problems without having to calculate any inverse Laplace transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We will illustrate this by considering the escape of the Brownian particle from one of the ends at x = 0, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For simplicity, we again assume that the particle starts in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Let Q(x0, t) denote the survival probability that a particle starting at x0 ∈ (0, a1) has not been absorbed at either end over the interval [0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It follows that Q(x0, t) = ˆ L 0 ρ(x, t)dx = m−1 � j=0 ˆ aj+1 aj ρj(x, t)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1) 11 (We drop the explicit dependence of ρ and ρj on the initial position x0 for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') Differentiating both sides of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1) with respect to t and using equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1a)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1c) shows that dQ(x0, t) dt = m � j=1 ˆ aj aj−1 ∂ρj(x, t) ∂t dx = m � j=1 ˆ aj aj−1 Dj ∂2ρj(x, t) ∂x2 dx = m � j=1 Dj �∂ρj(aj, t) ∂x − ∂ρj(aj−1, t) ∂x � = Dm ∂ρm(am, t) ∂x − D1 ∂ρ1(a0, t) ∂t ≡ −Jm(x0, t) − J0(x0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2) We have used flux continuity across each interior interface so that the survival proba- bility decreases at a rate equal to the sum of the outward fluxes at the ends x = 0, L, which are denoted by J0 and JL respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Laplace transforming equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2) and imposing the initial condition Q(x0, 0) = 1 gives s �Q(x0, s) − 1 = − �J0(x0, s) − �JL(x0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3) Assuming that κ0+κm > 0, the particle is eventually absorbed at one of the ends with probability one, which means that limt→∞ Q(x0, t) = lims→0 s �Q(x0, s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hence, �J0(x0, 0)+ �Jm(x0, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Let π0(x0) and πL(x0) denote the splitting probabilities for absorption at x = 0 and x = L, respectively, and denote the corresponding conditional MFPTs by T0(x0) and TL(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It can then be shown that π0(x0) = �J0(x0, 0), πL(x0) = �JL(x0, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4) and π0(x0)T0(x0) = − ∂ ∂s �J0(x0, s) ���� s=0 , πL(x0)TL(x0) = − ∂ ∂s �JL(x0, s) ���� s=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5) Hence, analyzing the statistics of escape from the domain [0, L] reduces to determining the small-s behavior of the solutions ∂x�ρ1(0, s) and ∂x�ρm(L, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We will proceed using the renewal equation approach of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Identical layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A considerable simplification of the iterative equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='22) occurs in the case of identical layers with Dj = D, κj = κ and aj = ja for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) for partially reflected BM is now the same in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' That is, �pj(x, s|x0) = �p(x − (j − 1)a, s|x0 − (j − 1)a) for x, x0 ∈ [aj−1, aj] with �p(x, s|x0) = \uf8f1 \uf8f2 \uf8f3 AF(x, s)F(x0, s), a ≤ x ≤ x0 AF(x0, s)F(x, s), x0 ≤ x ≤ a , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6) F(x, s) = √ sD cosh( � s/D[x − a]) + 2κ sinh( � s/D[x − a]), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7a) F(x, s) = √ sD cosh( � s/D[a − x]) + 2κ sinh( � s/D[a − x]), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7b) A = 1 √ sD 1 4κ √ sD cosh( � s/Da) + [sD + 4κ2] sinh( � s/Da) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7c) 12 In addition equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='22)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='26) for identical layers imply that N(s) = W(s)m−3, W(s) = � 0 1 −1 −g(a, s) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) with g(y, s) ≡ 2κ�p(a, s|y) − 1 κ�p(a, s|0) = 2g0(y, s) − g1(s), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9) where g0(y, s) ≡ �p(a, s|y) �p(a, s|0) = √ sD cosh( � s/Dy) + 2κ sinh( � s/Dy) √ sD , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10a) g1(s) ≡ 1 κ�p(a, s|0) = 4κ √ sD cosh( � s/Da) + [sD + 4κ2] sinh( � s/Da) κ √ sD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10b) The matrix W(s) can be diagonalized according to W(s) = UWd(s)U†, Wd(s) = diag(λ+(s), λ−(s)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='11) with λ±(s) = −g(a, s) ± � g(a, s)2 − 4 2 , λ+ + λ− = −g, λ+λ− = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='12) and U = � 1 1 λ+ λ− � , U† = � 1 1−λ2 + − λ+ 1−λ2 + 1 1−λ2 − − λ− 1−λ2 − � , U†U = UU† = � 1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='13) Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='11) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='23) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='26) gives � 1, g(a, s) � U(s)Wd(s)m−3U†(s) � Σ1(s) Σ2(s) � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14) and Σm−1(s) = � 0, 1 � U(s)Wd(s)m−3U†(s) � Σ1(s) Σ2(s) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15) with Σ2(s) = −g0(x0, s) κ − g(a, s)Σ1(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='16) In addition, from equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15c) we have �J0(x0, s) = f(x0, s) + κf(a, s)Σ1(s), �JL(x0, s) = κf(a, s)Σm−1(s), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17) where D∂x�p(0, s|y) = f(y, s) ≡ 2κ[ √ sD cosh( � s/D[a − y]) + 2κ sinh( � s/D[a − y])] 4κ √ sD cosh( � s/Da) + [sD + 4κ2] sinh( � s/Da) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='18) 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9 1 κ = 100 κ = 10 κ = 1 κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1 splitting probability initial position x0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Splitting probabilities for escape from a three-layer, homogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Plots of π0(x0) and πL(x0) as a function of x0 for various rates κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Other parameters are D = 1 and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' and D∂x�p(L, s|L − a) = −f(a, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For the sake of illustration, consider three layers (m = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14) implies that for κ > 0 Σ1(s) = 1 κ g(a, s)g0(x0, s) 1 − g(a, s)2 , Σ2(s) = − 1 κ g0(x0, s) 1 − g(a, s)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='19) Using the limits lim s→0 g0(y, s) = 1 + 2κy/D, lim s→0 g1(s) = 4(1 + κa/D), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='20) lim s→0 g(y, s) = −2(1 + 2κ[a − y]/D), lim s→0 f(y, s) = (1 + 2κ[a − y]/D) 2(1 + κa/D) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='21) we can thus determine the splitting probabilities π0(x0) and πL(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Example plots of π0(x0) and πL(x0) as a function of x0 ∈ [0, a] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2 for a = D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It can be checked that π0(x0) + πL(x0) = 1 for all x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Moreover, in the limit κ → ∞, we see that π0(0) → 1 and πL(0) → 0 as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Also note that for x0 < 1/2 (x0 > 1/2), π0(x0) is an increasing (a decreasing) function of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Large number of layers (m → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For a large number of layers (m ≫ 1) we have Wm−3 d = � λm−3 + 0 0 λm−3 − � = λm−3 − � ǫ 0 0 1 � , ǫ = �λ+ λ− �m−3 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='22) with |ǫ| ≪ 1 since |λ−| > |λ+|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It follows that N(s) = U(s)Wd(s)m−3U†(s) = λ−(s)m−3{M0(s) + ǫM1(s)}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='23) where M0 = 1 1 − λ2 − � 1 −λ− λ− −λ2 − � , M1 = 1 1 − λ2 + � 1 −λ+ λ+ −λ2 + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='24) 14 The next step is to introduce the series expansions Σj(s) = Σ(0) j (s) + ǫΣ(1) j (s) + O(ǫ2), j = 1, 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='25) with Σ(0) 2 (s) = −g0(x0, s) κ − g(a, s)Σ(0) 1 (s), Σ(n) 2 (s) = −g(a, s)Σ(n) 1 (s) for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='26) Substituting equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='25) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14) and collecting terms in powers of ǫ gives the O(1) and O(ǫ) equations � 1, g(a, s) � M0(s) � Σ(0) 1 (s) Σ(0) 2 (s) � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27a) � 1, g(a, s) � � M0(s) � Σ(1) 1 (s) Σ(1) 2 (s) � + M1(s) � Σ(0) 1 (s) Σ(0) 2 (s) �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27b) Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27a) has the solution Σ(0) 1 (s) = − λ−(s)g0(x0, s) κ(1 + g(a, s)λ−(s)) = g0(x0, s) κλ−(s) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28) so that Σ(1) 1 (s) = λ+(s)4 λ−(s)4 1 − λ+(s)2 1 − λ−(s)2 � Σ(0) 1 − g0(x0, s) κλ+(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='29) Finally, Σm−1(s) = (0, 1)N(s) � Σ1(s) Σ2(s) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='30) = λ−(s)m−3(0, 1){M0(s) + ǫM1(s)} � Σ(0) 1 (s) + ǫΣ(1) 1 (s) + O(ǫ2) Σ(0) 2 (s) + ǫΣ(1) 1 (s) + O(ǫ2) � = λ+(s)m−3(0, 1) � M0(s) � Σ(1) 1 (s) Σ(1) 2 (s) � + M1(s) � Σ(0) 1 (s) Σ(0) 2 (s) � + O(ǫ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We have used the fact the O(1) solution (Σ(0) 1 , Σ(0) 2 )⊤ is actually a null-vector of the matrix M0 so the leading contribution to Σm−1(s) is proportional to ǫλ−(s)m−3 = λ+(s)m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hence, Σm−1(s) → 0 as m → ∞ due to the fact that |λ+(s)| < 1 for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17) then imply that πm(x0) → 0 as m → ∞, with the rate of decay determined by λ+(0)m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Generalized model of multi-layer diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The analysis of the FPT problem in section 4 could also have been carried out using the solution of the diffusion equation constructed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' However, one advantage of the renewal approach is that it is based on snapping out BM, which can be used to generate sample paths of single-particle diffusion in a multi-layer medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rather than exploring numerical aspects here, we consider another advantage of the renewal approach, namely, it supports a more general model of semi-permeable membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This is based on an extension of snapping out BM that modifies the rule for killing each round of reflected BM within a layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We proceed by applying the encounter-based model of absorption [31, 32, 7, 8] to reflected BM in each of the layers separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Local time propagator for a single layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' As we mentioned in sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1, partially reflected BM in an interval can be implemented by introducing exponentially distributed local time thresholds at either end of the interval, which then determine when reflected BM is killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Here we generalize the killing mecha- nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Given the local times (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1b) of the j-th layer with totally reflecting boundaries, the local time propagator is defined according to [31] Pj(x, ℓ, ℓ′, t|x0)dx dℓ ℓ′ = P[x < X(t) < x + dx, ℓ < ℓ+ j−1 < ℓ + dℓ, ℓ′ < ℓ− j < ℓ′ + dℓ′|X(0) = x0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1) Next, for each interface we introduce a pair of independent identically distributed random local time thresholds �ℓ± j such that P[�ℓ± j > ℓ] ≡ Ψ± j (ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The special case of exponential distributions is given by equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The stochastic process in the j-th layer is then killed as soon as one of the local times ℓ+ j−1 and ℓ− j exceeds its corresponding threshold, which occurs at the FPT time Tj = min{τ + j , τ − j }, see equa- tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Since the corresponding local time thresholds �ℓ+ j−1 and �ℓ− j are statistically independent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' the relationship between the resulting probability density pj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' t|x0) for partially reflected BM in the j-th layer and Pj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ℓ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ℓ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' t|x0) can be established as follows: pj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' t|x0)dx = P[X(t) ∈ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' x + dx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' t < Tj|X0 = x0] = P[X(t) ∈ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' x + dx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ℓ+ j−1(t) < �ℓ+ j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ℓ− j (t) < �ℓ− j |X0 = x0] = ˆ ∞ 0 dℓψ+ j−1(ℓ) ˆ ∞ 0 dℓ′ψ− j (ℓ′)P[X(t) ∈ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' x + dx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ℓ+ j−1 < ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ℓ− j < ℓ′|X0 = x0] = ˆ ∞ 0 dℓψ+ j−1(ℓ) ˆ ∞ 0 dℓ′ψ− j (ℓ′) ˆ ℓ 0 dˆℓ ˆ ℓ′ 0 dˆℓ′[Pj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ˆℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ˆℓ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' t|x0)dx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We have also introduced the probability densities ψ± j (ℓ) = −∂ℓΨ± j (ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Reversing the orders of integration yields the result pj(x, t|x0) = ˆ ∞ 0 dℓΨ+ j−1(ℓ) ˆ ∞ 0 dℓ′Ψ− j (ℓ′)Pj(x, ℓ, ℓ′, t|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2) An evolution equation for the local time propagator can be derived as follows [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Since the local times only change at the boundaries x = aj−1, aj, the propagator satisfies the diffusion equation in the bulk of the domain ∂Pj ∂t = Dj ∂2Pj ∂x2 , x ∈ (aj−1, aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3) The nontrivial step is determining the boundary conditions at x = aj−1, aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Here we give a heuristic derivation based on a boundary layer construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For concreteness, consider the left-hand boundary layer [aj−1, aj−1 + h] and define ℓh j−1(t) = Dj h ˆ t 0 �ˆ h 0 δ(Xt′ − x)dx � dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4) By definition, hℓh j−1(t)/Dj is the residence or occupation time of the process X(t) in the boundary layer up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Although the width h and the residence time 16 in the boundary layer vanish in the limit h → 0, the rescaling by 1/h ensures that limh→0 ℓh j−1(t) = ℓ+ j−1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Moreover, from conservation of probability, the flux into the boundary layer over the residence time hδℓ/Dj generates a corresponding shift in the probability Pj within the boundary layer from ℓ → ℓ + δℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' That is, for ℓ > 0, −Jj(aj−1 + h, ℓ, ℓ′, t|x0)hδℓ = [Pj(aj−1, ℓ + δℓ, ℓ′, t|x0) − Pj(aj−1, ℓ, ℓ′, t|x0)]h, where Jj(x, ℓ, ℓ′, t|x0) = −D∂xPj(x, ℓ, ℓ′, t|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Dividing through by hδℓ and taking the limits h → 0 and δℓ → 0 yields −Jj(aj−1, ℓ, ℓ′, t|x0) = ∂ℓPj(aj−1, ℓ, ℓ′, t|x0), ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Moreover, when ℓ = 0 the probability flux Jj(aj−1, 0, ℓ′, t|x0) is identical to that of a Brownian particle with a totally absorbing boundary at x = aj−1, which we denote by Jj,∞(aj−1, ℓ′, t|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In addition, it can be shown that Pj(aj−1, 0, ℓ′, t|x0) = −Jj,∞(aj−1, ℓ′, t|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Applying a similar argument at the end x = aj, we obtain the pair of boundary conditions D∂xPj(aj−1, ℓ, ℓ′, t|x0) = −Pj(aj−1, 0, ℓ′, t|x0)δ(ℓ) + ∂Pj(aj−1, ℓ, ℓ′, t|x0) ∂ℓ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5a) −D∂xPj(aj, ℓ, ℓ′, t|x0) = −Pj(aj, ℓ, 0, t|x0)δ(ℓ′) + ∂Pj(aj, ℓ, ℓ′, t|x0) ∂ℓ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5b) The crucial step in the encounter-based approach is to note that for exponentially distributed local time thresholds, see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3), the right-hand side of equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2) reduces to a double Laplace transform of the local time propagator: pj(x, t|x0) = Pj(x, z+ j−1, z− j , t|x0), z+ j−1 = 2κj−1 Dj , z− j = 2κj Dj , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6) with Pj(x, z, z′, t|x0) ≡ ˆ ∞ 0 dℓe−zℓ ˆ ∞ 0 dℓ′e−z′ℓ′Pj(x, ℓ, ℓ′, t|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='7) Laplace transforming the propagator boundary conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5a) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='5b) then shows that the probability density pj of equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6) is the solution to the Robin BVP given by equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hence, the probability density of partially reflected BM in the j-th layer is equivalent to the doubly Laplace transformed local time propagator with the pair of Laplace variables z+ j−1 and z− j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Assuming that the Laplace transforms can be inverted, we can then incorporate non-exponential proba- bility distributions Ψ+ j−1(ℓ) and Ψ− j (ℓ′) such that the corresponding marginal density is now pj(x, t|x0) = ˆ ∞ 0 dℓΨ+ j−1(ℓ) ˆ ∞ 0 dℓ′Ψ− j (ℓ′)L−1 ℓ L−1 ℓ′ Pj(x, z, z′, t|x0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) where L−1 denotes the inverse Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' One major difference from the exponential case is that the stochastic process X(t) is no longer Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Killing time densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In order to sew together successive rounds of re- flected BM in the case of general distributions Ψj we will need the conditional FPT densities f + j−1(x0, t) and f − j (x0, t) for partially reflected BM in the j-th layer to be killed at the ends x = aj−1 and x = aj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The corresponding conditional 17 killing times were defined in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The FPT densities are given by the outward probability fluxes at the two ends: f + j−1(x0, t) = Dj∂xpj(aj−1, t|x0), f − j (x0, t) = −Dj∂xpj(aj, t|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9) As in previous sections, it is convenient to Laplace transform with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Laplace transforming equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) and using the Green’s function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8) gives �pj(x, s|x0) = ˆ ∞ 0 dℓΨ+ j−1(ℓ) ˆ ∞ 0 dℓ′Ψ− j (ℓ′)L−1 ℓ L−1 ℓ′ �Pj(x, z, z′, s|x0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10) where �Pj(x, z, z′, s|x0) = \uf8f1 \uf8f2 \uf8f3 Aj(z, z′, s)Fj(x, z, s)Fj(x0, z′, s), aj−1 ≤ x ≤ x0, Aj(z, z′, s)Fj(x0, z, s)Fj(x, z′, s), x0 ≤ x ≤ aj, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='11) with Fj(x, z, s) = � s/Dj cosh( � s/Dj[x − aj−1]) + z sinh( � s/Dj[x − aj−1]), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='12a) Fj(x, z′, s) = � s/Dj cosh( � s/Dj[aj − x]) + z′ sinh( � s/Dj[aj − x]), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='12b) Aj = 1 � sDj 1 (z + z′) � s/Dj cosh( � s/DjLj) + [s/Dj + zz′] sinh( � s/DjLj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='12c) Since �Pj(x, z, z, s|x0) satisfies the Robin boundary conditions Dj∂x �Pj(aj−1, z, z′, s|x0) = Djz �Pj(aj−1, z, z′, s|x0), Dj∂x �Pj(aj, z, z′, s|x0) = −Djz′ �Pj(aj, z, z′, s|x0), it follows that �f + j−1(x0, s) ≡ Dj∂x�pj(aj−1, s|x0) = Dj ˆ ∞ 0 dℓΨ+ j−1(ℓ) ˆ ∞ 0 dℓ′Ψ− j (ℓ′) � ∂ℓ �Pj(aj−1, ℓ, ℓ′, s|x0) + �Pj(aj−1, 0, ℓ′, s|x0) � = Dj ˆ ∞ 0 dℓψ+ j−1(ℓ) ˆ ∞ 0 dℓ′Ψ− j (ℓ′) �Pj(aj−1, ℓ, ℓ′, s|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='13) Similarly, �f − j (x0, s) ≡ −Dj∂x�pj(aj, s|x0) = Dj ˆ ∞ 0 dℓΨ+ j−1(ℓ) ˆ ∞ 0 dℓ′ψ− j (ℓ′) �Pj(aj, ℓ, ℓ′, s|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14) Evaluation of the FPT densities reduces to the problem of calculating the prop- agator �Pj(ak, ℓ, ℓ′, s|x0) by inverting the double Laplace transform �Pj(ak, z, z′, s|x0) with respect to z and z′, k = j − 1, j, and then evaluating the double integrals in equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In general, this is a non-trivial calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' However, a 18 major simplification occurs if we take one of the densities Ψ+ j−1 or Ψ− j to be an expo- nential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' First suppose that Ψ+ j−1(ℓ) = e−2κj−1ℓ/Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We then have a Robin boundary condition at x = aj−1, �f + j−1(x0, s) = 2κj−1�pj(aj−1, s|x0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='15) whereas �f − j (x0, s) = Dj ˆ ∞ 0 dℓ′ψ− j (ℓ′) �Pj(aj, z+ j−1, ℓ′, s|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='16) From equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10) we find that �Pj(aj, zj, z′, s|x0) = 1 Dj Λj(x0, s) z′ + hj(s), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17) where Λj(x0, s) = � s/Dj cosh( � s/Dj[x0 − aj−1]) + z+ j−1 sinh( � s/Dj[x0 − aj−1]) � s/Dj cosh( � s/DjLj) + z+ j−1 sinh( � s/DjLj , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='18) and hj(s) = � s/Dj � s/Dj tanh( � s/DjLj) + z+ j−1 � s/Dj + z+ j−1 tanh( � s/DjLj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='19) Inverting the Laplace transform with respect to z′ then gives �Pj(aj, z+ j−1, ℓ′, s|x0) = D−1 j Λj(x0, s)e−hj(s)ℓ′ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='20) and, hence, �f − j (x0, s) = Λj(x0, s) �ψ− j (hj(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='21) On the other hand, �pj(aj, s|x0) = D−1 j Λj(x0, s)�Ψ− j (hj(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='22) We thus obtain the following boundary condition at x = aj: �f − j (x0, s) = �K− j (s)�pj(aj, s|x0), �K− j (s) = Dj �ψ− j (hj(s)) �Ψ− j (hj(s)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='23) Finally, using the convolution theorem, the boundary condition at x = aj in the time domain takes the form Dj∂xpj(aj, t|x0) = − ˆ t 0 K− j (τ)pj(aj, t − τ|x0)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='24) That is, in the case of a non-Markovian density for killing partially reflected BM at one end of an interval, the corresponding boundary condition involves an effective time-dependent absorption rate K− j (t), which acts as a memory kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 19 Now suppose that Ψ− j (ℓ) = e−2κjℓ/Dj so that �f − j (x0, s) = 2κj�pj(aj, s|x0), �f + j−1(x0, s) = Dj ˆ ∞ 0 dℓψ+ j−1(ℓ) �Pj(aj−1, ℓ, z− j , s|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='25) From equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10) we have �Pj(aj, z, z− j , s|x0) = 1 Dj Λj(x0, s) z + hj(s), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='26) where Λj(x0, s) = � s/Dj cosh( � s/Dj[aj − x0]) + z− j sinh( � s/Dj[aj − x0]) � s/Dj cosh( � s/DjLj) + z− j sinh( � s/DjLj , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='27) and hj(s) = � s/Dj � s/Dj tanh( � s/DjLj) + z− j � s/Dj + z− j tanh( � s/DjLj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28) Using identical arguments to the previous case we find that the boundary condition at x = aj−1 is �f + j−1(x0, s) = �K+ j−1(s)�pj(aj−1, s|x0), �K+ j−1(s) = Dj �ψ+ j−1(hj(s)) �Ψ+ j−1(hj(s)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='29) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Generalized snapping out BM and the first renewal equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We now define a generalized snapping out BM by sewing together successive rounds of reflected BM along identical lines to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2, except that now each round is killed according to the general process introduced in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (For simplicity, we assume that the exterior boundaries at x = 0, L are totally reflecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=') Although each round of partially reflected Brownian motion is non-Markovian, all history is lost following absorption and restart so that we can construct a renewal equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' However, it is now more convenient to use a first rather than a last renewal equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Again we consider a general probability density φ(x0) of initial conditions x0 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Let f + j−1(t) and f − j (t) denote the conditional FPT densities for partially reflected BM in the j-th layer to be killed at the end x = aj−1 and x = aj, respectively, in the case of a general initial distribution φ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It follows that f + j−1(t) = ˆ aj aj−1 f + j−1(x0, t)φ(x0)dx0 = Dj ˆ aj aj−1 ∂xpj(aj−1, t|x0)φ(x0)dx0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='30a) f − j (t) = ˆ aj aj−1 f − j (x0, t)φ(x0)dx0 = −Dj ˆ aj aj−1 ∂xpj(aj, t|x0)φ(x0)dx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='30b) with f + j−1(x0, t) and f − j (x0, t) defined in equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We also set f + 1 (t) ≡ 0 and f − m(t) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Generalizing previous work [9, 10], the first renewal equation in the j-th layer, 1 ≤ j ≤ m, takes the form ρj(x, t) ≡ ˆ G ρj(x, t|x0)φ(x0)dx0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='31) = pj(x, t) + 1 2 m−1 � k=1 ˆ t 0 [ρj(x, t − τ|a− k ) + ρj(x, t − τ|a+ k )][f − k (τ) + f + k (τ)]dτ 20 for x ∈ (aj−1, aj) and pj(x, t) = ˆ aj aj−1 pj(x, t|x0)φ(x0)dx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='32) The first term on the right-hand side of equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='31) represents all sample trajecto- ries that start in the k-th layer and have not been absorbed at the ends x = ak−1, ak up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The integral term represents all trajectories that were first absorbed (stopped) at a semi-permeable interface at time τ and then switched to either posi- tively or negatively reflected BM state with probability 1/2, after which an arbitrary number of switches can occur before reaching x ∈ (aj−1, aj) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The probability that the first stopping event occurred at the k-th interface in the interval (τ, τ + dτ) is [f + k (τ) + f − k (τ)]dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Laplace transforming the renewal equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='31) with respect to time t gives �ρj(x, s) = �pj(x, s) + 1 2 m−1 � k=1 [�ρj(x, s|a− k ) + �ρj(x, s|a+ k )][ �f − k (s) + �f + k (s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='33) In order to determine the factors Σjk(x, s) = �ρj(x, s|a− k ) + �ρj(x, s|a+ k ), 1 ≤ k < m, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='34) we substitute into equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='33) the initial density φ(x0) = 1 2[δ(x0−a− k )+δ(x0−a+ k )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This gives Σjk(x, s) = �pj(x, s|ak)[δj,k + δj,k+1] + 1 2Σjk(x, s)[ �f − k (a− k , s) + �f + k (a+ k , s)] + 1 2Σj,k−1(x, s) �f + k−1(a− k , s) + 1 2Σj,k+1(x, s) �f − k+1(a+ k , s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='35) Comparison with equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='17c) implies that the above equation can be rewritten in the matrix form m−1 � l=1 Θkl(s)Σjl(s) = −�pj(x, s|ak)[δj,k + δj,k+1], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='36) where Θ(s) is a tridiagonal matrix with non-zero elements Θk,k(s) = dk(s) ≡ �f − k (a− k , s) + �f + k (a+ k , s) − 1, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='37a) Θk,k−1(s) = ck(s) ≡ �f + k−1(a− k , s), k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' m − 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='37b) Θk,k+1(s) = bk(s) ≡ �f − k+1(a+ k , s), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' , m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='37c) Assuming that the matrix Θ(s) is invertible, we obtain the formal solution Σjk(x, s) = −Θ −1 kj (s)�pj(x, s|aj) − Θ −1 k,j−1(s)�pj(x, s|aj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='38) Substituting into equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='33) yields the result �ρj(x, s) = �pj(x, s) + 1 2 m−1 � k=1 [Θ −1 kj (s)�pj(x, s|aj) + Θ −1 k,j−1(s)�pj(x, s|aj−1)] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='39) × [ �f − k (s) + �f + k (s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 21 Equivalence of first and last renewal equations for exponential killing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' An im- portant check of our analysis is to show that the solution (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='39) of the first renewal equation is equivalent to the solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='21) of the last renewal equation when each round of reflecting BM is killed according to an independent exponential distribu- tion for each local time threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Since �pj(x, s|x0) then satisfies Robin boundary conditions at x = aj−1, aj we find that Θk,k(s) = κk[�pk+1(ak, s|ak) + �pk(ak, s|ak)] − 1 = Θkk(s), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='40a) Θk,k−1(s) = κk−1�pk(ak−1, s|ak) = κk−1�pk(ak, s|ak−1) = Θk,k−1(s), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='40b) Θk,k+1(s) = κk+1�pk+1(ak+1, s|ak) = κk+1�pk+1(ak, s|ak+1) = Θk,k+1(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='40c) We have used two important properties of partially reflected BM: i) Symmetry of the Green’s function �p(x, s|x0) = �p(x0, s|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' ii) The solution for the functions Σjk(x, s) is obtained by introducing the initial con- ditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The FPT densities are thus evaluated at the initial points a± k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This means that when we impose the Robin boundary conditions we do not pick up the additional constant term in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' It follows that the solution (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='39) reduces to the form �ρj(x, s) = �pj(x, s) + m−1 � k=1 � �pj(x, s|aj−1)Θ−1 k,j−1(s) + �pj(x, s|aj)Θ−1 kj (s) � × κk[�pk(ak, s) + �pk+1(ak, s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='41) Finally, using the fact that κkΘ−1 kj (s) = κjΘ−1 jk (s), we recover the solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Non-exponential killing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The above analysis shows that the same solution struc- ture holds for both exponential and non-exponential killing, provided that we ex- press the tridiagonal matrix Θij in terms of the conditional FPT densities �f ± k (a± k , s), �f + k−1(a− k , s) and �f − k+1(a+ k , s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The latter are themselves determined from equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' One configuration that is analytically tractable is a 1D domain with a sequence of semi-permeable barriers whose distributions Ψ± j alternate be- tween exponential and non-exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For example, suppose Ψ− j (ℓ) = e−2κj/Dj and Ψ+ j (ℓ) = e−2κj/Dj+1 for all odd layers j = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=', whereas Ψ± j (ℓ) are non-exponential for even layers j = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='. Combining the analysis of the FPT densities in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2 with the analysis of the first renewal equation and its relationship with the last renewal equation, we obtain the following generalization of the interfacial boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='2b): Dj∂xρj(a− j , s) = Dj+1∂xρj+1(a+ j , s) = 1 2[ �K+ j (s)ρj+1(a+ j , s) − �K− j (s)ρj(a− j , s)], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='42) with �K± j (s) = 2κj for odd j and �K− j (s) = Dj �ψ− j (hj(s)) �Ψ− j (hj(s)) , �K+ j (s) = Dj+1 �ψ+ j (hj+1(s)) �Ψ+ j (hj+1(s)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='43) for even j, with hj(s) and hj+1(s) given by equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='19) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='28), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We thus have the setup shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Note, in particular, that the time-dependent permeability kernels of the even interfaces are asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 22 x = a1 x = 0 x = a4 x = a2 x = a3 K2-(t) κ1 κ1 K2+(t) K4-(t) κ2 κ2 K4+(t) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A 1D layered medium partitioned by a sequence of semi-permeable interfaces that alternate between symmetric constant permeabilities κj, j = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' and asymmetric time-dependent permeabilities K± j (t), j = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='. Permeability kernels for the gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For the sake of illustration, sup- pose that ψ± j (ℓ) for even j are given by the gamma distributions ψ± j (ℓ) = z± j (z± j ℓ)µ−1e−z± j ℓ Γ(µ) , µ > 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='44) where Γ(µ) is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The corresponding Laplace transforms are �ψ± j (z) = � z± j z± j + z �µ , �Ψ± j (z) = 1 − �ψ± j (z) z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='45) If µ = 1 then ψ± j reduce to the exponential distributions with constant reactivity κj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The parameter µ thus characterizes the deviation of ψ± j (ℓ) from the exponential case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' If µ < 1 (µ > 1) then ψ± j (ℓ) decreases more rapidly (slowly) as a function of the local time ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Substituting the gamma distributions into equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='43) yields �K− j (s) = Djhj(s)(z− j )µ [z− j + hj(s)]µ − (z− j )µ , �K+ j (s) = Dj+1hj+1(s)(z+ j )µ [z+ j + hj+1(s)]µ − (z+ j )µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='46) If µ = 1 then �K± j (s) = 2κj as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' On the other hand if µ = 2, say, then �K− j (s) = 2κj 2 + Djhj(s)/2κj , �K+ j (s) = 2κj 2 + Dj+1hj+1(s)/2κj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='47) The corresponding time-dependent kernels K± j (t) are normalizable since ˆ ∞ 0 K− j (t)dt = �K− j (0) = 2κj 2 + κ−1 j κj−1/[1 + 2κj−1Lj/Dj), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='48a) ˆ ∞ 0 K+ j (t)dt = �K+ j (0) = 2κj 2 + κ−1 j κj+1/[1 + 2κj+1Lj/Dj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='48b) However, the kernels are heavy-tailed with infinite moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For example, ⟨t⟩− ≡ 1 �K− j (0) ˆ ∞ 0 tK− j (t)dt = − 1 �K− j (0) lim s→0 ∂s �K− j (s) = 1 �K− j (0) lim s→0 Djh′ j(s)/2 [2 + Djhj(s)/2κj]2 = Dj 4κj �K− j (0) 2κj lim s→0 h′ j(s) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='49) That is, all moments are infinite since all derivatives of hj(s) are singular at s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' An analogous result was previously found for a single interface in 1D and 3D [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In this paper we developed a probabilistic framework for analyz- ing single-particle diffusion in heterogeneous multi-layered media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Our approach was based on a multi-layered version of snapping out BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We showed that the distribution of sample trajectories satisfied a last renewal equation that related the full probabil- ity density to the probability densities of partially reflected BM in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The renewal equation was solved using a combination of Laplace transforms and transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We also proved the equivalence of the renewal equation and the corre- sponding multi-layered diffusion equation in the case of constant permeabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We then used the renewal approach to incorporate a more general probabilistic model of semipermeable interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' This involved killing each round of partially reflected BM according to a non-Markovian encounter-based model of absorption at an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We constructed a corresponding first renewal equation that related the full probability density to the FPT densities for killing each round of reflected BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In particular, we showed that non-Markovian models of absorption can generate asymmetric, heavy- tailed time-dependent permeabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In developing the basic mathematical framework, we focused on relatively simple examples such as identical layers with constant permeabilities or alternating Marko- vian and non-Markovian interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' We also restricted our analysis to the Laplace domain rather than the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' However, it is clear that in order to apply the theory more widely, it will be necessary to develop efficient numerical schemes for solving the last or first renewal equations in Laplace space, and then inverting the Laplace transformed probability density to obtain the solution in the time do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In the case of non-Markovian models of absorption at both ends of a layer, it will also be necessary to compute the double inverse Laplace transform of the local time propagator and evaluate the resulting double integral in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Another computational issue is developing an efficient numerical scheme for simulating sample trajectories of snapping out BM in heterogeneous multi-layer media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Finally, from a modeling perspective, it would be interesting to identify plausible biophysical mechanisms underlying non-Markovian models of semi-permeable mem- branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' As previously highlighted within the context of encounter-based models of absorption [31, 32, 7, 8], various surface-based reactions are better modeled in terms of a reactivity that is a function of the local time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' For example, the surface may become progressively activated by repeated encounters with a diffusing particle, or an initially highly reactive surface may become less active due to multiple interactions with the particle (passivation) [4, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' REFERENCES [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Aho, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mattila, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' K¨uhn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Kek¨al¨ainen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Pulkkine, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Minussi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Vihinen- Ranta and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Timonen Diffusion through thin membranes: Modeling across scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' E 93 (2016) 043309 [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Alemany, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rose, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Garnier-Brun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Scott and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Doorly Random walk dif- fusion simulations in semi-permeable layered media with varying diffusivity Science Reports 12 (2022) 10759 [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Barbaro, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Giaconia and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Orioli A computer oriented method for the analysis of non steady state thermal behaviour of buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 23 (1988) 19-24 [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Bartholomew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mechanisms of catalyst deactivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A: Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 212 (2001) 17-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Borodin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Salminen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Handbook of Brownian Motion: Facts and Formulae Birkhauser Verlag, Basel-Boston-Berlin (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Bressloff Diffusion in cells with stochastically-gated gap junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 76 (2016) 1658-1682 24 [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Bressloff Diffusion-mediated absorption by partially reactive targets: Brownian function- als and generalized propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 55 (2022) 205001 [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Bressloff Spectral theory of diffusion in partially absorbing media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A 478 (2022) 20220319 [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Bressloff A probabilistic model of diffusion through a semipermeable barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A 478 (2022) 20220615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Bressloff Renewal equation for single-particle diffusion through a semipermeable inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' In press (2023) [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Brink and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Ramanan A model for the diffusion of fluorescent probes in the sep- tate giant axon of earthworm: axoplasmic diffusion and junctional membrane permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 48 (1985) 299-309 [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Bobrowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Semigroup-theoretic approach to diffusion in thin layers separated by semi- permeable membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 21 (2021) 1019-1057 [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Callaghan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Coy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Halpin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' MacGowan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Packer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Zelaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diffusion in porous systems and the influence of pore morphology in pulsed gradient spin- echo nuclear magnetic resonance studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 97 (1992) 651-662 [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Carr and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Turner A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 40 (2016) 7034-7050 [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Connors and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Long Electrical synapses in the mammalian brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Neurosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 27 (2004) 393-418 [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Coy and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Callaghan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Pulsed gradient spin echo nuclear magnetic resonance for molecules diffusing between partially reflecting rectangular barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 101 (1994) 4599-4609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' deMonte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Transient heat conduction in one-dimensional composites lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A natural analytic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 43 (2000) 3607-3619 [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diard, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Glandut, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Montella and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sanchez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' One layer, two layers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' An introduction to the EIS study of multilayer electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Part 1: Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Electroanal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 578 (2005) 247-257 [19] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Dudko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Berezhkovskii and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Weiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diffusion in the presence of periodically spaced permeable membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 121 (2004) 11283 [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Evans and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Martin Gap junctions: structure and function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Membr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 19 (2002) 121-136 [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Regev and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Farago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Application of underdamped Langevin dynamics simulations for the study of diffusion from a drug-eluting stent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A, Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 507 (2018) 231-239 [22] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Farago Algorithms for Brownian dynamics across discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 423 (2020) 109802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Filoche, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grebenkov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Andrade and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sapoval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Passivation of irregular surfaces accessed by diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 105 (2008) 7636-7640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [24] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Freger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diffusion impedance and equivalent circuit of a multilayer film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Electrochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 7 (2005) 957-961 [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Freidlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Functional Integration and Partial Differential Equations Annals of Mathematics Studies, Princeton University Press, Princeton (1985) New Jersey [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Goodenough and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Paul Gap junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Cold Spring Harb Perspect Biol 1 (2009) a002576 [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Graff, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Williford and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Burrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mechanisms of vapor permeation through multilayer barrier films: lag time versus equilibrium permeation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 96 (2004) 1840-1849 [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grebenkov Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena, in “Focus on Probability Theory”, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Velle LR pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 135-169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hauppauge: Nova Science Publishers (2006) [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grebenkov Pulsed-gradient spin-echo monitoring of restricted diffusion in multilayered structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Reson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 205 (2010) 181-195 [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grebenkov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Nguyen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Li Exploring diffusion across permeable barriers at high gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Narrow pulse approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Reson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 248 (2014) 153-163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grebenkov Paradigm shift in diffusion-mediated surface phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2020) 125, 078102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grebenkov An encounter-based approach for restricted diffusion with a gradient drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' (2022) 55 045203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grossel and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Depasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Alternating heat diffusion in thermophysical depth profiles: mul- tilayer and continuous descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' D: Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 31 (1998) 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [34] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Gurevich, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Lashkevich and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' delaCruz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Effective thermal parameters of layered films:an application to pulsed photothermal techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 52 (2009) 25 4302-4307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hahn and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Ozisik One-Dimensional Composite Medium Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 10 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 393-432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Wiley, Hoboken (2012) [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hickson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Barry and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mercer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Critical times in multilayer diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Part 1: Exact solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 52 (2009) 5776-5783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [37] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Hickson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Barry and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mercer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Critical times in multilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 2: Ap- proximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 52 (2009) 5784-5791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [38] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Ito and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' McKean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diffusion Processes and Their Sample Paths Springer-Verlag, Berlin (1965) [39] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Kay and Giuggioli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 4 (2022) L032039 [40] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Kenkre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Giuggiol and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Kalay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Molecular motion in cell membranes: analytic study of fence-hindered random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' E 77 (2008) 051907 [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Lejay The snapping out Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The Annals of Applied Probability 26 (2016) 1727-1742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Lejay Monte Carlo estimation of the mean residence time in cells surrounded by thin layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mathematics and Computers in Simulation 143 (2018) 65-77 [43] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Barbour and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Unified multilayer diffusion model and application to diffu- sion experiment in porous media by method of chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 43 (2009) 2412-2416 [44] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Lu and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Tervola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Transient heat conduction in the composites lab-analytical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 38 (2005) 81 [45] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Milshtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The solving of boundary value problems by numerical integration of stochastic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 38(1995) 77-85 [46] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Moutal and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Grebenkov Diffusion across semi-permeable barriers: spectral proper- ties, efficient computation, and applications J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 81 (2019) 1630-1654 [47] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Novikov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Fieremans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Jensen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Helpern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Random walks with barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 7 (2011) 508-514 [48] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Papanicolaou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' The probabilistic solution of the third boundary value problem for second order elliptic equations Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Fields 87 (1990) 27-77 [49] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Pontrelli and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' de Monte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mass diffusion through two-layer porous media: an applica- tion to the drug-eluting stent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 50 (2007) 3658-3669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Powles, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mallett, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Rickayzen and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Exact analytic solutions for dif- fusion impeded by an infinite array of partially permeable barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' A 436 (1992) 391 [51] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Ramanan and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Brink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Exact solution of a model of diffusion in an infinite chain or monlolayer of cells coupled by gap junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 58 (1990) 631-639 [52] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Shackelford and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Fickian diffusion of radio nuclides for engineered containment barriers: diffusion coefficients, porosities, and complicating issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Geol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 152 (2013) 133-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 123 [53] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Tanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Transient diffusion in a system partitioned by permeable barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' application to NMR measurements with a pulsed field gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 69 (1978) 1748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [54] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Todo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Oshizaka, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Kadhum and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Sugibayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Mathematical model to predict skin concentration after topical application of drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Pharmaceutics 5 (2013) 634-651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' [55] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Yates, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Papiernik, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Gao and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Gan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Analytical solutions for the transport of volatile organic chemicals in unsaturated layered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Water Resour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 36 (2000) 1993-2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfEwPV/content/2301.02895v1.pdf'} diff --git a/GdAzT4oBgHgl3EQfHfsT/content/tmp_files/2301.01044v1.pdf.txt b/GdAzT4oBgHgl3EQfHfsT/content/tmp_files/2301.01044v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d36ce29987e8becb8813681546c0588de34ad644 --- /dev/null +++ b/GdAzT4oBgHgl3EQfHfsT/content/tmp_files/2301.01044v1.pdf.txt @@ -0,0 +1,1583 @@ +Analysis of Label-Flip Poisoning Attack +on Machine Learning Based Malware Detector +Kshitiz Aryal +Department of Computer Science +Tennessee Technological University +Cookeville, TN, USA +karyal42@tntech.edu +Maanak Gupta +Department of Computer Science +Tennessee Technological University +Cookeville, TN, USA +mgupta@tntech.edu +Mahmoud Abdelsalam +Department of Computer Science +North Carolina A&T State University +Greensboro, NC, USA +mabdelsalam1@ncat.edu +Abstract—With the increase in machine learning (ML) applica- +tions in different domains, incentives for deceiving these models +have reached more than ever. As data is the core backbone of ML +algorithms, attackers shifted their interest towards polluting the +training data itself. Data credibility is at even higher risk with +the rise of state-of-art research topics like open design principles, +federated learning, and crowd-sourcing. Since the machine learn- +ing model depends on different stakeholders for obtaining data, +there are no existing reliable automated mechanisms to verify +the veracity of data from each source. +Malware detection is arduous due to its malicious nature with +the addition of metamorphic and polymorphic ability in the +evolving samples. ML has proven to solve the zero-day malware +detection problem, which is unresolved by traditional signature- +based approaches. The poisoning of malware training data can +allow the malware files to go undetected by the ML-based +malware detectors, helping the attackers to fulfill their malicious +goals. A feasibility analysis of the data poisoning threat in the +malware detection domain is still lacking. Our work will focus on +two major sections: training ML-based malware detectors and +poisoning the training data using the label-poisoning approach. +We will analyze the robustness of different machine learning +models against data poisoning with varying volumes of poisoning +data. +Index +Terms—Cybersecurity, +Poisoning +Attacks, +Machine +Learning, Malware Detectors, Adversarial Malware Analysis +I. INTRODUCTION +Machine Learning (ML) techniques have been emerging +rapidly, providing computational intelligence to various ap- +plications. The ability of machine learning to generalize to +unseen data has paved its way from labs to the real world. It +has already gained unprecedented success in many fields like +image processing [1], [2], natural language processing [3], [4], +recommendation systems used by Google, YouTube and Face- +book, cybersecurity [5], [6], robotics [7], drug research [8], [9], +and many other domains. ML-based systems are achieving +unparalleled performance through modern deep neural net- +works bringing revolutions in AI-based services. Recent works +have shown significant achievements in fields like self-driving +cars and voice-controlled systems used by tech giants like +autopilot in Tesla, Apple Siri, Amazon Alexa, and Microsoft +Cortana. With machine learning being applied to such critical +applications, continuous security threats are never a bombshell. +In addition to traditional security threats like malware at- +tack [10], phishing [11], man-in-the-middle attack [12], denial- +of-service [13], SQL injection [14], adversaries are finding +novel ways to sneak into ML models [15]. +Data poisoning and evasion attacks [16]–[20] are the latest +menaces against the security of machine learning models. +Poisoning attacks enable attackers to control the model’s +behavior by manipulating a model’s data, algorithms, or hyper- +parameters during the model training phase. On the other hand, +an evasion attack is carried out during the test time by manip- +ulating the test sample. Adversaries can craft legitimate inputs +imperceptible to humans but force models to make wrong +predictions. Szegedy et al. [21] discovered the vulnerability +of deep learning architecture against adversarial attacks, and +ever since, there have been several major successful adversar- +ial attacks against machine learning architectures [22], [23]. +Sophisticated attackers are motivated by very high incentives +to manipulate the result of the machine learning models. With +the current data scale with which machine learning models are +trained, it is impossible to verify each data point individually. +In most scenarios, it is unlikely that an attacker gets access +to training data. However, with many systems adopting online +learning [24], crowd-sourcing [25] for training data, open +design principles, and federated learning, poisoning attacks +already pose a serious threat to ML models [26]. There have +been instances [27] when big companies have been compro- +mised by a data poisoning attack. Malware public databases +like VirusTotal1, which rely on crowdsourced malware files +for training its algorithm, can be poisoned by attackers while +Google’s mail spam filter can be thrown out of track by wrong +reporting of spam emails. +Data poisoning relates to adding training data that either +leaves a backdoor on the model or negatively impacts the +model’s performance. Figure 1 shows the architecture of the +poisoning attack. In the given figure, the addition of poisoned +data in the training bag forces the model to learn and predict +so that attackers benefit from it. This type of poisoning is +not limited to particular domains but has extended across +all ML applications. Label flipping attack is carried out to +flip the prediction of machine learning detectors. Among +all the existing approaches, we chose one of the simplest +poisoning techniques called label poisoning. We swap the +1https://www.virustotal.com/ +arXiv:2301.01044v1 [cs.CR] 3 Jan 2023 + +Fig. 1. General architecture for Poisoning Machine Learning Models +existing training data labels in label poisoning to check the +ML models’ robustness. +In this work, we perform a comparative analysis of different +machine learning-based malware detectors’ robustness against +label-flipping data poisoning attacks. Unlike the existing ap- +proaches, we are demonstrating the impact of simple label- +switching data poisoning in different malware detectors. We +will first train eight different ML models widely used to detect +malware, namely Stochastic Gradient Descent (SGD), Random +Forest (RF), Logistic Regression (LR), K-Nearest Neighbor +Classifier (KNN), Linear Support Vector Machine (SVM), +Decision Tree (DT), Perceptron, and Multi-Layer Perceptron +(MLP). This will be followed by poisoning 10% and 20% of +training data by flipping the label of data samples. All of the +models are retrained after data poisoning, and the performance +of each model is evaluated. The major contributions of this +paper are as follows. +• We taxonomize the existing data poisoning attacks on ma- +chine learning models in terms of domains, approaches, and +targets. +• We provide threat modeling for adversarial poisoning attacks +against malware detectors. The threat is modeled in terms of +the attack surface, the attacker’s knowledge, the attacker’s +capability, and adversarial goals. +• We train eight different machine learning-based malware +detectors from malware data obtained from VirusTotal and +VirusShare2. We compare the performance of these malware +detectors with training and testing data in terms of accuracy, +precision, and recall. +• Finally, we show a simple label-switching approach to +poison the data without any knowledge of training models. +2https://virusshare.com/ +Fig. 2. Taxonomy of poisoning attack on attack domain, approach and target +The performance of malware detectors is analyzed while +poisoning 10% and 20% of the total training data. +The rest of the paper is organized as follows. The existing +literature for data poisoning attacks in different domains, +including malware, is discussed in Section II. Section III +provides the threat modeling for data poisoning attacks. An +overview of ML algorithms that are used to train the malware +detector in this paper is discussed in Section IV. Section V dis- +cusses experimental methodology elaborating on the algorithm +and the testbed used for the experiment. The evaluation and +discussion on the performed experiments are given in Section +VI. Finally, Section VII concludes this work. +II. LITERATURE REVIEW +Data poisoning attacks have been used against the machine +learning domain for a long time. The existing literature on +data poisoning attacks can be taxonomized in terms of attack +domains, approach, and the target (victim), as illustrated in +Figure 2. The recently trending technologies like crowd- +sourcing and federated learning are always vulnerable as the +veracity of individual data can never be verified. The recent +victims of poisoning attacks have spread in security, network, +and speech recognition domains. We also classified the major +approaches that are taken to produce or optimize the poisoning +attacks in Figure 2. The existing data poisoning approaches +have targeted almost all the machine learning algorithms +ranging from traditional algorithms like regression to modern +deep neural network architectures. +Table I summarizes the existing literature on poisoning +attacks. Biggio et al. [43] attacked a support vector machine +using gradient ascent. To make poisoning attacks closer to +the real world, Yang et al. [44] used a generative adversarial +network with an autoencoder to poison deep neural nets. +Gongalez et al. [45] extended poisoning from binary learning +to multi-class problems. Shafahi et al. [28] proposed a targeted +clean label poisoning attack on neural networks using an +optimization-based crafting method. Shen et al. [31] performed +an imperceptible poisoning attack on a deep neural network +by clogging the back-propagation from gradient tensors during +training while also minimizing the gradient norm. Jiang et + +Machine learning +Machine learning with +without Data Poisoning +Data Poisoning +Data +Data +Collection +Collection +Poisoned +直盲自 +自自自 +Data +18168 +0100 +1818 +0100 +Preprocessing +Preprocessing +Data +110 +0110 +Data +Training +Training Data with +Data +Poisoned dataset +Training +Training +Poisoned Mode +Trained +Prediction +Poisoned +Model +Prediction ModelData Poisoning +Y +Domains +Approach +Target +> Image +Gradient +Neural +Network +Crowd Sourcing +Reinforcement learning +Support Vector +Machine +> Graph +Label Flipping +Regression +Federated Learning +Generative Adversarial +Network +Truth-Finder +Dawid-Skene +Security +Empirical Inverstigations +Graph +Recommendation +FakeUsers Insertion +Embedding +Spectrum +Online learning +Network +SpeechRecognitionTABLE I +DATA POISONING ATTACKS +Domains +Approach +Target +Publications +Image +Crowd +Sourcing +Graph +Federated +Learning +Security +Online +Learning +Gradient +Reinforcement +Learning +Label +Flipping +GAN +Others +Neural +Network +SVM +Regression +Graph +Embedding +Customized +Shafahi et al. [28] +√ +√ +√ +Liu et al. [29] +√ +√ +√ +√ +Cao et al. [30] +√ +√ +√ +Shen et al. [31] +√ +√ +√ +Zhang et al. [32] +√ +√ +√ +Jiang et al. [33] +√ +√ +√ +Kwon et al. [34] +√ +√ +√ +Zhang et al. [35] +√ +√ +√ +Bagdasaryal et al. [36] +√ +√ +√ +√ +Li et al. [37] +√ +√ +√ +Sasaki et al. [38] +√ +√ +√ +Zhang et al. [39] +√ +√ +√ +√ +Lovisotto et al. [40] +√ +√ +√ +Li et al. [41] +√ +√ +√ +√ +Kravchik et al. [42] +√ +√ +√ +This Work +√ +√ +√ +√ +√ +Domains:Poisoning domain for crafted attack, Approach: Approach to poison the training data, Target: Target of poisoning attack +al. [33] performed a flexible poisoning attack against linear and +logistic regression. Kwon et al. [34] could selectively poison +particular classes against deep neural networks. Cao et al. [30] +proposed a distributed label-flipping poisoning approach to +poison the DL model in federated architecture. Miao et al. [46] +poisoned Dawid-Skene [47] model by exploiting the reliability +degree of workers. Fang et al. [48] proposed a poisoning attack +against a graph-based recommendation system by maximizing +the hit ratio of target items using fake users. +In the given Table I, we can observe that only a handful +of works have been carried out in the security domain. +Sasaki et al. [38] proposed an attack framework for backdoor +embedding, which prevented the detection of specific types of +malware. They generated poisoning samples by solving an op- +timization problem and tested it against a logistic regression- +based malware detector. To poison the Android malware de- +tectors, Lie et al. [41] experimented backdoor poisoning attack +against Drebin [49], DroidCat [50], MamaDroid [51] and +DroidAPIMiner [52]. Kravchik et al. [42] attacked the cyber +attack detectors deployed in the industrial control system. The +back gradient optimization techniques used to pollute the train- +ing data successfully poison the neural network-based model. +These works have focused their approach on some algorithm +testing against some defense mechanism. However, none of +the works compared the feebleness of multiple algorithms +against data poisoning attacks. In this work, we demonstrate +the effectiveness of label switch poisoning of the training +data against eight machine learning algorithms widely used +in malware detectors. +III. THREAT MODEL: KNOW THE ADVERSARY +All security threats are defined in terms of their goals and +attack capabilities. Modeling the threat allows for identifying +and better understanding the risk arriving with a threat. A +poisoning attack is performed by manipulating the training +data either at the initial learning or incremental learning +Fig. 3. Threat model for poisoning attack +period. The threat model of a poisoning attack reflects the +attacker’s knowledge, goal, capabilities, and attack surface, as +shown in Figure 3. +Attack Surface: Attack surface denotes how the adversary +attacks the model under analysis. Machine learning algorithms +require data to pass through different stages in the pipeline, +and each stage offers some kind of vulnerability. In this work, +we are only concerned about poisoning attacks which make +the training data an attack surface. +Attacker’s Knowledge: The attacker’s knowledge is the +amount of information about the model under attack that +an attacker has. Based on the amount of knowledge of the +attacker, the poisoning approach is determined. Attacker’s +knowledge can be broadly classified into the two following +categories: +• White box model: In the white box model, an attacker has +complete information about the underlying target model, +such as the algorithm used, training data, hyper-parameters, +and gradient information. It’s easier to carry out a white box +attack due to the information available that helps the attacker +to create a worst-case scenario for the target model. +• Black box model: In the black-box model, an attacker only + +Threat Model +Attack Surface +Attacker's +Attacker's +Capability +Attacker's Goal +Knowledge +Training +White Box +Data +Untargeted +data +model +Injection +Misclassifcation +Black Box +Data +Targeted +model +Modification +Misclassification +Logic +Confidence +Corruption +Reductionhas information about the model’s input and output. An +attacker has no information about the internal structure of +the model. Black-box models can also be divided further +into complete black-box models and gray-box models. In +the gray box model, the model’s performance for each input +the attacker provides can be known. As such, the gray box +attack is considered to be relatively easier than the complete +black box model. +In this paper, we perform a black box attack on different +malware detection models. Our experiments will prove the vul- +nerability of these models to random label poisoning attacks +without having any information about the models. +Attacker’s Capability: The attacker’s capability represents +the ability of an adversary to manipulate the data and model in +different stages of the ML pipeline. It defines the sections that +can be manipulated, the mechanism used for manipulation, and +constraints to the attacker. Poisoning can be carried out in a +well-controlled environment if the attacker has complete infor- +mation about the underlying model and training data. Attacker +capabilities can be classified into the following categories: +• Data Injection: It is the ability to insert new data into the +training dataset, leading machine learning models to learn +on contaminated data. +• Data Modification: It is the ability to access and modify +the training data as well as the data labels. Label flipping +is a well-known approach carried out in poisoning attack +domains. +• Logic Corruption: It is the ability to manipulate the logic of +ML models. This ability is out of scope for data poisoning +and is considered a model poisoning approach. +Adversarial Goals: The attacker’s objective is to deceive the +ML model by injecting poisoned data. However, poisoning +training data might differ depending on the goals of an +attacker. Attacker goals can be categorized as: +• Untargeted Misclassification: +An attacker tries to change +the model’s output to a value different than the original +prediction. Untargeted misclassification is a relatively easier +goal for attackers. +• Targeted Misclassification: +An attacker’s goal is to add a +certain backdoor in the models so that particular samples +are classified to a chosen class. +• Confidence Reduction: An attacker can also poison training +data to reduce the confidence of the machine learning model +for a particular prediction. In this approach, changing the +classification label is unnecessary, but reducing the confi- +dence score is enough to meet the attacker’s goal. +Our paper aims to cause the malware detector models to +misclassify. However, since we are dealing with binary clas- +sification, it can be considered either targeted or untargeted +misclassification. +IV. OVERVIEW OF MACHINE LEARNING ALGORITHMS +Almost all of the ML architectures have already been +victimized by data poisoning attacks. In this section, we will +brief some ML architectures in which we performed data +poisoning attacks later in this paper. +Stochastic Gradient Descent: +Stochastic gradient descent +(SGD) is derived from the gradient descent algorithm, which +is a popular ML optimization technique. A gradient gives the +slope of the function and measures the degree of change of +a variable in response to the changes of another variable. +Starting from an initial value, gradient descent runs iteratively +to find the optimal values of the parameters, which are the +minimal possible value of the given cost function. In Stochastic +Gradient Descent, a few samples are randomly selected in +place of the whole dataset for each iteration. The term batch +determines the number of samples to calculate each iteration’s +gradient. In normal gradient descent optimization, a batch is +taken to be the whole dataset leading to the problem when the +dataset gets big. Stochastic gradient descent considers a small +batch in each iteration to lower the computing cost of the +gradient descent approach while working with a large dataset. +Random Forest: +A random forest is a supervised ML +algorithm that is constructed from an ensemble of decision tree +algorithms. Its ensemble nature helps to provide a solution to +complex problems. The random forest is made up of a large +number of decision trees that have been trained via bagging +or bootstrap aggregation. The average mean of the output +of constituent decision trees is the random forest’s ultimate +forecast. The precision of the output improves as the number +of decision trees used grows. A random forest overcomes the +decision tree algorithm’s limitations by eliminating over-fitting +and enhancing precision. +Logistic Regression: The probability for classification prob- +lems is modeled using logistic regression, which divides +them into two possible outcomes. For classification, logistic +regression is an extension of the linear regression model. For +regression tasks, linear regression works well; however, it fails +to replicate for classification. The linear model considers the +class a number and finds the optimum hyperplane that mini- +mizes the distances between the points and the hyperplane. As +it interpolates between the points, it cannot be interpreted as +probabilities. Because there is no relevant threshold for class +separation, logistic regression is applied. It is a widely used +classification algorithm due to its ease of implementation and +strong performance in linearly separable classes. +K-Nearest Neighbors (KNN) Classifier: The KNN algorithm +relies on the assumption that similar things exist in close +proximity. It is a non-parametric and lazy learning algorithm. +KNN does not carry any assumption for underlying data +distribution. It does not require training data points for model +generation, as all the training data are used in a testing phase. +This results in faster training and a slower testing process. The +costly testing phase will consume more time and memory. In +KNN, K is the number of nearest neighbors and is generally +considered odd. KNN, however, suffers from the curse of +dimensionality. With increased feature dimension, it requires +more data and becomes prone to overfitting. +Support Vector Machine (SVM): A support vector machine + +Algorithm 1: Data Poisoning Algorithm +Input: Non-poisoned feature set +Output: Poisoned feature set +Data: Static features obtained from malware and +benign training set +1 for all the samples do +2 +Train the machine learning models and measure +the performance +3 +for 10% each of Malware and Benign data do +4 +if Training label is not flipped then +5 +label=Get training label of given data +6 +if label==0 then +7 +Flip the label to 1 +8 +else if label==1 then +9 +Flip the label to 0 +10 +Train all the models and measure the performance +11 +for 20% each of Malware and Benign data do +12 +if Training label is not flipped then +13 +label=Get training label of given data +14 +if label==0 then +15 +Flip the label to 1 +16 +else if label==1 then +17 +Flip the label to 0 +18 +Train all the models and measure the performance +is a popular supervised ML algorithm applied in both classi- +fication and regression tasks. SVM aims to find a hyperplane +that classifies the data points. In SVM, there are several pos- +sible hyperplanes, and we need to determine the optimal hy- +perplane that maximizes the margin between the two classes. +Hyperplanes are the decision boundary for SVM, where data +points near to hyperplane are the support vectors. Due to its +effectiveness in high dimensional spaces and memory-efficient +properties, it is widely adopted in different domains. +Multi-Layer Perceptron: +The term ’Perceptron’ is derived +from the ability to perceive, see, and recognize images in a +human-like manner. A perceptron machine is based on the +neuron, a basic unit of computation, with a cell receiving a +series of pairs of inputs and weights. Although the perceptron +was originally thought to represent any circuit and logic, non- +linear data cannot be represented by a perceptron with only one +neuron. Multi-Layer Perceptron was developed to overcome +this limitation. In multi-layer perceptron, the mapping between +input and output is non-linear. It has input and output layers +and several hidden layers stacked with numerous neurons. +Because the inputs are merged with the initial weights in +a weighted sum and applied to the activation function, the +multi-layer perceptron falls under the category of feedforward +algorithms. Each linear combination is propagated to the +following layer, unlike with a perceptron. +V. EXPERIMENTAL METHODOLOGY +In this paper, we are using the label-flipping approach to +poison the training data. With source class CS and a target +class CT from a set of classes C, the dataset DI is poisoned. +The detailed poisoning performed in the paper is shown in +Algorithm 1. We perform a label poisoning attack of differ- +ent volumes to training data without guiding the poisoning +mechanism through machine learning architecture or the loss +function. It is an efficient way to showcase the ability of +random poisoning to hamper the model’s performance. We are +training all eight malware detector models three times in total. +As illustrated in Algorithm 1, we begin the model training +with clean data without adding any noise. After recording the +model’s performance on clean data, we proceed towards the +first stage of poisoning our data. We take 10% of shuffled +training data belonging to each malware and benign class, and +we change their labels. We retrain all the models and again +measure the performance of the models. We repeat the same +operation with 20% of shuffled training data. The percentage +of poisoned data is taken randomly for this experimental +purpose, as the goal is to show the impact on the models. +The algorithm we followed in carrying out this experiment is +not a novel approach but a generic approach to poison the +data. +A. Experimental Environment and Dataset +All the experiments are performed in Google-Colab us- +ing Google’s GPU. All the implementation will be worked +around using python libraries and Scikit-Learn. The training +dataset [53] is obtained from the Kaggle repository, where +data are collected from VirusTotal and VirusShare. The dataset +comprises windows PE malware and benign files processed +through static executable analysis. The dataset comprises +216,352 files (75,503 benign files and 140,849 malware files) +with 54 features. +VI. EVALUATION RESULTS AND ANALYSIS +A. Data Pre-processing and Transformation +We begin our experiment by loading data from Kaggle +dataset [53]. To clean the data, we followed two different +approaches. First, we ignored rows that are missing more than +50% of data, whereas we replaced the null values with the +arithmetic mean value of the column for rows with less than +50% missing values. Second, we normalized the data by scal- +ing the values from 0 to 1. Afterward, 85% of data were used +for training purposes while the remaining 15% were used for +testing purposes. We trained selected eight machine learning +models with standard hyper-parameters for each model. We +didn’t tweak many machine learning parameters to fine-tune +the detection accuracy, resulting in significant overfitting in a +few models. +B. Performance Indicators +We evaluated the malware detectors’ performance using the +following metrics: + +TABLE II +MALWARE DETECTION TRAINING RESULT +Algorithm +Clean Data +Training Data +Testing Data +Accuracy +Precision +Recall +F1 +Accuracy +Precision +Recall +F1 +Stochastic Gradient Descent +93.41 +92.49 +88.29 +90.34 +72.98 +58.6 +78.77 +67.20 +Decision Tree +99.96 +99.98 +99.91 +99.94 +59.65 +44.5 +59.85 +51.05 +Random Forest +99.97 +99.92 +99.97 +99.94 +83.65 +98.82 +54.12 +69.94 +Logistic Regression +93.2 +92.21 +87.94 +90.02 +92.33 +92.24 +85.36 +88.67 +KNN Classifier +98.38 +97.33 +98.05 +97.69 +97.42 +96.38 +96.25 +96.31 +Support Vector Machine +93.15 +92.44 +87.51 +89.91 +92.03 +90.89 +85.94 +88.34 +Perceptron +90.93 +88.6 +84.91 +86.72 +75.39 +60.28 +87.86 +71.50 +Multi-Layer Perceptron +91.28 +91.07 +83.16 +86.94 +71.93 +57.45 +77.66 +66.04 +TABLE III +MALWARE DETECTION PERFORMANCE WITH 10% POISONING DATA +Algorithm +10% Poisoned Data +Training Data +Testing Data +Accuracy +Precision +Recall +F1 +Accuracy +Precision +Recall +F1 +Stochastic Gradient Descent +85.12 +82.49 +77.14 +79.73 +72.39 +64.23 +61.38 +62.77 +Decision Tree +96.77 +99.44 +92.01 +95.58 +51.92 +38.33 +43.98 +40.96 +Random Forest +96.77 +98.92 +92.51 +95.61 +80.13 +82.68 +60.22 +69.68 +Logistic Regression +84.51 +82.29 +75.39 +78.69 +83.26 +81.06 +72.91 +76.77 +KNN Classifier +89.49 +85.47 +87.1 +86.28 +86.59 +83.1 +81.15 +82.11 +Support Vector Machine +84.75 +82.84 +75.42 +78.96 +66.99 +63.14 +31.16 +41.73 +Perceptron +77.94 +67.78 +79.69 +73.25 +40.16 +25.89 +31 +73.25 +Multi-Layer Perceptron +83.85 +82.72 +72.58 +77.32 +83.33 +82.81 +70.74 +76.30 +Accuracy = +TP + TN +TP + TN + FP + FN +Precision = +TP +TP + FP , Recall = +TP +TP + FN +F1-score = 2 ∗ (Precision ∗ Recall) +Precision + Recall +A positive outcome corresponds to a malware sample, while +a negative result corresponds to a benign example. TP, TN, +FP, and FN are true positives, true negatives, false positives, +and false negatives, respectively. Accuracy is the percentage +of correct predictions on the given data. Precision measures +the ratio between true positives and all the positives. Recall +provides the ability of our model to predict true positives +correctly. The F1 score is the harmonic mean, the combination +of a classifier’s precision and recall. +C. Results and Discussion +Table II shows the accuracy, precision, and recall for train- +ing and testing data. Stochastic Gradient Descent, Decision +Trees, Random Forest, and Perceptron looked overfitted to +training data compared to other models. Since the data volume +is a little bit high, decision tree-based classifiers are prone to +overfitting problems. We used shallow layer neural networks +leading perceptron to overfit in the data. However, classifiers +like logistic regression, KNN classifier, and Support Vector +Machine have shown the best performance in all three metrics. +We have compared the performance of both the training and +testing sets as we have only poisoned the training data while +preserving the test data from attack. +We flipped the labels of 10% training data as a poisoning +attack. On poisoning 10% of total data, the performance metric +for each detector is displayed in Table III. The results show the +robustness of decision trees and random forest-based malware +detectors compared to other malware detectors. We further +poisoned 20% of total training data to see the impact of +increased poisoned data in each model, whose results are +shown in Table IV. The left-most confusion matrix in each of +the figures from Figure 4 to Figure 11 shows the number of +TP, TN, FP, and FN for each classifier on clean data, whereas +the middle and right one shows results with 10% and 20% +poisoning, respectively. In the confusion matrix, label ’0’ is +for malware, and label ’1’ is for benign samples. The top-left +corner in the confusion matrix gives True Positive, the top- +right corner gives False Positive, the bottom-left gives False +Negative, and the bottom-right corner gives True Negative +samples. +D. Analysis and Observations +The goal of this work is to show the vulnerability of popular +machine-learning models that are used for malware detection. +Results in Tables II, III and IV reflect the limitations of +all the experimented machine learning models even with the +basic label poisoning attack. Figure 12 shows the ROC curve, +comparing the models’ performance on the clean data, 10% +and 20% poisoned data. In the ROC curve, the blue curve +corresponds to the performance of clean data, the orange +curve corresponds to 10% poisoned data, and the green curve +corresponds to the 20% poisoned data. The curve closest to +the top-left corner is the one performing best. We can infer +from the graph that logistic regression, K-Nearest Neighbors, + +TABLE IV +MALWARE DETECTION PERFORMANCE WITH 20% POISONING DATA +Algorithm +20% Poisoned data +Training Data +Testing Data +Accuracy +Precision +Recall +F1 +Accuracy +Precision +Recall +F1 +Stochastic Gradient Descent +78.56 +75.65 +70.21 +72.83 +62.69 +54.86 +50.72 +52.71 +Decision Tree +96.54 +93.54 +98.34 +95.88 +40.26 +34.25 +49.67 +40.54 +Random Forest +96.54 +93.04 +98.94 +95.90 +72.8 +68.77 +61.66 +65.02 +Logistic Regression +78.38 +74.3 +72.13 +73.20 +77.58 +75.1 +76.78 +75.93 +KNN Classifier +87.41 +82.48 +87.94 +85.12 +82.15 +76.16 +82.2 +79.06 +Support Vector Machine +78.58 +74.45 +72.6 +73.51 +75.39 +74.74 +60.37 +66.79 +Perceptron +75.16 +68.58 +72.57 +72.57 +49.37 +38.28 +38.28 +38.28 +Multi Layer Perceptron +77.6 +75.45 +67.1 +71.03 +76.85 +74.81 +65.66 +69.94 +Fig. 4. Confusion Matrix for Stochastic Gradient Descent Based Malware Detector +Fig. 5. Confusion Matrix for Decision Tree Based Malware Detector +Fig. 6. Confusion Matrix for Random Forest-Based Malware Detector +Fig. 7. Confusion Matrix for Logistic Regression Based Malware Detector +Support Vector Machine, and Multi-Layer Perceptron are the +best models on the clean data. However, the distance between +the three curves represents the robustness of the model toward +the poisoning attack. If the separation between the curves of +clean data and poisoning data is low, it infers that the poisoning +attack has a minimal impact on the model’s performance. In +the ROC graph, we can observe that Random Forest, Logistic +Regression, K-Nearest Neighbors, and Multi-Layer Perceptron + +ConfusionMatrixforSGD +Predicted Class +12000 +12148 +1884 +10000 +Actual Class +8000 +6000 +1522 +6082 +4000 +2000 +0 +1Confusion Matrix for SGD with 10% Poisoning +PredictedClass +10000 +IClass +0 +11778 +1652 +8000 +Actual +6000 +2495 +5711 +4000 +2000 +0 +1Confusion Matrix for SGD with 20% Poisoning +PredictedClass +10000 +IClass +0 +10830 +1936 +8000 +Actual +6000 +3294 +5576 +4000 +2000 +0 +1ConfusionMatrixforDT +Predicted Class +7000 +7593 +6439 +Actual Class +6000 +5000 +2677 +4927 +4000 +-3000 +0 +1Confusion Matrix for DT with 10% Poisoning +PredictedClass +8000 +Class +0 +4915 +8515 +7000 +Actual +6000 +5000 +4763 +3443 +4000 +0 +1Confusion Matrix for DT with 20% Poisoning +PredictedClass +7000 +IClass +0 +5440 +7326 +6000 +Actual +5000 +4945 +3925 +4000 +0 +1ConfusionMatrixforRF +Predicted Class +12500 +14000 +32 +10000 +Actual Class +7500 +5000 +4405 +3199 +-2500 +0 +-Confusion Matrix for RF with 10% Poisoning +PredictedClass +12000 +12145 +1285 +10000 +ActualClass +8000 +6000 +3293 +4913 +4000 +2000 +0 +1Confusion Matrix for RF with 20% Poisoning +PredictedClass +10000 +ActualClass +0 +10036 +2730 +8000 +6000 +3154 +5716 +4000 +0 +1ConfusionMatrixfor LR +PredictedClass +12500 +13492 +540 +10000 +Actual Class +7500 +-5000 +1119 +6485 +2500 +0 +1Confusion Matrix for LR with 10% Poisoning +PredictedClass +12000 +10000 +Actual Class +0 +12032 +1398 +8000 +6000 +2223 +5983 +4000 +2000 +0 +1Confusion Matrix for LR with 20% Poisoning +PredictedClass +10000 +IClass +0 +10774 +1992 +8000 +Actual +6000 +2859 +6011 +4000 +2000 +0 +1Fig. 8. Confusion Matrix for KNN Based Malware Detector +Fig. 9. Confusion Matrix for Support Vector Machine-Based Malware Detector +Fig. 10. Confusion Matrix for Perceptron Based Malware Detector +Fig. 11. Confusion Matrix for Multi-Layer Perceptron Based Malware Detector +have their graphs close to each other, proving their robustness +against poisoned data. Random Forest’s robustness can be +attributed to its ensemble nature which helps it to capture +better insights about the data. The robustness of logistic +regression and K-Nearest Neighbors can be due to the low +dimensionality of our training data. Further, we can observe +the performance of models, like SVM and perceptron, doing +better with the 20% poisoned data than with 10% poisoned +data. The gain in the performance of these models is due +to unrestricted data poisoning. Since we are not guiding our +poisoning approach according to the models, further adding +poisoning data after some threshold point slightly improves the +models’ performance. In the end, even the least sophisticated +attacks, like label poisoning, are causing the performance +decay of the models to a large extent. This further alerts us +toward the catastrophic consequences of more sophisticated +attacks like gradients and reinforcement learning. +VII. CONCLUSION +In this work, we perform a feasibility analysis of label- +flip poisoning attacks on ML-based malware detectors. We +evaluated eight different ML models that are widely used in +malware detection. Spotting the lack of poisoning attacks work +in the malware domain, this paper analyses the robustness +of ML-based malware detectors against different volumes of +poisoned data. We observed the decay in performance of all +the models while poisoning 10% and 20% of total training +data. The significant decrease in the performance of the models +shows the severe vulnerability of malware detectors to guided +poisoning approaches. We also observed differences in the +effect of poisoning attacks across the different models. Our +work is carried out within the limited scope of one generic +poisoning algorithm and a single malware dataset. There are +few future research directions that are clearly visible. The +malware detectors can be tested against many advanced poi- +soning approaches using numerous datasets from the industry. + +ConfusionMatrixforPerceptron +Predicted Class +7000 +7831 +6201 +Actual Class +6000 +5000 +4000 +2139 +5465 +3000 +1ConfusionMatrixforPerceptronwith10%Poisoning +PredictedClass +7000 +IClass +0 +7734 +5696 +6000 +Actual +5000 +4897 +3309 +4000 +0 +1Confusion Matrix for Perceptron with 20% Poisoning +PredictedClass +10000 +ActualClass +0 +1393 +11373 +8000 +6000 +4507 +4363 +4000 +2000 +0 +1ConfusionMatrixforMLE +Predicted Class +12500 +13412 +620 +10000 +Actual Class +7500 +5000 +1277 +6327 +2500 +0 +1Confusion Matrix for MLP with 10% Poisoning +PredictedClass +12000 +1205 +10000 +IClass +0 +12225 +8000 +Actual +6000 +2401 +5805 +4000 +2000 +0 +1Confusion Matrix for MLP with 20% Poisoning +PredictedClass +10000 +IClass +0 +10805 +1961 +8000 +Actual +6000 +3046 +5824 +4000 +2000 +0 +1ConfusionMatrixforKNN +Predicted Class +12500 +13767 +265 +10000 +Actual Class +7500 +5000 +261 +7343 +2500 +0 +1Confusion Matrix for KNN with 10% Poisoning +PredictedClass +10000 +IClass +0 +11860 +1570 +8000 +Actual +6000 +1531 +6675 +4000 +2000 +0 +1Confusion Matrix for KNN with 20% Poisoning +PredictedClass +10000 +IClass +0 +10484 +2282 +8000 +Actual +6000 +1578 +7292 +4000 +2000 +0 +1ConfusionMatrixforSVM +Predicted Class +12500 +13371 +661 +10000 +Actual Class +7500 +5000 +1061 +6543 +2500 +0 +1Confusion Matrix for SVM with 10% Poisoning +PredictedClass +10000 +IClass +0 +11989 +1441 +8000 +Actual +6000 +5718 +2488 +-4000 +2000 +0 +1Confusion Matrix for SVM with 20% Poisoning +PredictedClass +10000 +IClass +0 +10955 +1811 +8000 +Actual +6000 +3510 +5360 +-4000 +2000 +0 +1Fig. 12. ROC Curve for Malware detectors under Poisoning Environments +The poisoning can be tested in a more real environment by +poisoning the executable files. The research community still +lacks exhaustive studies on the vulnerabilities of malware +detectors and how to make detectors more robust against these +poisoning attacks. +REFERENCES +[1] D. Ciregan, U. Meier, and J. Schmidhuber, “Multi-column deep neural +networks for image classification,” in 2012 IEEE Conference on Com- +puter Vision and Pattern Recognition, 2012, pp. 3642–3649. +[2] J. Schmidhuber, U. Meier, and D. Ciresan, “Multi-column deep neural +networks for image classification,” in 2012 IEEE Conference on Com- +puter Vision and Pattern Recognition. +IEEE Computer Society, 2012. +[3] K. Chowdhary, “Natural Language Processing,” Fundamentals of Arti- +ficial Intelligence, pp. 603–649, 2020. +[4] J. Hirschberg and C. D. Manning, “Advances in Natural Language +Processing,” Science, vol. 349, no. 6245, pp. 261–266, 2015. +[5] C.-F. Tsai, Y.-F. Hsu, C.-Y. Lin, and W.-Y. Lin, “Intrusion detection by +machine learning: A review,” Expert Systems with Applications, vol. 36, +no. 10, pp. 11 994–12 000, 2009. +[6] N. Peiravian and X. Zhu, “Machine learning for android malware detec- + +KNNCleanData,auc=0.971460832443106 +KNN 10% Poison, auc=0.8552929416737186 +KNN 20% Poison, auc=0.8216704426092349SVM Clean Data, auc=0.9073593040810904 +SVM 10% Poison, auc=0.6002161213967441 +SVM 20% Poison, auc=0.73096877256933DT Clean Data, auc=0.5872804184858597 +DT 10% Poison, auc=0.5037422175699491 +DT 20% Poison, auc=0.4169380476360457MLp CleanData,auc=0.8939386759774156 +MLP 10% Poison, auc=0.8088423576886244 +MLP 20% Poison, auc=0.7514920551542543Perceptron Clean Data, auc=0.7824525005443335 +Perceptron 10% Poison, auc=0.3838246137390344 +Perceptron 20% Poison, auc=0.476954003915064SGD Clean Data, auc=0.7482652186900373 +SGD 10% Poison, auc=0.7025164286923702 +SGD 20% Poison, auc=0.6086523161420353RF Clean Data.auC=0.7456458394939469 +RF 10% Poison, auc=0.7625879961069476 +RF 20% Poison, auc=0.711092219132663LR Clean Data, auc=0.9073593040810904 +LR 10% Poison, auc=0.8125026745227009 +LR 20% Poison, auc=0.7608190424784267tion using permission and api calls,” in 2013 IEEE 25th International +Conference on Tools with Artificial Intelligence, 2013, pp. 300–305. +[7] J. Kober and J. Peters, “Learning motor primitives for robotics,” in 2009 +IEEE International Conference on Robotics and Automation, 2009. +[8] R. Manicavasaga, P. B. Lamichhane, P. Kandel, and D. A. Talbert, “Drug +repurposing for rare orphan diseases using machine learning techniques,” +in The International FLAIRS Conference Proceedings, vol. 35, 2022. +[9] A. Dhakal, C. McKay, J. J. Tanner, and J. Cheng, “Artificial intelligence +in the prediction of protein–ligand interactions: recent advances and +future directions,” Briefings in Bioinformatics, vol. 23, no. 1, p. bbab476, +2022. +[10] M. H. R. Khouzani, S. Sarkar, and E. Altman, “Maximum Damage +Malware Attack in Mobile Wireless Networks,” IEEE/ACM Transactions +on Networking, vol. 20, no. 5, pp. 1347–1360, 2012. +[11] S. Gupta, A. Singhal, and A. Kapoor, “A literature survey on social +engineering attacks: Phishing attack,” in 2016 International Conference +on Computing, Communication and Automation (ICCCA), 2016. +[12] F. Callegati, W. Cerroni, and M. Ramilli, “Man-in-the-Middle Attack to +the HTTPS Protocol,” IEEE Security Privacy, vol. 7, no. 1, 2009. +[13] C. Schuba, I. Krsul, M. Kuhn, E. Spafford, A. Sundaram, and D. Zam- +boni, “Analysis of a denial of service attack on TCP,” in Proceedings. +1997 IEEE Symposium on Security and Privacy (Cat. No.97CB36097), +1997. +[14] W. G. Halfond, J. Viegas, A. Orso et al., “A classification of sql-injection +attacks and countermeasures,” in Proceedings of the IEEE International +Symposium on Secure Software Engineering, vol. 1. +IEEE, 2006. +[15] I. Yilmaz and R. Masum, “Expansion of cyber attack data from +unbalanced +datasets +using +generative +techniques,” +arXiv +preprint +arXiv:1912.04549, 2019. +[16] B. Kolosnjaji, A. Demontis, B. Biggio, D. Maiorca, G. Giacinto, +C. Eckert, and F. Roli, “Adversarial malware binaries: Evading deep +learning for malware detection in executables,” in IEEE European Signal +Processing Conference, 2018, pp. 533–537. +[17] F. Kreuk, A. Barak, S. Aviv-Reuven, M. Baruch, B. Pinkas, and +J. Keshet, “Adversarial examples on discrete sequences for beating +whole-binary malware detection,” arXiv preprint :1802.04528, 2018. +[18] L. Demetrio, B. Biggio, G. Lagorio, F. Roli, and A. Armando, “Explain- +ing Vulnerabilities of Deep Learning to Adversarial Malware Binaries,” +arXiv preprint arXiv:1901.03583, 2019. +[19] O. Suciu, S. E. Coull, and J. Johns, “Exploring adversarial examples +in malware detection,” in 2019 IEEE Security and Privacy Workshops, +2019. +[20] K. Aryal, M. Gupta, and M. Abdelsalam, “A Survey on Adversarial +Attacks for Malware Analysis,” arXiv preprint arXiv:2111.08223, 2021. +[21] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, +and R. Fergus, “Intriguing properties of neural networks,” arXiv preprint +arXiv:1312.6199, 2013. +[22] S. M. P. Dinakarrao, S. Amberkar, S. Bhat, A. Dhavlle, H. Sayadi, +A. Sasan, H. Homayoun, and S. Rafatirad, “Adversarial attack on +microarchitectural events based malware detectors,” in Proceedings of +the 56th Annual Design Automation Conference 2019, 2019, pp. 1–6. +[23] W. Hu and Y. Tan, “Generating adversarial malware examples for black- +box attacks based on GAN,” arXiv preprint arXiv:1702.05983, 2017. +[24] S. Shalev-Shwartz et al., “Online learning and online convex optimiza- +tion,” Foundations and Trends® in Machine Learning, vol. 4, 2012. +[25] A. Rai, K. K. Chintalapudi, V. N. Padmanabhan, and R. Sen, “Zee: +Zero-effort Crowdsourcing for Indoor Localization,” in Proceedings of +the 18th Annual International Conference on Mobile Computing and +Networking, 2012, pp. 293–304. +[26] K. Bonawitz, H. Eichner, W. Grieskamp, D. Huba, A. Ingerman, +V. Ivanov, C. Kiddon, J. Koneˇcn`y, S. Mazzocchi, B. McMahan et al., +“Towards federated learning at scale: System design,” Proceedings of +Machine Learning and Systems, vol. 1, pp. 374–388, 2019. +[27] “Tay, Microsoft’s AI chatbot, gets a crash course in racism from +Twitter,” +https://www.theguardian.com/technology/2016/mar/24/ +tay-microsofts-ai-chatbot-gets-a-crash-course-in-racism-from-twitter, +2016. +[28] A. Shafahi, W. R. Huang, M. Najibi, O. Suciu, C. Studer, T. Dumitras, +and T. Goldstein, “Poison Frogs! Targeted Clean-Label Poisoning At- +tacks on Neural Networks,” Advances in Neural Information Processing +Systems, vol. 31, 2018. +[29] X. Liu, S. Si, X. Zhu, Y. Li, and C.-J. Hsieh, “A unified framework for +data poisoning attack to graph-based semi-supervised learning,” arXiv +preprint arXiv:1910.14147, 2019. +[30] D. Cao, S. Chang, Z. Lin, G. Liu, and D. Sun, “Understanding distributed +poisoning attack in federated learning,” in 2019 IEEE 25th International +Conference on Parallel and Distributed Systems (ICPADS). IEEE, 2019. +[31] J. Shen, X. Zhu, and D. Ma, “TensorClog: An imperceptible poisoning +attack on deep neural network applications,” IEEE Access, vol. 7, 2019. +[32] J. Zhang, J. Chen, D. Wu, B. Chen, and S. Yu, “Poisoning Attack +in Federated Learning using Generative Adversarial Nets,” in 2019 +18th IEEE International Conference On Trust, Security And Privacy In +Computing And Communications/13th IEEE International Conference +On Big Data Science And Engineering (TrustCom/BigDataSE). +IEEE, +2019. +[33] W. Jiang, H. Li, S. Liu, Y. Ren, and M. He, “A Flexible Poisoning Attack +Against Machine Learning,” in ICC 2019-2019 IEEE International +Conference on Communications (ICC). +IEEE, 2019, pp. 1–6. +[34] H. Kwon, H. Yoon, and K.-W. Park, “Selective poisoning attack on deep +neural network to induce fine-grained recognition error,” in IEEE Sec- +ond International Conference on Artificial Intelligence and Knowledge +Engineering, 2019, pp. 136–139. +[35] H. Zhang, T. Zheng, J. Gao, C. Miao, L. Su, Y. Li, and K. Ren, “Data +poisoning attack against knowledge graph embedding,” in Proceedings +of the Twenty-Eighth International Joint Conference on Artificial +Intelligence, IJCAI-19. +International Joint Conferences on Artificial +Intelligence Organization, 7 2019, pp. 4853–4859. [Online]. Available: +https://doi.org/10.24963/ijcai.2019/674 +[36] E. Bagdasaryan, A. Veit, Y. Hua, D. Estrin, and V. Shmatikov, “How to +backdoor federated learning,” in International Conference on Artificial +Intelligence and Statistics. +PMLR, 2020, pp. 2938–2948. +[37] M. Li, Y. Sun, H. Lu, S. Maharjan, and Z. Tian, “Deep reinforcement +learning for partially observable data poisoning attack in crowdsensing +systems,” IEEE Internet of Things Journal, vol. 7, 2020. +[38] S. Sasaki, S. Hidano, T. Uchibayashi, T. Suganuma, M. Hiji, and S. Kiy- +omoto, “On embedding backdoor in malware detectors using machine +learning,” in IEEE International Conference on Privacy, Security and +Trust, 2019, pp. 1–5. +[39] X. Zhang, X. Zhu, and L. Lessard, “Online Data Poisoning Attack,” in +Learning for Dynamics and Control. +PMLR, 2020, pp. 201–210. +[40] G. Lovisotto, S. Eberz, and I. Martinovic, “Biometric backdoors: A +poisoning attack against unsupervised template updating,” in 2020 IEEE +European Symposium on Security and Privacy (EuroS&P). IEEE, 2020. +[41] C. Li, X. Chen, D. Wang, S. Wen, M. E. Ahmed, S. Camtepe, and +Y. Xiang, “Backdoor attack on machine learning based android malware +detectors,” IEEE Trans. on Dependable and Secure Computing, 2021. +[42] M. Kravchik, B. Biggio, and A. Shabtai, “Poisoning attacks on cyber +attack detectors for industrial control systems,” in Proceedings of the +36th Annual ACM Symposium on Applied Computing, 2021. +[43] B. Biggio, B. Nelson, and P. Laskov, “Poisoning attacks against support +vector machines,” arXiv preprint arXiv:1206.6389, 2012. +[44] C. Yang, Q. Wu, H. Li, and Y. Chen, “Generative Poisoning Attack +Method Against Neural Networks,” preprint arXiv:1703.01340, 2017. +[45] L. Mu˜noz-Gonz´alez, B. Biggio, A. Demontis, A. Paudice, V. Wongras- +samee, E. C. Lupu, and F. Roli, “Towards Poisoning of Deep Learning +Algorithms with Back-gradient Optimization,” in Proceedings of the +10th ACM workshop on Artificial Intelligence and Security, 2017. +[46] C. Miao, Q. Li, L. Su, M. Huai, W. Jiang, and J. Gao, “Attack under +Disguise: An Intelligent Data Poisoning Attack Mechanism in Crowd- +sourcing,” in Proceedings of the 2018 World Wide Web Conference, +2018. +[47] A. P. Dawid and A. M. Skene, “Maximum Likelihood Estimation of +Observer Error-Rates Using the EM Algorithm,” Journal of the Royal +Statistical Society: Series C (Applied Statistics), vol. 28, no. 1, 1979. +[48] M. Fang, G. Yang, N. Z. Gong, and J. Liu, “Poisoning attacks to +graph-based recommender systems,” in Proceedings of the 34th Annual +Computer Security Applications Conference, 2018, pp. 381–392. +[49] D. Arp, M. Spreitzenbarth, M. Hubner, H. Gascon, K. Rieck, and +C. Siemens, “Drebin: Effective and explainable detection of android +malware in your pocket.” in NDSS, vol. 14, 2014, pp. 23–26. +[50] H. Cai, N. Meng, B. Ryder, and D. Yao, “Droidcat: Effective android +malware detection and categorization via app-level profiling,” IEEE +Transactions on Information Forensics and Security, vol. 14, no. 6, 2018. +[51] E. Mariconti, L. Onwuzurike, P. Andriotis, E. De Cristofaro, G. Ross, +and G. Stringhini, “Mamadroid: Detecting android malware by building +markov chains of behavioral models,” preprint arXiv:1612.04433, 2016. + +[52] Y. Aafer, W. Du, and H. Yin, “Droidapiminer: Mining api-level features +for robust malware detection in android,” in International Conference +on Security and Privacy in Communication Systems. +Springer, 2013. +[53] “Malware +detection,” +https://www.kaggle.com/competitions/ +malware-detection/data. + diff --git a/GdAzT4oBgHgl3EQfHfsT/content/tmp_files/load_file.txt b/GdAzT4oBgHgl3EQfHfsT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4c52a69b48156b40fda3b02668c3a9aa6c7026c0 --- /dev/null +++ b/GdAzT4oBgHgl3EQfHfsT/content/tmp_files/load_file.txt @@ -0,0 +1,1376 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf,len=1375 +page_content='Analysis of Label-Flip Poisoning Attack on Machine Learning Based Malware Detector Kshitiz Aryal Department of Computer Science Tennessee Technological University Cookeville, TN, USA karyal42@tntech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='edu Maanak Gupta Department of Computer Science Tennessee Technological University Cookeville, TN, USA mgupta@tntech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='edu Mahmoud Abdelsalam Department of Computer Science North Carolina A&T State University Greensboro, NC, USA mabdelsalam1@ncat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='edu Abstract—With the increase in machine learning (ML) applica- tions in different domains, incentives for deceiving these models have reached more than ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' As data is the core backbone of ML algorithms, attackers shifted their interest towards polluting the training data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Data credibility is at even higher risk with the rise of state-of-art research topics like open design principles, federated learning, and crowd-sourcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Since the machine learn- ing model depends on different stakeholders for obtaining data, there are no existing reliable automated mechanisms to verify the veracity of data from each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Malware detection is arduous due to its malicious nature with the addition of metamorphic and polymorphic ability in the evolving samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' ML has proven to solve the zero-day malware detection problem, which is unresolved by traditional signature- based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The poisoning of malware training data can allow the malware files to go undetected by the ML-based malware detectors, helping the attackers to fulfill their malicious goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' A feasibility analysis of the data poisoning threat in the malware detection domain is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Our work will focus on two major sections: training ML-based malware detectors and poisoning the training data using the label-poisoning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We will analyze the robustness of different machine learning models against data poisoning with varying volumes of poisoning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Index Terms—Cybersecurity, Poisoning Attacks, Machine Learning, Malware Detectors, Adversarial Malware Analysis I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' INTRODUCTION Machine Learning (ML) techniques have been emerging rapidly, providing computational intelligence to various ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The ability of machine learning to generalize to unseen data has paved its way from labs to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It has already gained unprecedented success in many fields like image processing [1], [2], natural language processing [3], [4], recommendation systems used by Google, YouTube and Face- book, cybersecurity [5], [6], robotics [7], drug research [8], [9], and many other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' ML-based systems are achieving unparalleled performance through modern deep neural net- works bringing revolutions in AI-based services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Recent works have shown significant achievements in fields like self-driving cars and voice-controlled systems used by tech giants like autopilot in Tesla, Apple Siri, Amazon Alexa, and Microsoft Cortana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' With machine learning being applied to such critical applications, continuous security threats are never a bombshell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In addition to traditional security threats like malware at- tack [10], phishing [11], man-in-the-middle attack [12], denial- of-service [13], SQL injection [14], adversaries are finding novel ways to sneak into ML models [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Data poisoning and evasion attacks [16]–[20] are the latest menaces against the security of machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Poisoning attacks enable attackers to control the model’s behavior by manipulating a model’s data, algorithms, or hyper- parameters during the model training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' On the other hand, an evasion attack is carried out during the test time by manip- ulating the test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Adversaries can craft legitimate inputs imperceptible to humans but force models to make wrong predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [21] discovered the vulnerability of deep learning architecture against adversarial attacks, and ever since, there have been several major successful adversar- ial attacks against machine learning architectures [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sophisticated attackers are motivated by very high incentives to manipulate the result of the machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' With the current data scale with which machine learning models are trained, it is impossible to verify each data point individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In most scenarios, it is unlikely that an attacker gets access to training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' However, with many systems adopting online learning [24], crowd-sourcing [25] for training data, open design principles, and federated learning, poisoning attacks already pose a serious threat to ML models [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' There have been instances [27] when big companies have been compro- mised by a data poisoning attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Malware public databases like VirusTotal1, which rely on crowdsourced malware files for training its algorithm, can be poisoned by attackers while Google’s mail spam filter can be thrown out of track by wrong reporting of spam emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Data poisoning relates to adding training data that either leaves a backdoor on the model or negatively impacts the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Figure 1 shows the architecture of the poisoning attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In the given figure, the addition of poisoned data in the training bag forces the model to learn and predict so that attackers benefit from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' This type of poisoning is not limited to particular domains but has extended across all ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Label flipping attack is carried out to flip the prediction of machine learning detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Among all the existing approaches, we chose one of the simplest poisoning techniques called label poisoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We swap the 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='com/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='01044v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='CR] 3 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' General architecture for Poisoning Machine Learning Models existing training data labels in label poisoning to check the ML models’ robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In this work, we perform a comparative analysis of different machine learning-based malware detectors’ robustness against label-flipping data poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Unlike the existing ap- proaches, we are demonstrating the impact of simple label- switching data poisoning in different malware detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We will first train eight different ML models widely used to detect malware, namely Stochastic Gradient Descent (SGD), Random Forest (RF), Logistic Regression (LR), K-Nearest Neighbor Classifier (KNN), Linear Support Vector Machine (SVM), Decision Tree (DT), Perceptron, and Multi-Layer Perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' This will be followed by poisoning 10% and 20% of training data by flipping the label of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' All of the models are retrained after data poisoning, and the performance of each model is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The major contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We taxonomize the existing data poisoning attacks on ma- chine learning models in terms of domains, approaches, and targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We provide threat modeling for adversarial poisoning attacks against malware detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The threat is modeled in terms of the attack surface, the attacker’s knowledge, the attacker’s capability, and adversarial goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We train eight different machine learning-based malware detectors from malware data obtained from VirusTotal and VirusShare2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We compare the performance of these malware detectors with training and testing data in terms of accuracy, precision, and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Finally, we show a simple label-switching approach to poison the data without any knowledge of training models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 2https://virusshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='com/ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Taxonomy of poisoning attack on attack domain, approach and target The performance of malware detectors is analyzed while poisoning 10% and 20% of the total training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The existing literature for data poisoning attacks in different domains, including malware, is discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Section III provides the threat modeling for data poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' An overview of ML algorithms that are used to train the malware detector in this paper is discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Section V dis- cusses experimental methodology elaborating on the algorithm and the testbed used for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The evaluation and discussion on the performed experiments are given in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Finally, Section VII concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' LITERATURE REVIEW Data poisoning attacks have been used against the machine learning domain for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The existing literature on data poisoning attacks can be taxonomized in terms of attack domains, approach, and the target (victim), as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The recently trending technologies like crowd- sourcing and federated learning are always vulnerable as the veracity of individual data can never be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The recent victims of poisoning attacks have spread in security, network, and speech recognition domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We also classified the major approaches that are taken to produce or optimize the poisoning attacks in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The existing data poisoning approaches have targeted almost all the machine learning algorithms ranging from traditional algorithms like regression to modern deep neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Table I summarizes the existing literature on poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Biggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [43] attacked a support vector machine using gradient ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' To make poisoning attacks closer to the real world, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [44] used a generative adversarial network with an autoencoder to poison deep neural nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Gongalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [45] extended poisoning from binary learning to multi-class problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Shafahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [28] proposed a targeted clean label poisoning attack on neural networks using an optimization-based crafting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [31] performed an imperceptible poisoning attack on a deep neural network by clogging the back-propagation from gradient tensors during training while also minimizing the gradient norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Jiang et ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Machine learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Machine learning with ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='直盲自 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='自自自 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='18168 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='0100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='1818 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='0100 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Poisoned ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Prediction ModelData Poisoning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Domains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='> Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Neural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Crowd Sourcing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Reinforcement learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Support Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='> Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Label Flipping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Federated Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Generative Adversarial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Truth-Finder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Dawid-Skene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Empirical Inverstigations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Recommendation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='FakeUsers Insertion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Spectrum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Online learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='SpeechRecognitionTABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='DATA POISONING ATTACKS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Domains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Publications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Crowd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Sourcing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Federated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Online ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Reinforcement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Flipping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='GAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Others ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Neural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='SVM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Customized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Shafahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [28] √ √ √ Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [29] √ √ √ √ Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [30] √ √ √ Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [31] √ √ √ Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [32] √ √ √ Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [33] √ √ √ Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [34] √ √ √ Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [35] √ √ √ Bagdasaryal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [36] √ √ √ √ Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [37] √ √ √ Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [38] √ √ √ Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [39] √ √ √ √ Lovisotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [40] √ √ √ Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [41] √ √ √ √ Kravchik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [42] √ √ √ This Work √ √ √ √ √ Domains:Poisoning domain for crafted attack, Approach: Approach to poison the training data, Target: Target of poisoning attack al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [33] performed a flexible poisoning attack against linear and logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [34] could selectively poison particular classes against deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [30] proposed a distributed label-flipping poisoning approach to poison the DL model in federated architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Miao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [46] poisoned Dawid-Skene [47] model by exploiting the reliability degree of workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [48] proposed a poisoning attack against a graph-based recommendation system by maximizing the hit ratio of target items using fake users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In the given Table I, we can observe that only a handful of works have been carried out in the security domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [38] proposed an attack framework for backdoor embedding, which prevented the detection of specific types of malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' They generated poisoning samples by solving an op- timization problem and tested it against a logistic regression- based malware detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' To poison the Android malware de- tectors, Lie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [41] experimented backdoor poisoning attack against Drebin [49], DroidCat [50], MamaDroid [51] and DroidAPIMiner [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kravchik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [42] attacked the cyber attack detectors deployed in the industrial control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The back gradient optimization techniques used to pollute the train- ing data successfully poison the neural network-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' These works have focused their approach on some algorithm testing against some defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' However, none of the works compared the feebleness of multiple algorithms against data poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In this work, we demonstrate the effectiveness of label switch poisoning of the training data against eight machine learning algorithms widely used in malware detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' THREAT MODEL: KNOW THE ADVERSARY All security threats are defined in terms of their goals and attack capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Modeling the threat allows for identifying and better understanding the risk arriving with a threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' A poisoning attack is performed by manipulating the training data either at the initial learning or incremental learning Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Threat model for poisoning attack period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The threat model of a poisoning attack reflects the attacker’s knowledge, goal, capabilities, and attack surface, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Attack Surface: Attack surface denotes how the adversary attacks the model under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Machine learning algorithms require data to pass through different stages in the pipeline, and each stage offers some kind of vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In this work, we are only concerned about poisoning attacks which make the training data an attack surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Attacker’s Knowledge: The attacker’s knowledge is the amount of information about the model under attack that an attacker has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Based on the amount of knowledge of the attacker, the poisoning approach is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Attacker’s knowledge can be broadly classified into the two following categories: White box model: In the white box model, an attacker has complete information about the underlying target model, such as the algorithm used, training data, hyper-parameters, and gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It’s easier to carry out a white box attack due to the information available that helps the attacker to create a worst-case scenario for the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=" Black box model: In the black-box model, an attacker only Threat Model Attack Surface Attacker's Attacker's Capability Attacker's Goal Knowledge Training White Box Data Untargeted data model Injection Misclassifcation Black Box Data Targeted model Modification Misclassification Logic Confidence Corruption Reductionhas information about the model’s input and output." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' An attacker has no information about the internal structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Black-box models can also be divided further into complete black-box models and gray-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In the gray box model, the model’s performance for each input the attacker provides can be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' As such, the gray box attack is considered to be relatively easier than the complete black box model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In this paper, we perform a black box attack on different malware detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Our experiments will prove the vul- nerability of these models to random label poisoning attacks without having any information about the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Attacker’s Capability: The attacker’s capability represents the ability of an adversary to manipulate the data and model in different stages of the ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It defines the sections that can be manipulated, the mechanism used for manipulation, and constraints to the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Poisoning can be carried out in a well-controlled environment if the attacker has complete infor- mation about the underlying model and training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Attacker capabilities can be classified into the following categories: Data Injection: It is the ability to insert new data into the training dataset, leading machine learning models to learn on contaminated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Data Modification: It is the ability to access and modify the training data as well as the data labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Label flipping is a well-known approach carried out in poisoning attack domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Logic Corruption: It is the ability to manipulate the logic of ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' This ability is out of scope for data poisoning and is considered a model poisoning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Adversarial Goals: The attacker’s objective is to deceive the ML model by injecting poisoned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' However, poisoning training data might differ depending on the goals of an attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Attacker goals can be categorized as: Untargeted Misclassification: An attacker tries to change the model’s output to a value different than the original prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Untargeted misclassification is a relatively easier goal for attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Targeted Misclassification: An attacker’s goal is to add a certain backdoor in the models so that particular samples are classified to a chosen class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confidence Reduction: An attacker can also poison training data to reduce the confidence of the machine learning model for a particular prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In this approach, changing the classification label is unnecessary, but reducing the confi- dence score is enough to meet the attacker’s goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Our paper aims to cause the malware detector models to misclassify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' However, since we are dealing with binary clas- sification, it can be considered either targeted or untargeted misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' OVERVIEW OF MACHINE LEARNING ALGORITHMS Almost all of the ML architectures have already been victimized by data poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In this section, we will brief some ML architectures in which we performed data poisoning attacks later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Stochastic Gradient Descent: Stochastic gradient descent (SGD) is derived from the gradient descent algorithm, which is a popular ML optimization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' A gradient gives the slope of the function and measures the degree of change of a variable in response to the changes of another variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Starting from an initial value, gradient descent runs iteratively to find the optimal values of the parameters, which are the minimal possible value of the given cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In Stochastic Gradient Descent, a few samples are randomly selected in place of the whole dataset for each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The term batch determines the number of samples to calculate each iteration’s gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In normal gradient descent optimization, a batch is taken to be the whole dataset leading to the problem when the dataset gets big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Stochastic gradient descent considers a small batch in each iteration to lower the computing cost of the gradient descent approach while working with a large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Random Forest: A random forest is a supervised ML algorithm that is constructed from an ensemble of decision tree algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Its ensemble nature helps to provide a solution to complex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The random forest is made up of a large number of decision trees that have been trained via bagging or bootstrap aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The average mean of the output of constituent decision trees is the random forest’s ultimate forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The precision of the output improves as the number of decision trees used grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' A random forest overcomes the decision tree algorithm’s limitations by eliminating over-fitting and enhancing precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Logistic Regression: The probability for classification prob- lems is modeled using logistic regression, which divides them into two possible outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' For classification, logistic regression is an extension of the linear regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' For regression tasks, linear regression works well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' however, it fails to replicate for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The linear model considers the class a number and finds the optimum hyperplane that mini- mizes the distances between the points and the hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' As it interpolates between the points, it cannot be interpreted as probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Because there is no relevant threshold for class separation, logistic regression is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It is a widely used classification algorithm due to its ease of implementation and strong performance in linearly separable classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' K-Nearest Neighbors (KNN) Classifier: The KNN algorithm relies on the assumption that similar things exist in close proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It is a non-parametric and lazy learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' KNN does not carry any assumption for underlying data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It does not require training data points for model generation, as all the training data are used in a testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' This results in faster training and a slower testing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The costly testing phase will consume more time and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In KNN, K is the number of nearest neighbors and is generally considered odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' KNN, however, suffers from the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' With increased feature dimension, it requires more data and becomes prone to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Support Vector Machine (SVM): A support vector machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Algorithm 1: Data Poisoning Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Input: Non-poisoned feature set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Output: Poisoned feature set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Data: Static features obtained from malware and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='benign training set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='1 for all the samples do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Train the machine learning models and measure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='the performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='for 10% each of Malware and Benign data do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='if Training label is not flipped then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='label=Get training label of given data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='if label==0 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Flip the label to 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='else if label==1 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Flip the label to 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Train all the models and measure the performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='for 20% each of Malware and Benign data do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='if Training label is not flipped then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='label=Get training label of given data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='if label==0 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Flip the label to 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='else if label==1 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Flip the label to 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Train all the models and measure the performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='is a popular supervised ML algorithm applied in both classi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='fication and regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' SVM aims to find a hyperplane that classifies the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In SVM, there are several pos- sible hyperplanes, and we need to determine the optimal hy- perplane that maximizes the margin between the two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hyperplanes are the decision boundary for SVM, where data points near to hyperplane are the support vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Due to its effectiveness in high dimensional spaces and memory-efficient properties, it is widely adopted in different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Multi-Layer Perceptron: The term ’Perceptron’ is derived from the ability to perceive, see, and recognize images in a human-like manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' A perceptron machine is based on the neuron, a basic unit of computation, with a cell receiving a series of pairs of inputs and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Although the perceptron was originally thought to represent any circuit and logic, non- linear data cannot be represented by a perceptron with only one neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Multi-Layer Perceptron was developed to overcome this limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In multi-layer perceptron, the mapping between input and output is non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It has input and output layers and several hidden layers stacked with numerous neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Because the inputs are merged with the initial weights in a weighted sum and applied to the activation function, the multi-layer perceptron falls under the category of feedforward algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Each linear combination is propagated to the following layer, unlike with a perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' EXPERIMENTAL METHODOLOGY In this paper, we are using the label-flipping approach to poison the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' With source class CS and a target class CT from a set of classes C, the dataset DI is poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The detailed poisoning performed in the paper is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We perform a label poisoning attack of differ- ent volumes to training data without guiding the poisoning mechanism through machine learning architecture or the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' It is an efficient way to showcase the ability of random poisoning to hamper the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We are training all eight malware detector models three times in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' As illustrated in Algorithm 1, we begin the model training with clean data without adding any noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' After recording the model’s performance on clean data, we proceed towards the first stage of poisoning our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We take 10% of shuffled training data belonging to each malware and benign class, and we change their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We retrain all the models and again measure the performance of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We repeat the same operation with 20% of shuffled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The percentage of poisoned data is taken randomly for this experimental purpose, as the goal is to show the impact on the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The algorithm we followed in carrying out this experiment is not a novel approach but a generic approach to poison the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Experimental Environment and Dataset All the experiments are performed in Google-Colab us- ing Google’s GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' All the implementation will be worked around using python libraries and Scikit-Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The training dataset [53] is obtained from the Kaggle repository, where data are collected from VirusTotal and VirusShare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The dataset comprises windows PE malware and benign files processed through static executable analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The dataset comprises 216,352 files (75,503 benign files and 140,849 malware files) with 54 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' EVALUATION RESULTS AND ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Data Pre-processing and Transformation We begin our experiment by loading data from Kaggle dataset [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' To clean the data, we followed two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' First, we ignored rows that are missing more than 50% of data, whereas we replaced the null values with the arithmetic mean value of the column for rows with less than 50% missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Second, we normalized the data by scal- ing the values from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Afterward, 85% of data were used for training purposes while the remaining 15% were used for testing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We trained selected eight machine learning models with standard hyper-parameters for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We didn’t tweak many machine learning parameters to fine-tune the detection accuracy, resulting in significant overfitting in a few models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Performance Indicators We evaluated the malware detectors’ performance using the following metrics: TABLE II MALWARE DETECTION TRAINING RESULT Algorithm Clean Data Training Data Testing Data Accuracy Precision Recall F1 Accuracy Precision Recall F1 Stochastic Gradient Descent 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='41 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='49 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='29 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='34 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='98 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='77 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='20 Decision Tree 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='96 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='91 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='65 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='85 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='05 Random Forest 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='92 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='65 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='82 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='12 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 Logistic Regression 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='21 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='02 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='33 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='24 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='36 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='67 KNN Classifier 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='38 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='33 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='05 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='69 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='42 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='38 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='25 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='31 Support Vector Machine 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='15 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='44 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='51 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='91 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='03 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='89 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='34 Perceptron 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='93 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='91 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='72 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='39 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='28 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='86 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='50 Multi-Layer Perceptron 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='28 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='07 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='16 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='93 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='45 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='66 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='04 TABLE III MALWARE DETECTION PERFORMANCE WITH 10% POISONING DATA Algorithm 10% Poisoned Data Training Data Testing Data Accuracy Precision Recall F1 Accuracy Precision Recall F1 Stochastic Gradient Descent 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='12 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='49 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='14 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='73 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='39 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='23 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='38 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='77 Decision Tree 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='77 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='44 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='01 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='58 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='92 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='33 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='98 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='96 Random Forest 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='77 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='92 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='51 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='61 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='13 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='68 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='22 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='68 Logistic Regression 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='51 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='29 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='39 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='69 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='26 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='06 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='91 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='77 KNN Classifier 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='49 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='47 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='28 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='59 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='15 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='11 Support Vector Machine 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='75 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='84 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='42 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='96 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='99 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='14 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='16 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='73 Perceptron 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='78 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='69 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='25 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='16 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='89 31 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='25 Multi-Layer Perceptron 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='85 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='72 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='58 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='32 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='33 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='81 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='30 Accuracy = TP + TN TP + TN + FP + FN Precision = TP TP + FP , Recall = TP TP + FN F1-score = 2 ∗ (Precision ∗ Recall) Precision + Recall A positive outcome corresponds to a malware sample, while a negative result corresponds to a benign example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' TP, TN, FP, and FN are true positives, true negatives, false positives, and false negatives, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Accuracy is the percentage of correct predictions on the given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Precision measures the ratio between true positives and all the positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Recall provides the ability of our model to predict true positives correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The F1 score is the harmonic mean, the combination of a classifier’s precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Results and Discussion Table II shows the accuracy, precision, and recall for train- ing and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Stochastic Gradient Descent, Decision Trees, Random Forest, and Perceptron looked overfitted to training data compared to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Since the data volume is a little bit high, decision tree-based classifiers are prone to overfitting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We used shallow layer neural networks leading perceptron to overfit in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' However, classifiers like logistic regression, KNN classifier, and Support Vector Machine have shown the best performance in all three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We have compared the performance of both the training and testing sets as we have only poisoned the training data while preserving the test data from attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We flipped the labels of 10% training data as a poisoning attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' On poisoning 10% of total data, the performance metric for each detector is displayed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The results show the robustness of decision trees and random forest-based malware detectors compared to other malware detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We further poisoned 20% of total training data to see the impact of increased poisoned data in each model, whose results are shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The left-most confusion matrix in each of the figures from Figure 4 to Figure 11 shows the number of TP, TN, FP, and FN for each classifier on clean data, whereas the middle and right one shows results with 10% and 20% poisoning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In the confusion matrix, label ’0’ is for malware, and label ’1’ is for benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The top-left corner in the confusion matrix gives True Positive, the top- right corner gives False Positive, the bottom-left gives False Negative, and the bottom-right corner gives True Negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Analysis and Observations The goal of this work is to show the vulnerability of popular machine-learning models that are used for malware detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Results in Tables II, III and IV reflect the limitations of all the experimented machine learning models even with the basic label poisoning attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Figure 12 shows the ROC curve, comparing the models’ performance on the clean data, 10% and 20% poisoned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In the ROC curve, the blue curve corresponds to the performance of clean data, the orange curve corresponds to 10% poisoned data, and the green curve corresponds to the 20% poisoned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The curve closest to the top-left corner is the one performing best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We can infer from the graph that logistic regression, K-Nearest Neighbors, TABLE IV MALWARE DETECTION PERFORMANCE WITH 20% POISONING DATA Algorithm 20% Poisoned data Training Data Testing Data Accuracy Precision Recall F1 Accuracy Precision Recall F1 Stochastic Gradient Descent 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='56 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='65 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='21 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='83 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='69 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='86 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='72 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='71 Decision Tree 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='54 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='54 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='34 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='88 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='26 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='25 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='67 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='54 Random Forest 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='54 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='04 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='90 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='77 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='66 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='02 Logistic Regression 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='38 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='13 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='20 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='58 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='78 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='93 KNN Classifier 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='41 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='48 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='12 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='15 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='16 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='06 Support Vector Machine 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='58 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='45 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='51 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='39 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='74 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='37 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='79 Perceptron 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='58 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='57 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='57 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='37 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='28 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='28 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='28 Multi Layer Perceptron 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='45 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='03 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='85 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='81 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='66 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='94 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confusion Matrix for Stochastic Gradient Descent Based Malware Detector Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confusion Matrix for Decision Tree Based Malware Detector Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confusion Matrix for Random Forest-Based Malware Detector Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confusion Matrix for Logistic Regression Based Malware Detector Support Vector Machine, and Multi-Layer Perceptron are the best models on the clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' However, the distance between the three curves represents the robustness of the model toward the poisoning attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' If the separation between the curves of clean data and poisoning data is low, it infers that the poisoning attack has a minimal impact on the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In the ROC graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' we can observe that Random Forest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Logistic Regression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' K-Nearest Neighbors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' and Multi-Layer Perceptron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='ConfusionMatrixforSGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Predicted Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='12000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='12148 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='IClass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='10830 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='1936 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Actual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6000 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confusion Matrix for Support Vector Machine-Based Malware Detector Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confusion Matrix for Perceptron Based Malware Detector Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Confusion Matrix for Multi-Layer Perceptron Based Malware Detector have their graphs close to each other, proving their robustness against poisoned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Random Forest’s robustness can be attributed to its ensemble nature which helps it to capture better insights about the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The robustness of logistic regression and K-Nearest Neighbors can be due to the low dimensionality of our training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Further, we can observe the performance of models, like SVM and perceptron, doing better with the 20% poisoned data than with 10% poisoned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The gain in the performance of these models is due to unrestricted data poisoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Since we are not guiding our poisoning approach according to the models, further adding poisoning data after some threshold point slightly improves the models’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' In the end, even the least sophisticated attacks, like label poisoning, are causing the performance decay of the models to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' This further alerts us toward the catastrophic consequences of more sophisticated attacks like gradients and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' CONCLUSION In this work, we perform a feasibility analysis of label- flip poisoning attacks on ML-based malware detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We evaluated eight different ML models that are widely used in malware detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Spotting the lack of poisoning attacks work in the malware domain, this paper analyses the robustness of ML-based malware detectors against different volumes of poisoned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We observed the decay in performance of all the models while poisoning 10% and 20% of total training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The significant decrease in the performance of the models shows the severe vulnerability of malware detectors to guided poisoning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' We also observed differences in the effect of poisoning attacks across the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Our work is carried out within the limited scope of one generic poisoning algorithm and a single malware dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' There are few future research directions that are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' The malware detectors can be tested against many advanced poi- soning approaches using numerous datasets from the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='ConfusionMatrixforPerceptron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='Predicted Class ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ciregan, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Meier, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Schmidhuber, “Multi-column deep neural networks for image classification,” in 2012 IEEE Conference on Com- puter Vision and Pattern Recognition, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 3642–3649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Schmidhuber, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Meier, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ciresan, “Multi-column deep neural networks for image classification,” in 2012 IEEE Conference on Com- puter Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' IEEE Computer Society, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Chowdhary, “Natural Language Processing,” Fundamentals of Arti- ficial Intelligence, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 603–649, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hirschberg and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Manning, “Advances in Natural Language Processing,” Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 349, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 6245, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 261–266, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Tsai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hsu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lin, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lin, “Intrusion detection by machine learning: A review,” Expert Systems with Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 11 994–12 000, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Peiravian and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zhu, “Machine learning for android malware detec- KNNCleanData,auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='971460832443106 KNN 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8552929416737186 KNN 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8216704426092349SVM Clean Data, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='9073593040810904 SVM 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6002161213967441 SVM 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='73096877256933DT Clean Data, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='5872804184858597 DT 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='5037422175699491 DT 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='4169380476360457MLp CleanData,auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8939386759774156 MLP 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8088423576886244 MLP 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7514920551542543Perceptron Clean Data, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7824525005443335 Perceptron 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='3838246137390344 Perceptron 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='476954003915064SGD Clean Data, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7482652186900373 SGD 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7025164286923702 SGD 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6086523161420353RF Clean Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='auC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7456458394939469 RF 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7625879961069476 RF 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='711092219132663LR Clean Data, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='9073593040810904 LR 10% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='8125026745227009 LR 20% Poison, auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='7608190424784267tion using permission and api calls,” in 2013 IEEE 25th International Conference on Tools with Artificial Intelligence, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 300–305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kober and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Peters, “Learning motor primitives for robotics,” in 2009 IEEE International Conference on Robotics and Automation, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Manicavasaga, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lamichhane, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kandel, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Talbert, “Drug repurposing for rare orphan diseases using machine learning techniques,” in The International FLAIRS Conference Proceedings, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 35, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Dhakal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' McKay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Tanner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Cheng, “Artificial intelligence in the prediction of protein–ligand interactions: recent advances and future directions,” Briefings in Bioinformatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' bbab476, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Khouzani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sarkar, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Altman, “Maximum Damage Malware Attack in Mobile Wireless Networks,” IEEE/ACM Transactions on Networking, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1347–1360, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Singhal, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kapoor, “A literature survey on social engineering attacks: Phishing attack,” in 2016 International Conference on Computing, Communication and Automation (ICCCA), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Callegati, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Cerroni, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ramilli, “Man-in-the-Middle Attack to the HTTPS Protocol,” IEEE Security Privacy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Schuba, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Krsul, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kuhn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Spafford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sundaram, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zam- boni, “Analysis of a denial of service attack on TCP,” in Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1997 IEEE Symposium on Security and Privacy (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='97CB36097), 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Halfond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Viegas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Orso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=', “A classification of sql-injection attacks and countermeasures,” in Proceedings of the IEEE International Symposium on Secure Software Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' IEEE, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [15] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Yilmaz and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Masum, “Expansion of cyber attack data from unbalanced datasets using generative techniques,” arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='04549, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kolosnjaji, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Demontis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Biggio, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Maiorca, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Giacinto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Eckert, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Roli, “Adversarial malware binaries: Evading deep learning for malware detection in executables,” in IEEE European Signal Processing Conference, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 533–537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kreuk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Barak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Aviv-Reuven, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Baruch, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Pinkas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Keshet, “Adversarial examples on discrete sequences for beating whole-binary malware detection,” arXiv preprint :1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='04528, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Demetrio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Biggio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lagorio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Roli, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Armando, “Explain- ing Vulnerabilities of Deep Learning to Adversarial Malware Binaries,” arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='03583, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [19] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Suciu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Coull, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Johns, “Exploring adversarial examples in malware detection,” in 2019 IEEE Security and Privacy Workshops, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [20] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Aryal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Gupta, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Abdelsalam, “A Survey on Adversarial Attacks for Malware Analysis,” arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='08223, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Szegedy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zaremba, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sutskever, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Bruna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Erhan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Goodfellow, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Fergus, “Intriguing properties of neural networks,” arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6199, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Dinakarrao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Amberkar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Bhat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Dhavlle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sayadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sasan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Homayoun, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Rafatirad, “Adversarial attack on microarchitectural events based malware detectors,” in Proceedings of the 56th Annual Design Automation Conference 2019, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [23] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Tan, “Generating adversarial malware examples for black- box attacks based on GAN,” arXiv preprint arXiv:1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='05983, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Shalev-Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=', “Online learning and online convex optimiza- tion,” Foundations and Trends® in Machine Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 4, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Rai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Chintalapudi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Padmanabhan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sen, “Zee: Zero-effort Crowdsourcing for Indoor Localization,” in Proceedings of the 18th Annual International Conference on Mobile Computing and Networking, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 293–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Bonawitz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Eichner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Grieskamp, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Huba, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ingerman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ivanov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kiddon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Koneˇcn`y, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Mazzocchi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=', “Towards federated learning at scale: System design,” Proceedings of Machine Learning and Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 374–388, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [27] “Tay, Microsoft’s AI chatbot, gets a crash course in racism from Twitter,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='theguardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='com/technology/2016/mar/24/ tay-microsofts-ai-chatbot-gets-a-crash-course-in-racism-from-twitter, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Shafahi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Najibi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Suciu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Studer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Dumitras, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Goldstein, “Poison Frogs!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Targeted Clean-Label Poisoning At- tacks on Neural Networks,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [29] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Si, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Li, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hsieh, “A unified framework for data poisoning attack to graph-based semi-supervised learning,” arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='14147, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Cao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Chang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Liu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sun, “Understanding distributed poisoning attack in federated learning,” in 2019 IEEE 25th International Conference on Parallel and Distributed Systems (ICPADS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Shen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zhu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ma, “TensorClog: An imperceptible poisoning attack on deep neural network applications,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 7, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Chen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Yu, “Poisoning Attack in Federated Learning using Generative Adversarial Nets,” in 2019 18th IEEE International Conference On Trust, Security And Privacy In Computing And Communications/13th IEEE International Conference On Big Data Science And Engineering (TrustCom/BigDataSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [33] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ren, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' He, “A Flexible Poisoning Attack Against Machine Learning,” in ICC 2019-2019 IEEE International Conference on Communications (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kwon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Yoon, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Park, “Selective poisoning attack on deep neural network to induce fine-grained recognition error,” in IEEE Sec- ond International Conference on Artificial Intelligence and Knowledge Engineering, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 136–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [35] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Miao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Li, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ren, “Data poisoning attack against knowledge graph embedding,” in Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' International Joint Conferences on Artificial Intelligence Organization, 7 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 4853–4859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='24963/ijcai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='2019/674 [36] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Bagdasaryan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Veit, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hua, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Estrin, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Shmatikov, “How to backdoor federated learning,” in International Conference on Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 2938–2948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Maharjan, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Tian, “Deep reinforcement learning for partially observable data poisoning attack in crowdsensing systems,” IEEE Internet of Things Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 7, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Sasaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hidano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Uchibayashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Suganuma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hiji, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kiy- omoto, “On embedding backdoor in malware detectors using machine learning,” in IEEE International Conference on Privacy, Security and Trust, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [39] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Zhu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lessard, “Online Data Poisoning Attack,” in Learning for Dynamics and Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 201–210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lovisotto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Eberz, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Martinovic, “Biometric backdoors: A poisoning attack against unsupervised template updating,” in 2020 IEEE European Symposium on Security and Privacy (EuroS&P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Wen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ahmed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Camtepe, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Xiang, “Backdoor attack on machine learning based android malware detectors,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' on Dependable and Secure Computing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Kravchik, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Biggio, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Shabtai, “Poisoning attacks on cyber attack detectors for industrial control systems,” in Proceedings of the 36th Annual ACM Symposium on Applied Computing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [43] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Biggio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Nelson, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Laskov, “Poisoning attacks against support vector machines,” arXiv preprint arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='6389, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Chen, “Generative Poisoning Attack Method Against Neural Networks,” preprint arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='01340, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [45] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Mu˜noz-Gonz´alez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Biggio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Demontis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Paudice, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Wongras- samee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Lupu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Roli, “Towards Poisoning of Deep Learning Algorithms with Back-gradient Optimization,” in Proceedings of the 10th ACM workshop on Artificial Intelligence and Security, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [46] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Miao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Su, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Huai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Jiang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Gao, “Attack under Disguise: An Intelligent Data Poisoning Attack Mechanism in Crowd- sourcing,” in Proceedings of the 2018 World Wide Web Conference, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Dawid and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Skene, “Maximum Likelihood Estimation of Observer Error-Rates Using the EM Algorithm,” Journal of the Royal Statistical Society: Series C (Applied Statistics), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 1, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Fang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Yang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Gong, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Liu, “Poisoning attacks to graph-based recommender systems,” in Proceedings of the 34th Annual Computer Security Applications Conference, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 381–392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Arp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Spreitzenbarth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Hubner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Gascon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Rieck, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Siemens, “Drebin: Effective and explainable detection of android malware in your pocket.” in NDSS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 14, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 23–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Cai, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Meng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ryder, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Yao, “Droidcat: Effective android malware detection and categorization via app-level profiling,” IEEE Transactions on Information Forensics and Security, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' 6, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [51] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Mariconti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Onwuzurike, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Andriotis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' De Cristofaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Ross, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Stringhini, “Mamadroid: Detecting android malware by building markov chains of behavioral models,” preprint arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='04433, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [52] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Aafer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Du, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Yin, “Droidapiminer: Mining api-level features for robust malware detection in android,” in International Conference on Security and Privacy in Communication Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content=' [53] “Malware detection,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} +page_content='com/competitions/ malware-detection/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfHfsT/content/2301.01044v1.pdf'} diff --git a/GtAzT4oBgHgl3EQfUvxQ/vector_store/index.faiss b/GtAzT4oBgHgl3EQfUvxQ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..773ee10abb2e926bcc2bde79938e31a6bdedda67 --- /dev/null +++ b/GtAzT4oBgHgl3EQfUvxQ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:42001c617a2cfa4034219c76568dbfe5ea05626070f92a6f3d8fca8b9cdf868d +size 4063277 diff --git a/I9E1T4oBgHgl3EQfGAMz/content/tmp_files/2301.02908v1.pdf.txt b/I9E1T4oBgHgl3EQfGAMz/content/tmp_files/2301.02908v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..53fccf6219482970ae89f873e3c2ffdb96f843b5 --- /dev/null +++ b/I9E1T4oBgHgl3EQfGAMz/content/tmp_files/2301.02908v1.pdf.txt @@ -0,0 +1,1168 @@ +arXiv:2301.02908v1 [math.FA] 7 Jan 2023 +DYNAMICAL PROPERTIES AND SOME CLASSES OF +NON-POROUS SUBSETS OF LEBESGUE SPACES +STEFAN IVKOVI´C, SERAP ¨OZTOP, AND SEYYED MOHAMMAD TABATABAIE∗ +Abstract. In this paper, we introduce several classes of non-σ-porous +subsets of a general Lebesgue space. Also, we study some linear dynam- +ics of operators and show that the set of all non-hypercyclic vectors of a +sequences of weighted translation operators on Lp-spaces is not σ-porous. +1. Introduction +σ-porous sets, as a collection of very thin subsets of metric spaces, were +introduced and studied first time in [8] through a research on boundary be- +havior of functions, and then were applied in differentiation and Banach spaces +theories in [3, 14]. The concepts related to porosity have been active topics in +recent decades because they can be adapted for many known notions in several +kind of metric spaces; see the monograph [21]. σ-porous subsets of R are null +and of first category, while in every complete metric space without any isolated +points these two categories are different [20]. On the other hand, linear dy- +namics including hypercyclicity in operator theory received attention during +the last years; see books [2, 11] and for instance [6, 16, 17]. Recently, F. Bayart +in [1] through study of hypercyclic shifts (which was previously studied in [15]; +see also [10]) proved that the set of non-hypercyclic vectors of some classes of +weighted shift operators on ℓ2(Z) is a non-σ-porous set. This would be a new +example of a first category set which is not σ-porous. In this work, by some +idea from the proof of [1, Theorem1] first we introduce a class of non-σ-porous +subsets of general Lebesgue spaces, and then we develop the main result of [1] +to sequences of weighted translation operators on general Lebesgue spaces in +the context of discrete groups and hypergroups. In particular, we prove that +if p ≥ 1, K is a discrete hypergroup, (an) is a sequence with distinct terms in +K, and w : K → (0, ∞) is a bounded measurable function such that +� +n∈N +1 +w(a0)w(a1) . . . w(an)χ{an+1} ∈ Lp(K), +then the set of all non-hypercyclic vectors of the sequence (Λn)n is not σ- +porous, where the operators Λn are given in Definition 3.8. Also, we study +2010 Mathematics Subject Classification. 47A16, 28A05, 43A15, 43A62. +∗Corresponding author. +Key words and phrases. non-σ-porous sets, Lebesgue spaces, σ-porous operators, locally +compact groups, locally compact hypergroups, hypercyclic vectors. +1 + +2 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +non-σ-porosity of non-hypercyclic vectors of weighted composition operators +on L∞(Ω) for a general measure space Ω equipped with a nonnegative Radon +measure and on Lp(R, τ), where τ is the Lebesgue measure on R. We show +that if G is a locally compact group, µ is a left Haar measure on G, a ∈ G, +and w : G → (0, ∞) be a weight such that +� +1 +w(a)w(a2) . . . w(an) +� +n ∈ L∞(G, µ), +then the set of all non-hypercyclic vectors of the weighted translation operator +Ta,w,∞ on L∞(G, µ) is not σ-porous. +2. Non-σ-porous subsets of Lebesgue spaces +In this section, we will introduce some classes of non-σ-porous subsets of +Lebesgue spaces related to a fixed function. First, we recall the definition of +the main notion of this paper. +Definition 2.1. Let 0 < λ < 1. A subset E of a metric space X is called +λ-porous at x ∈ E if for each δ > 0 there is an element y ∈ B(x; δ) \ {x} such +that +B(y; λ d(x, y)) ∩ E = ∅. +E is called λ-porous if it is λ-porous at every element of E. Also, E is called +σ-λ-porous if it is a countable union of λ-porous subsets of X. +The following lemma plays a key role in the proof of main results of this +section. This fact is a special case of [19, Lemma2]; see also [1, Lemma2]. +Lemma 2.2. Let F be a non-empty family of non-empty closed subsets of a +complete metric space X such that for each F ∈ F and each x ∈ X and r > 0 +with B(x; r) ∩ F ̸= ∅, there exists an element J ∈ F such that +∅ ̸= J ∩ B(x; r) ⊆ F ∩ B(x; r) +and F ∩B(x; r) is not λ-porous at all elements of J ∩B(x; r). Then, every set +in F is not σ-λ-porous. +The next result is a development of of [1, Theorem1]. Same as [1], the proof +of this theorem is based on Lemma 2.2. +Theorem 2.3. Let p ≥ 1, Ω be a locally compact Hausdorff space, µ be a +nonnegative Radon measure on Ω, and A ⊆ Ω be a Borel set such that +|f|χA ≤ ∥f∥p a.e. +(f ∈ Lp(Ω, µ)). +(2.1) +Then, for each measurable function g on Ω with gχA ∈ Lp(Ω, µ), the set +Γg := +� +f ∈ Lp(Ω, µ) : |f| ≥ |g|χA a.e. +� +is not σ-porous in Lp(Ω, µ). + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +3 +Proof. Fix an arbitrary number 0 < λ ≤ 1 +2, and pick 0 < β < λ. Denote +F := +� +Γg : gχA ∈ Lp(Ω, µ) +� +. +We will show that the collection F satisfies the conditions of Lemma 2.2. +Let g ∈ Lp(Ω, µ). Without lossing the generality, we can assume that g is a +nonnegative function. Trivially, Γg ̸= ∅. Let (fn) be a sequence in Γg and +fn → f in Lp(Ω, µ). Then, by (2.1), |f| ≥ gχA a.e., and so f ∈ Γg. Therefore, +every element of the collection F is a closed subset of Lp(Ω, µ). Now, assume +that f ∈ Lp(Ω, µ) and r > 0 with B(f; r) ∩ Γg ̸= ∅. We find a measurable +function h with 0 ≤ hχA ∈ Lp(Ω, µ) such that +∅ ̸= B(f; r) ∩ Γh ⊆ B(f; r) ∩ Γg, +and B(f; r) ∩ Γg is not λ-porous at elements of B(f; r) ∩ Γh. +Since +� +|f| + β−1gχA +�p ∈ L1(Ω, µ) and µ is a Radon measure, the mapping +ν defined by +ν(B) := +� +B +� +|f| + β−1gχA +�p dµ +(for every Borel set B ⊆ Ω) +is a Radon measure [9]. Hence, there are some 0 < ǫ < 1, a function k ∈ +B(f; r) ∩ Γg and a compact subset D of Ω with µ(D) > 0 such that +∥k − f∥p < ǫ1/p r. +and +� +Dc +� +|f| + β−1gχA +�p dµ < (1 − ǫ) rp. +(2.2) +Pick some α with +∥k − f∥p < α < ǫ1/p r, +and denote +δ := ǫ1/p r − α +2µ(D) +1 +p +. +Now, we define two functions h, ξ : Ω → C by +h := (gχA + δ)χD + β−1gχA χΩ\D +and +ξ := (|k| + δ)η χD + hχΩ\D, +where +η(x) := + + + +k(x) +|k(x)|, +if k(x) ̸= 0 +1, +if k(x) = 0 +for all x ∈ Ω. Since D is compact, we have hχA ∈ Lp(Ω, µ). Also, for each +x ∈ D, +|k(x) − ξ(x)| = +��k(x) − +� +|k(x)| + δ +� +η(x) +�� += +��k(x) − k(x) − δ η(x) +�� += δ, + +4 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +and therefore +∥(ξ − k) χD∥p = δ µ(D) +1 +p = ǫ1/p r − α +2 +. +This implies that +∥(ξ − f) χD∥p ≤ ∥(ξ − k) χD∥p + ∥(k − f) χD∥p +≤ ǫ1/p r − α +2 ++ α < ǫ1/p r. +Hence, +∥ξ − f∥p +p = +� +D +|ξ − f|p dµ + +� +Ω\D +|ξ − f|p dµ +< ǫ rp + +� +Ω\D +|β−1gχA − f|p dµ +≤ ǫ rp + +� +Ω\D +(β−1gχA + |f|)p dµ +< ǫ rp + (1 − ǫ) rp = rp, +and so, ξ ∈ B(f; r). Moreover, +|ξ(x)| = |k(x)| + δ ≥ g(x) + δ = h(x) +a.e. on D ∩ A, +and for each x ∈ (Ω \ D) ∩ A we have |ξ(x)| = h(x). This shows that ξ ∈ Γh, +and so +∅ ̸= B(f; r) ∩ Γh ⊆ B(f; r) ∩ Γg +because h ≥ g. Now, let u ∈ B(f; r)∩Γh and put r′ := min{δ, λ (r−∥f −u∥p)}. +Let v ∈ B(u; r′). We define the function γ : Ω → C by +γ(x) := + + + + + +v(x), +if x ∈ D +� +|v(x)| + β|u(x) − v(x)| +� +θ(x), +if x ∈ Ω \ D +where +θ(x) := + + + +v(x) +|v(x)|, +if v(x) ̸= 0 +1, +if v(x) = 0. +Therefore, for each x ∈ Ω \ D we have +|γ(x) − v(x)| = β |u(x) − v(x)| +and +|γ(x)| ≥ β |u(x)|. +Easily, +∥γ − v∥p +p = ∥(γ − v) χD∥p +p + ∥(γ − v) χΩ\D∥p +p += ∥(γ − v) χΩ\D∥p +p += βp ∥(u − v) χΩ\D∥p +p +≤ βp ∥u − v∥p +p < λp ∥u − v∥p +p, + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +5 +and hence, +γ ∈ B +� +v; λ ∥u − v∥p +� +⊆ B(f; r). +In addition, +|γ(x)| ≥ β |u(x)| ≥ β h(x) = g(x) +for a.e. x ∈ (Ω \ D) ∩ A +and +|γ(x)| = |v(x)| ≥ |u(x)| − δ ≥ g(x) +for a.e. x ∈ D ∩ A, +because ∥u − v∥p ≤ δ and also |u| ≥ h. Therefore, +B +� +v; λ ∥u − v∥p +� +∩ B(f; r) ∩ Γg ̸= ∅. +and this competes the proof. +□ +Remark 2.4. Note that, in general, the condition (2.1) in the statement of +Theorem 2.3 does not implies that Ω is a discrete space. In particular, if in +the condition (2.1) we set A := Ω, then it implies that Lp(Ω, µ) ⊆ L∞(Ω, µ), +and this inclusion is equivalent to +α := inf{µ(E) : µ(E) > 0} > 0, +(2.3) +and equivalently, for each q > p, Lp(Ω, µ) ⊆ Lq(Ω, µ); see [18]. If in addition, +suppµ = Ω, then the condition (2.3) implies that for each x ∈ Ω, +µ({x}) = inf{µ(F) : F is a compact neighborhood of x} > 0. +Specially, if Ω is a locally compact group (or hypergroup) and µ is a left Haar +measure of it, then the condition (2.1) implies that Ω is a discrete topological +space. +The next result is a direct conclusion of Theorem 2.3. +Corollary 2.5. Let Ω be a discrete topological space and ϕ := (ϕj)j∈Ω ⊆ +[1, ∞) such that for each j, ϕj ≥ 1. Put µϕ := � +j∈Ω ϕj δj, where δj is the +point-mass measure at j. Then, for each g ∈ Lp(Ω, µϕ), the set +Γg := +� +f ∈ Lp(Ω, µϕ) : |f| ≥ |g| +� +is not σ-porous in Lp(Ω, µϕ). +Proof. Just note that for each k ∈ Ω and f ∈ Lp(Ω, µϕ), +∥f∥p +p = +� +j∈Ω +|f(j)|p µϕ({j}) ≥ |f(k)| ϕk ≥ |f(k)|p. +□ +In particular, if a set is endowed with the counting measure, we get the fact. +Corollary 2.6. Let p ≥ 1 and A be a non-empty set. Then, for each g ∈ +ℓp(A), the set +Γg := +� +f ∈ ℓp(A) : |f| ≥ |g| +� +is not σ-porous in ℓp(A). +The situation for L∞-spaces is different. + +6 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +Theorem 2.7. Let Ω be a locally compact Hausdorff space and µ be a non- +negative Radon measure on Ω. Then, for each g ∈ L∞(Ω, µ), the set +Γg := +� +f ∈ L∞(Ω, µ) : |f| ≥ |g| a.e. +� +is not σ-porous in L∞(Ω, µ). +Proof. Same as the proof of Theorem 2.3 fix 0 < λ ≤ 1 +2, and set +F := +� +Γg : g ∈ L∞(Ω, µ) +� +. +This collection satisfies the conditions of Lemma 2.2. Trivially, Γg is a closed +subset of L∞(Ω, µ) for all g ∈ L∞(Ω, µ). Let Assume that 0 ≤ g ∈ L∞(Ω, µ), +and let f ∈ L∞(Ω, µ) and r > 0. +If B(f; r) ∩ Γg ̸= ∅, we choose some +k ∈ B(f; r) ∩ Γg and we find some ε ∈ (0, 1) such that ∥k − f∥∞ < εr. Pick +some δ ∈ (0, (1 − ε)r), and set +h := g + δ +and +ξ := (|k| + δ)η, +where η is as in the proof of Theorem 2.3. Then, we get +∥ξ − f∥∞ ≤ ∥ξ − k∥∞ + ∥k − f∥∞ ≤ δ + εr < (1 − ε)r + εr = r, +so ξ ∈ B(f; r), and ξ ∈ Γh since |k| ≥ g. Next, let u ∈ B(f; r) ∩ Γh, and set +r′ := min{δ, λ (r − ∥f − u∥∞)}. Pick some v ∈ B(u; r′). Then, ∥u − v∥∞ < +r′ ≤ δ, so |v| ≥ |u| − δ ≥ h − δ = g a.e., so v ∈ Γg. Thus, +v ∈ B(v; λ∥u − v∥∞) ∩ B(f; r) ∩ Γg. +This completes the proof. +□ +Remark 2.8. The main Theorem 2.3 is valid also for the sequence space c0, +because the sequences with finitely many non-zero coefficients approximate +sequences in c0. +At the end of this section, we give a class of non-σ-porous subsets of the +Lp-space on real line. In the proof of this result, which is also based on Lemma +2.2, we apply some functions defined in the proof of Theorem 2.3. +Theorem 2.9. Let p ≥ 1, and τ be the Lebesgue measure on R. For each +g ∈ Lp(R, τ) put +Θg := +� +f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥gχ[m,m+1]∥p for all m ∈ Z +� +. +Then, Θg is not σ-porous in Lp(R, τ). +Proof. Let 0 < λ ≤ 1 +2, and 0 < β < λ. Denote +F := +� +Θg : g ∈ Lp(R, τ) +� +. +We prove that the collection F satisfies the conditions of Lemma 2.2. Let +0 ≤ g ∈ Lp(R, τ). Then, easily Θg ̸= ∅ and it is closed in Lp(R, τ). Now, +assume that f ∈ Lp(R, τ) and r > 0 with B(f; r)∩Θg ̸= ∅. Then, there exists + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +7 +a large enough number N ∈ N, some 0 < ǫ < 1 and a function k ∈ B(f; r)∩Θg +such that +∥k − f∥p < ǫ +1 +p r +and +� +[−N,N]c +� +|f| + β−1g +�p dτ < (1 − ǫ) rp. +Pick some α with ∥k − f∥p < α < ǫ +1 +p r, and denote δ := ǫ +1 +p r−α +2(2N) +1 +p . Put +A1 := {m ∈ [N] : g = 0 a.e. on [m, m + 1]}, +A2 := [N] \ A1 +and +B1 := {m ∈ [N] : k = 0 a.e. on [m, m + 1]}, +B2 := [N] \ B1, +where [N] := {−N, . . . , N − 1}, and then define +ρ := +� +m∈A1 +χ[m,m+1] + +� +m∈A2 +gχ[m,m+1] +∥gχ[m,m+1]∥p +, +and +η := +� +m∈B1 +χ[m,m+1] + +� +m∈B2 +kχ[m,m+1] +∥kχ[m,m+1]∥p +. +Now, we define h, ξ : R → C by +h := gχ[−N,N] + δρ + β−1g χ[−N,N]c +and +ξ := |k| χ[−N,N] + δη + hχ[−N,N]c. +Clearly, h ∈ Lp(R, τ). For each x ∈ [−N, N] we have |k(x) − ξ(x)| = δ |η(x)|, +and so +∥(k − ξ)χ[−N,N]∥p +p = δp ∥ηχ[−N,N]∥p +p += δp +� +m∈[N] +∥ηχ[m,m+1]∥p +p += δp 2N. +Hence, ∥(k − ξ)χ[−N,N]∥p = δ (2N) +1 +p . Now, similar to the proof of Theorem +2.3 we have ξ ∈ B(f; r). Moreover, +∥ξχ[m,m+1]∥p = ∥kχ[m,m+1]∥p + δ ≥ ∥gχ[m,m+1]∥p + δ = ∥hχ[m,m+1]∥p +for all m ∈ [N]. And also for each m /∈ [N], +∥ξχ[m,m+1]∥p = ∥hχ[m,m+1]∥p ≥ ∥gχ[m,m+1]∥p. +So, +ξ ∈ B(f; r) ∩ Θh ⊆ B(f; r) ∩ Θg. + +8 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +Now, let u ∈ B(f; r) ∩ Θh and put r′ := min{δ, λ (r − ∥f − u∥p)}. Let v ∈ +B(u; r′). We define the function γ : R → C by +γ(x) := + + + + + +v(x), +if x ∈ [−N, N] +� +|v(x)| + β|u(x) − v(x)| +� +θ(x), +if x ∈ [−N, N]c +where +θ(x) := + + + +v(x) +|v(x)|, +if v(x) ̸= 0 +1, +if v(x) = 0. +Similar to the proof of Theorem 2.3, we have γ ∈ B +� +v; λ ∥u − v∥p +� +. Now, for +each m /∈ [N], +|γ|χ(m,m+1) = (|v| + β|u − v|)χ(m,m+1) ≥ β|u|χ(m,m+1). +Hence, +∥γχ[m,m+1]∥p ≥ β ∥uχ[m,m+1]∥p ≥ β ∥hχ[m,m+1]∥p +since u ∈ B(f; r) ∩ Θh. However, in this case we have (m, m + 1) ∈ [−N, N]c, +so hχ(m,m+1) = β−1gχ(m,m+1). Thus, β∥hχ[m,m+1]∥p = ∥gχ[m,m+1]∥p. If m ∈ +[N], we have γχ[m,m+1] = vχ[m,m+1] because γχ[−N,N] = vχ[−N,N] and [m, m+ +1] ⊆ [−N, N]. We get +�� ∥uχ[m,m+1]∥p − ∥vχ[m,m+1]∥p +�� ≤ ||(u − v)χ[m,m+1]∥p ≤ ∥u − v∥p < δ +because v ∈ B(u; r′), hence +∥γχ[m,m+1]∥p = ∥vχ[m,m+1]∥p +≥ ∥uχ[m,m+1]∥p − δ +≥ ∥hχ[m,m+1]∥p − δ += ∥gχ[m,m+1]∥p. +Therefore, +γ ∈ B +� +v; λ ∥u − v∥p +� +∩ B(f; r) ∩ Θg, +and the proof is complete. +□ +3. Applications +In this section, we will apply the results of the previous section, to prove that +the set of all non-hypercyclic vectors of some sequences of weighted translation +operators is non-σ-porous. +Definition 3.1. Let X be a Banach space. A sequence (Tn)n∈N0 of operators +in B(X) is called hypercyclic if there is an element x ∈ X (called hypercyclic +vector) such that the orbit {Tn(x) : n ∈ N0} is dense in X. The set of all +hypercyclic vectors of a sequence (Tn)n∈N0 is denoted by HC((Tn)n∈N0). An +operator T ∈ B(X) is called hypercyclic if the sequence (T n)n∈N0 is hyper- +cyclic. + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +9 +Let G be a locally compact group and a ∈ G. +Then, for each function +f : G → C we define Laf : G → C by Laf(x) := f(a−1x) for all x ∈ G. Note +that if p ≥ 1, then the left translation operator +La : Lp(G) → Lp(G), +f �→ Laf +is not hypercyclic because ∥La∥ ≤ 1. Hypercyclicity of weigted translation +operators on Lp(G) and regarding an aperiodic element a was studied in [5] +(an element a ∈ G is called aperiodic if the closed subgroup of G generated by +a is not compact). +Definition 3.2. Let G be a locally compact group with a left Haar measure +µ. Fix p ≥ 1. We denote Lp(G) := Lp(G, µ). Assume that w : G → (0, ∞) +is a bounded measurable function (called a weight) and a ∈ G. Then, the +weighted translation operator Ta,w,p : Lp(G) → Lp(G) is defined by +Ta,w,p(f) := w Laf, +(f ∈ Lp(G)). +For each n ∈ N we denote ϕn := w Law . . . Lan−1w, where a0 := e, the +identity element of G. +Theorem 3.3. Let p ≥ 1, G be a discrete group and a ∈ G. Let µ be a left +Haar measure on G with µ({e}) ≥ 1 and (γn)n be an unbounded sequence of +non-negative integers. Let w : G → (0, ∞) be a bounded function such that for +some finite nonempty set F ⊆ G and some N > 0 we have +aγnF ∩ F = ∅ +(n ≥ N), +and +β := inf +� γn +� +k=1 +w(akt) : n ≥ N, t ∈ F +� +> 0. +Then, the set +Λ := +� +f ∈ Lp(G, µ) : ∥T γn +a,w,pf − χF ∥p ≥ µ(F) +1 +p for all n ≥ N +� +is non-σ-porous. + +10 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +Proof. Let Γ := {f ∈ Lp(G, µ) : |f| ≥ +1 +βχF }. Then, Γ is not σ-porous in +Lp(G, µ) thanks to Theorem 2.7. Also, for each f ∈ Γ and n ≥ N we have +∥T γn +a,w,pf − χF∥p +p = +� +G +| +n +� +k=1 +w(a−γn+kx) f(a−γnx) − χF (x)|p dµ(x) += +� +G +| +γn +� +k=1 +w(akx) f(x) − χF(aγnx)|p dµ(x) += +� +G +| +γn +� +k=1 +w(akx) f(x) − χa−γnF (x)|p dµ(x) +≥ +� +F +| +γn +� +k=1 +w(akx) f(x) − χa−γnF (x)|p dµ(x) += +� +F +| +γn +� +k=1 +w(akx) f(x)|p dµ(x) +≥ +� +F +|β 1 +β |p dµ(x) += µ(F). +This completes the proof. +□ +Example 3.4. Let G be the additive group Z with the counting measure. +Let F be a finite non-empty subset of Z. Put N := max{|j| : j ∈ F}. If +w := (wn)n∈Z ⊆ (0, ∞) is a bounded sequence with wn ≥ 1 for all n ≥ N. +Then the required conditions in the previous theorem hold with respect to F +and a := 1. +The following fact is a direct conclusion of the previous theorem. +Corollary 3.5. Let p ≥ 1, G be a discrete group and a ∈ G with infinite order. +Let µ be the counting measure on G and (γn)n be an unbounded sequence of +non-negative integers. Let w : G → (0, ∞) be a bounded function such that for +some t ∈ G, +inf +� γn +� +k=1 +w(akt) : n ∈ N +� +> 0. +Then, the set +� +f ∈ Lp(G, µ) : ∥T γn +a,wf − χ{t}∥p ≥ 1 for all n +� +is non-σ-porous. +Theorem 3.6. Let p ≥ 1, G be a discrete group, and a ∈ G. Let µ be a left +Haar measure on G with µ({e}) ≥ 1. Let (γn)n be an unbounded sequence of +non-negative integers and let w : G → (0, ∞) be a bounded function such that +inf +n∈N +γn +� +k=1 +w(ak) > 0. + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +11 +Then, the set +Γ := +� +f ∈ Lp(G, µ) : |f(e)| inf +n∈N +γn +� +k=1 +w(ak) ≥ 1 +� +is non-σ-porous. In particular, setting Tn := T γn +a,w,p for all n, the set of all +non-hypercyclic vectors of the sequence (Tn)n is not σ-porous in Lp(G, µ). +Proof. Since µ({e}) ≥ 1, applying Theorem 2.3 the set Γ is non-σ-porous, +because +[ inf +n∈N +γn +� +k=1 +w(ak)]−1 χ{e} ∈ Lp(G, µ). +Let f ∈ Γ. If n is a nonnegative integer, then for every x in G we have +∥Tnf∥p ≥ +��ϕγn(x) Laγn f(x) +��, +and so setting x = aγn we have +∥Tnf∥p ≥ +��ϕn(aγn) Laγn f(aγn) +�� += +� γn +� +k=1 +w(ak) +� +|f(e)| +≥ |f(e)| inf +m∈N +γm +� +k=1 +w(ak) ≥ 1. +This implies that the set {Tnf : n ∈ N} is not dense in Lp(G, µ), and so Γ +is a subset of the set of all non-hypercyclic vectors of T. This completes the +proof. +□ +Now, we recall the definition of hypergroups which are generalizations of +locally compact groups; see the monograph [4] and the basic paper [12] for +more details. In locally compact hypergroups the convolution of two Dirac +measures is not necessarily a Dirac measure. +Let K be a locally compact +Hausdorff space. We denote by M(K) the space of all regular complex Borel +measures on K, and by δx the Dirac measure at the point x. The support of +a measure µ ∈ M(K) is denoted by supp(µ). +Definition 3.7. Suppose that K is a locally compact Hausdorff space, (µ, ν) �→ +µ∗ν is a bilinear positive-continuous mapping from M(K)×M(K) into M(K) +(called convolution), and x �→ x− is an involutive homeomorphism on K (called +involution) with the following properties: +(i) +M(K) with ∗ is a complex associative algebra; +(ii) +if x, y ∈ K, then δx∗δy is a probability measure with compact support; +(iii) +the mapping (x, y) �→ supp(δx ∗ δy) from K × K into C(K) is contin- +uous, where C(K) is the set of all non-empty compact subsets of K +equipped with Michael topology; +(iv) there exists a (necessarily unique) element e ∈ K (called identity) such +that for all x ∈ K, δx ∗ δe = δe ∗ δx = δx; + +12 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +(v) for all x, y ∈ K, e ∈ supp(δx ∗ δy) if and only if x = y−; +Then, K ≡ (K, ∗,− , e) is called a locally compact hypergroup. +A nonzero nonnegative regular Borel measure m on K is called the (left) +Haar measure if for each x ∈ K, δx∗m = m. For each x, y ∈ K and measurable +function f : K → C we denote +f(x ∗ y) := +� +K +f d(δx ∗ δy), +while this integral exists. +Definition 3.8. Suppose that a := (an)n∈N0 is a sequence in a hypergroup +K, and w is a weight function on K. For each n ∈ N0 we define the bounded +linear operator Λn+1 on Lp(K) by +Λn+1f(x) := w(a0 ∗ x) w(a1 ∗ x) . . . w(an ∗ x) f(an+1 ∗ x) +(f ∈ Lp(K)) +for all x ∈ K. Also, we assume that Λ0 is the identity operator on Lp(K). +Some linear dynamical properties of this sequence of operators were studied +in [13]. The sequence {Λn}n is a generalization of the usual powers of a single +weighted translation operator on Lp(G), where G is a locally compact group. +In fact, any locally compact group G with the mapping +µ ∗ ν �→ +� +G +� +G +δxydµ(x)dν(y) +(µ, ν ∈ M(G)) +as convolution, and x �→ x−1 from G onto G as involution is a locally compact +hypergroup. Let η := (an)n∈N0 be a sequence in G, and w be a weight on G. +Then for each f ∈ Lp(G), n ∈ N0 and x ∈ G, we have +Λn+1f(x) = w(a0x) w(a1x) . . . w(anx) f(an+1x). +In particular, let a ∈ G and for each n ∈ N0, put an := a−n. Then, Λn = T n +a,w,p +for all n ∈ N. In this case, the operator Ta,w,p is hypercyclic if and only if the +sequence (Λn)n is hypercyclic. +Let K be a discrete hypergroup with the convolution ∗ between Radon +measures of K and the involution ·− : K → K. Then, by [12, Theorem7.1A], +the measure µ on K given by +µ({x}) := +1 +δx ∗ δx−({e}), +(x ∈ K) +(3.1) +is a left Haar measure on K. +Proposition 3.9. Let K be a discrete hypergroup, µ be the Haar measure +(3.1), and p ≥ 1. Then for each g ∈ Lp(K, µ), the set +� +f ∈ Lp(K, µ) : |f| ≥ |g| +� +is not σ-porous in Lp(K, µ). + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +13 +Proof. Just note that for each x ∈ K we have µ({x}) ≥ 1 because +1 = δx ∗ δx−(K) ≥ δx ∗ δx−({e}). +Hence, the measure space (K, µ) satisfies the condition of Corollary 2.5. +□ +Let a := (an)n∈N be a sequence in a discrete hypergroup K such that +an ̸= am for each m ̸= n, and let w : K → (0, ∞) be bounded. We define +ha,w : K → C by +ha,w := +� +n∈N0 +1 +w(a0)w(a1) . . . w(an)χ{an+1}. +Theorem 3.10. Let p ≥ 1, and K be a discrete hypergroup endowed with the +left Haar measure (3.1). Let a := (an)n∈N0 ⊆ K with distinct terms, and w be +a weight on K such that ha,w ∈ Lp(K). Then, the set of all non-hypercyclic +vectors of the sequence (Λn)n is not σ-porous. +Proof. First, thanks to Proposition 3.9, the set +E := +� +f ∈ Lp(K) : |f(an+1)| ≥ +1 +w(a0)w(a1) . . . w(an) for all n +� +is not σ-porous because it equals to the set +� +f ∈ Lp(K) : |f| ≥ ha,w +� +. Now, +for each f ∈ E, +∥Λn+1f∥p ≥ sup +x∈K +w(a0 ∗ x) w(a1 ∗ x) . . . w(an ∗ x) |f(an+1 ∗ x)| +≥ w(a0) w(a1) . . . w(an) |f(an+1)| ≥ 1 +for all n ∈ N0. This implies that 0 does not belong to the closure of {Λnf : +n ∈ N} in Lp(K), and so E ⊆ [HC((Λn)n)]c. This completes the proof. +□ +Since any group is a hypergroup, we can give the fact below. +Corollary 3.11. Let p ≥ 1, and G be a discrete group. Let a ∈ G be of +infinite order, (γn)n∈N0 ⊆ N be with distinct terms and w : G → (0, ∞) be a +weight such that +� +1 +w(aγ0)w(aγ1) . . . w(aγn) +� +n ∈ ℓp(G). +Then, the set of all non-hypercyclic vectors of the sequence (T γn +a,w,p)n is not +σ-porous in ℓp(G). +Now, we can write the next corollary which is a generalization of [1, Theorem +1]. +Corollary 3.12. Let p ≥ 1, (γn)n ⊆ N be strictly increasing and (wn)n∈Z be +a bounded sequence in (0, ∞) such that +� +1 +wγ0wγ1wγ2 . . . wγn +� +n ∈ ℓp(Z). + +14 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +Then, the set of all non-hypercyclic vectors of the sequence (Tn)n is not σ- +porous, where +(Tn+1a)k := wγ0wγ1wγ2 . . . wγnak+γn+1 +(k ∈ N0) +for all a := (aj)j ∈ ℓp(Z). +Applying Theorem 2.7 we can speak regarding some more general situation +in the case of p = ∞. Let Ω be a locally compact Hausdorff space endowed with +a nonnegative Radon measure µ. Let w : Ω → (0, ∞) be a bounded measurable +function, and α : Ω → Ω be a bi-measurable mapping such that ∥f ◦ α±1∥∞ = +∥f∥∞ for all f ∈ L∞(Ω, µ). Then, we define Tα,w,∞ : L∞(Ω, µ) → L∞(Ω, µ) +by +Tα,w,∞(f) := w (f ◦ α) +(f ∈ L∞(Ω, µ)). +If Ω be a locally compact group and a ∈ Ω, setting αa(x) := ax for all x ∈ Ω, +we denote Ta,w,∞ := Tαa,w,∞. Note that α−1 means the inverse function of α, +and for each k ∈ N, α−k := (α−1)k. +Theorem 3.13. Let Tα,w,∞ be the weighted composition operator defined as +above and let {γn}n ⊆ N be a fixed unbounded sequence. Suppose that there +exists a sequence {An}n of disjoint subsets of Ω with µ(An) > 0 for all n such +that +jα,w := +� +n∈N +1 +(w ◦ α−γn) (w ◦ α−γn+1) . . . (w ◦ α−1)χAn ∈ L∞(Ω, µ). +Then, the set {f ∈ L∞(Ω, µ) : ∥T γn +α,w,∞(f)∥∞ ≥ 1 for all n} is not σ-porous. +In particular, the set of all non-hypercyclic vectors of the sequence {T γn +α,w,∞}n +is not σ-porous. +Proof. Let E := {f ∈ L∞(Ω, µ) : |f| ≥ jα,w}. Then, E is not σ-porous thanks +to Theorem 2.7. For each f ∈ E and n ∈ N we have +∥T γn +α,w,∞(f)∥∞ = ∥ +γn +� +k=1 +(w ◦ αγn−k) (f ◦ αγn)∥∞ += ∥ +γn +� +k=1 +(w ◦ α−k) f∥∞ +≥ ∥ +γn +� +k=1 +(w ◦ α−k) χAn f∥∞ +≥ ∥ +γn +� +k=1 +(w ◦ α−k) χAn jα,w∥∞ += 1. +This completes the proof. +□ + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +15 +Corollary 3.14. Let G be a locally compact group and µ be a left Haar mea- +sure on G. Let a ∈ G and w : G → (0, ∞) be a bounded measurable function. +Then, if +� +1 +w(a)w(a2) . . . w(an) +� +n ∈ L∞(G, µ), +then the set of all non-hypercyclic vectors of the operator Ta,w,∞ on L∞(G, µ) +is not σ-porous. +Corollary 3.15. If (wn)n∈Z is a bounded sequence such that +� +1 +w1 . . . wn +� +n ∈ ℓ∞, +then the set of all non-hypercyclic vectors of the sequence (Tγn,w)n is not σ- +porous in ℓ∞. +Theorem 3.16. Let Tα,w,∞ be the weighted composition operator on L∞(Ω, µ) +and let F ⊆ Ω be a Borel set with 0 < µ(F) < ∞. Let there exists a constant +N > 0 such that for all n ≥ N, +αn(F) ∩ F = ∅, +(3.2) +and +β := inf{ +n +� +k=1 +(w ◦ α−k)(t) : n ≥ N, t ∈ F} ̸= 0. +Then, the set +{f ∈ L∞(Ω, µ) : ∥T n +α,w,∞f − χF∥∞ ≥ 1 for all n ≥ N} +is not σ-porous in L∞(Ω, µ). +Proof. Let Γ := {f ∈ L∞(Ω, µ) : |f| ≥ 1 +βχF }. Then by Theorem 2.7, Γ is not +σ-porous in L∞(Ω, µ). Also, for each f ∈ Γ we have +∥T n +α,w,∞f − χF ∥∞ = ∥ +n +� +k=1 +(w ◦ αn−k) (f ◦ αn) − χF∥∞ += ∥ +n +� +k=1 +(w ◦ α−k) f − χF ◦ αn∥∞ += ∥ +n +� +k=1 +(w ◦ α−k) f − χαn(F )∥∞ +≥ ∥ +n +� +k=1 +(w ◦ α−k) fχF − χαn(F )χF∥∞ += ∥ +n +� +k=1 +(w ◦ α−k) fχF∥∞ +≥ β∥fχF∥∞ ≥ β 1 +β ∥χF∥∞ = 1. + +16 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +This completes the proof. +□ +Example 3.17. Let Ω := R and µ be the Lebesgue measure. Put α(t) := t−1 +for all t ∈ R and F := [0, 1]. If w ∈ Cb(R) such that |w(t)| ≥ 1 for all t ≥ k > 0 +and inf{|w(t)| : t ∈ [0, 1]} > 0, then the required conditions in the previous +theorem hold with respect to F. +With some similar proof, one can prove the next fact without the condition +(3.2). +Theorem 3.18. Let Tα,w,∞ be the weighted composition operator on L∞(Ω, µ) +and let F ⊆ Ω be a Borel set with 0 < µ(F) < ∞ such that +inf{ +n +� +k=1 +(w ◦ α−k)(t) : n ≥ N, t ∈ F} ̸= 0. +Then, the set +{f ∈ L∞(Ω, µ) : ∥T n +α,w,∞f∥∞ ≥ 1 for all n ≥ N} +is not σ-porous in L∞(Ω, µ). In particular, the set of all non-hypercyclic vec- +tors of the operator Tα,w,∞ is not σ-porous. +In sequel, we find some application for Theorem 2.9 regarding hypercyclicity +of shift operators on Lp(R, τ). +Theorem 3.19. Consider the weighted translation operator Tα,w on Lp(R, τ) +given by Tα,wf := w · (f ◦α), where 0 < w, w−1 ∈ Cb(R) and α(t) = t + 1. For +each n ∈ N put An := [n, n + 1] = αn([0, 1]). Set +yα,w := +� +n∈N +1 +inft∈An +�n +k=1(w ◦ α−k)(t)χAn +and assume that yα,w ∈ Lp(R, τ) (in particular inft∈An +�n +k=1(w ◦ α−k)(t) > 0 +for all n ∈ N). Then, the set +{f ∈ Lp(R, τ) : ∥T n +α,w(f)∥p ≥ 1 for all n ∈ N} +is not σ-porous. +Proof. By Theorem 2.9, the set +E := {f ∈ Lp(R, τ) : ∥fχAn∥p ≥ ∥yα,wχAn∥p for all n ∈ N} +is not σ-porous, because it equals to +{f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥yα,wχ[m,m+1]∥p for all m ∈ Z}, + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +17 +as yα,wχ[m,m+1] = 0 for all m ∈ Z with m ≤ 0. Now, note that for each f ∈ E +and n ∈ N, +∥T n +α,w(f)∥p +p = +� +R +� n +� +k=1 +(w ◦ αn−k)(t) +�p +|(f ◦ αn)(t)|p dτ += +� +R +� n +� +k=1 +(w ◦ α−k)(t) +�p +|f(t)|p dτ +≥ +� +An +� n +� +k=1 +(w ◦ α−k)(t) +�p +|f(t)|p dτ +≥ inf +t∈An +� n +� +k=1 +(w ◦ α−k)(t) +�p +∥yα,wχAn∥p +p += inf +t∈An +� n +� +k=1 +(w ◦ α−k)(t) +�p +1 +inft∈An [�n +k=1(w ◦ α−k)(t)]p τ(An) = 1. +□ +Assume now that there exists some l ∈ Z such that +β := inf{ +n +� +k=1 +(w ◦ α−k)(t) : t ∈ [l, l + 1], n ∈ N} > 0. +Put +F := {f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥ 1 +β χ[l,l+1]χ[m,m+1]∥p for all m ∈ Z}. +So by Theorem 2.9, F is not σ-porous. For every f ∈ F we have +∥T n +α,w(f)∥p +p = +� +R +� n +� +k=1 +(w ◦ αn−k)(t) +�p +|(f ◦ αn)(t)|p dτ += +� +R +� n +� +k=1 +(w ◦ α−k)(t) +�p +|f(t)|p dτ +≥ +� +[l,l+1] +� n +� +k=1 +(w ◦ α−k)(t) +�p +|f(t)|p dτ +≥ 1. +Hence, the set +{f ∈ Lp(R, τ) : ∥T n +α,w(f)∥p ≥ 1 for all n ∈ N} +is not σ-porous. +Next, suppose that α is an aperiodic function on R (this means that for each +compact set C ⊂ R, there exists a constant N > 0 such that αn(C) ∩ C = ∅ +for all n ≥ N) and β > 0, where β is as above. Then, the set +{f ∈ Lp(R, τ) : ∥T n +α,w(f) − χ[l,l+1]∥p ≥ 1 for all n ≥ N} + +18 +S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE +is not σ-porous. Indeed, for all f ∈ F, and n ≥ N we have +∥T n +α,w(f) − χ[l,l+1]∥p ≥ ∥ +n +� +k=1 +(w ◦ α−k) fχ[l,l+1]∥p +by the similar calculations as in the proof of Theorem 3.16. However, +∥ +n +� +k=1 +(w ◦ α−k) fχ[l,l+1]∥p ≥ β ∥fχ[l,l+1]∥p ≥ 1. +References +1. F. Bayart, Porosity and hypercyclic operators, Proc. Amer. Math. Soc. 133(11) (2005) +3309-3316. +2. F. Bayart and ´E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math. +179, Cambridge University Press, Cambridge, 2009. +3. C. L. Belna, M. J. Evans and P.D.Humke, Symmetric and ordinary differentiation, Proc. +Amer. Math. Soc. 72(2) (1978) 261-267. +4. W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, +De Kruyter, Berlin, 1995. +5. C-C. Chen and C-H. Chu, Hypercyclic weighted translations on groups, Proc. Amer. +Math. Soc. 139 (2011) 2839-2846. +6. C-C. Chen, S. ¨Oztop and S. M. Tabatabaie, Disjoint dynamics on weighted Orlicz spaces, +Disjoint dynamics on weighted Orlicz spaces, Complex Anal. Oper. Theory, 14(72) +(2020). https://doi.org/10.1007/s11785-020-01034-x +7. C.-C. Chen and S. M. Tabatabaie, Chaotic operators on hypergroups, Oper. Mat. 12(1) +(2018) 143-156. +8. E. P. Dolˇzenko, Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. +Mat. 31 (1967) 3-14. +9. G.B. Folland, Real Analysis, Modern Techniques and Their Applications; Second Edition, +John Wiley and Sons, Inc. New York (1999). +10. K-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000) +47-68. +11. K-G. Grosse-Erdmann and A. Peris, Linear Chaos, Universitext, Springer, 2011. +12. R. I. Jewett, Spaces with an abstract convolution of measures, Adv. Math., 18 (1975) +1-101. +13. V. Kumar and S. M. Tabatabaie, Hypercyclic sequences of weighted translations on hy- +pergroups, Semigroup Forum, 103 (2021) 916–934. +14. D. Preiss and L. Zaj´ıˇcek, Fr´echet differentiation of convex functions in a Banach space +with a separable dual, Proc. Amer. Math. Soc. 91(2) (1984) 202–204. +15. H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995) 993-1004. +16. Y. Sawano, S. M. Tabatabaie and F. Shahhoseini, Disjoint Dynamics of Weighted +Translations +on +Solid +Spaces, +Topology +Appl. +298, +107709, +14 +pp. +(2021) +DOI:10.1016/J.TOPOL.2021.107709 +17. S. M. Tabatabaie and S. Ivkovic, Linear dynamics of cosine operator functions on solid +Banach function spaces, Positivity, 25 (2021) 1437–1448. +18. A. Villani, Another note on the inclusion Lp(µ) ⊂ Lq(µ), Amer. Math. Monthly, 92 +(1985) 485–487. +19. L. Z´ajiˇcek, Porosity and σ-porous, Real Anal. Exchange 13 (1987/1988) 314-350. +20. L. Zaj´ıˇcek, Small non-σ-porous sets in topologically complete metric spaces, Colloq. +Math. 77(2) (1998) 293-304. +21. L. Z´ajiˇcek, On σ-porous sets in abstract spaces, Abstr. Appl. Anal. 5 (2005) 509-534. + +DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS +19 +Mathematical Institute of the Serbian Academy of Sciences and Arts, p.p. +367, Kneza Mihaila 36, 11000 Beograd, Serbia. +Email address: stefan.iv10@outlook.com +Department of Mathematics, Faculty of Science, Istanbul University, Istan- +bul, Turkey +Email address: oztops@istanbul.edu.tr +Department of Mathematics, University of Qom, Qom, Iran. +Email address: sm.tabatabaie@qom.ac.ir + diff --git a/I9E1T4oBgHgl3EQfGAMz/content/tmp_files/load_file.txt b/I9E1T4oBgHgl3EQfGAMz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d77296452179c38d54a885d2d93456ce30c5b189 --- /dev/null +++ b/I9E1T4oBgHgl3EQfGAMz/content/tmp_files/load_file.txt @@ -0,0 +1,675 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf,len=674 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='02908v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='FA] 7 Jan 2023 DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS OF LEBESGUE SPACES STEFAN IVKOVI´C, SERAP ¨OZTOP, AND SEYYED MOHAMMAD TABATABAIE∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In this paper, we introduce several classes of non-σ-porous subsets of a general Lebesgue space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Also, we study some linear dynam- ics of operators and show that the set of all non-hypercyclic vectors of a sequences of weighted translation operators on Lp-spaces is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Introduction σ-porous sets, as a collection of very thin subsets of metric spaces, were introduced and studied first time in [8] through a research on boundary be- havior of functions, and then were applied in differentiation and Banach spaces theories in [3, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The concepts related to porosity have been active topics in recent decades because they can be adapted for many known notions in several kind of metric spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' see the monograph [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' σ-porous subsets of R are null and of first category, while in every complete metric space without any isolated points these two categories are different [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' On the other hand, linear dy- namics including hypercyclicity in operator theory received attention during the last years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' see books [2, 11] and for instance [6, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Recently, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Bayart in [1] through study of hypercyclic shifts (which was previously studied in [15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' see also [10]) proved that the set of non-hypercyclic vectors of some classes of weighted shift operators on ℓ2(Z) is a non-σ-porous set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This would be a new example of a first category set which is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In this work, by some idea from the proof of [1, Theorem1] first we introduce a class of non-σ-porous subsets of general Lebesgue spaces, and then we develop the main result of [1] to sequences of weighted translation operators on general Lebesgue spaces in the context of discrete groups and hypergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In particular, we prove that if p ≥ 1, K is a discrete hypergroup, (an) is a sequence with distinct terms in K, and w : K → (0, ∞) is a bounded measurable function such that � n∈N 1 w(a0)w(a1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an)χ{an+1} ∈ Lp(K), then the set of all non-hypercyclic vectors of the sequence (Λn)n is not σ- porous, where the operators Λn are given in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Also, we study 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 47A16, 28A05, 43A15, 43A62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' non-σ-porous sets, Lebesgue spaces, σ-porous operators, locally compact groups, locally compact hypergroups, hypercyclic vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 1 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE non-σ-porosity of non-hypercyclic vectors of weighted composition operators on L∞(Ω) for a general measure space Ω equipped with a nonnegative Radon measure and on Lp(R, τ), where τ is the Lebesgue measure on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We show that if G is a locally compact group, µ is a left Haar measure on G, a ∈ G, and w : G → (0, ∞) be a weight such that � 1 w(a)w(a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an) � n ∈ L∞(G, µ), then the set of all non-hypercyclic vectors of the weighted translation operator Ta,w,∞ on L∞(G, µ) is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Non-σ-porous subsets of Lebesgue spaces In this section, we will introduce some classes of non-σ-porous subsets of Lebesgue spaces related to a fixed function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' First, we recall the definition of the main notion of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let 0 < λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' A subset E of a metric space X is called λ-porous at x ∈ E if for each δ > 0 there is an element y ∈ B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' δ) \\ {x} such that B(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' λ d(x, y)) ∩ E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' E is called λ-porous if it is λ-porous at every element of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Also, E is called σ-λ-porous if it is a countable union of λ-porous subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The following lemma plays a key role in the proof of main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This fact is a special case of [19, Lemma2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' see also [1, Lemma2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let F be a non-empty family of non-empty closed subsets of a complete metric space X such that for each F ∈ F and each x ∈ X and r > 0 with B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ F ̸= ∅, there exists an element J ∈ F such that ∅ ̸= J ∩ B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ⊆ F ∩ B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) and F ∩B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) is not λ-porous at all elements of J ∩B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, every set in F is not σ-λ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The next result is a development of of [1, Theorem1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Same as [1], the proof of this theorem is based on Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, Ω be a locally compact Hausdorff space, µ be a nonnegative Radon measure on Ω, and A ⊆ Ω be a Borel set such that |f|χA ≤ ∥f∥p a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (f ∈ Lp(Ω, µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1) Then, for each measurable function g on Ω with gχA ∈ Lp(Ω, µ), the set Γg := � f ∈ Lp(Ω, µ) : |f| ≥ |g|χA a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' � is not σ-porous in Lp(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Fix an arbitrary number 0 < λ ≤ 1 2, and pick 0 < β < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Denote F := � Γg : gχA ∈ Lp(Ω, µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We will show that the collection F satisfies the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let g ∈ Lp(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Without lossing the generality, we can assume that g is a nonnegative function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Trivially, Γg ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let (fn) be a sequence in Γg and fn → f in Lp(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1), |f| ≥ gχA a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=', and so f ∈ Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Therefore, every element of the collection F is a closed subset of Lp(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, assume that f ∈ Lp(Ω, µ) and r > 0 with B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We find a measurable function h with 0 ≤ hχA ∈ Lp(Ω, µ) such that ∅ ̸= B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γh ⊆ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg, and B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg is not λ-porous at elements of B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Since � |f| + β−1gχA �p ∈ L1(Ω, µ) and µ is a Radon measure, the mapping ν defined by ν(B) := � B � |f| + β−1gχA �p dµ (for every Borel set B ⊆ Ω) is a Radon measure [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Hence, there are some 0 < ǫ < 1, a function k ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg and a compact subset D of Ω with µ(D) > 0 such that ∥k − f∥p < ǫ1/p r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' and � Dc � |f| + β−1gχA �p dµ < (1 − ǫ) rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2) Pick some α with ∥k − f∥p < α < ǫ1/p r, and denote δ := ǫ1/p r − α 2µ(D) 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, we define two functions h, ξ : Ω → C by h := (gχA + δ)χD + β−1gχA χΩ\\D and ξ := (|k| + δ)η χD + hχΩ\\D, where η(x) := \uf8f1 \uf8f2 \uf8f3 k(x) |k(x)|, if k(x) ̸= 0 1, if k(x) = 0 for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Since D is compact, we have hχA ∈ Lp(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Also, for each x ∈ D, |k(x) − ξ(x)| = ��k(x) − � |k(x)| + δ � η(x) �� = ��k(x) − k(x) − δ η(x) �� = δ, 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE and therefore ∥(ξ − k) χD∥p = δ µ(D) 1 p = ǫ1/p r − α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This implies that ∥(ξ − f) χD∥p ≤ ∥(ξ − k) χD∥p + ∥(k − f) χD∥p ≤ ǫ1/p r − α 2 + α < ǫ1/p r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Hence, ∥ξ − f∥p p = � D |ξ − f|p dµ + � Ω\\D |ξ − f|p dµ < ǫ rp + � Ω\\D |β−1gχA − f|p dµ ≤ ǫ rp + � Ω\\D (β−1gχA + |f|)p dµ < ǫ rp + (1 − ǫ) rp = rp, and so, ξ ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Moreover, |ξ(x)| = |k(x)| + δ ≥ g(x) + δ = h(x) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' on D ∩ A, and for each x ∈ (Ω \\ D) ∩ A we have |ξ(x)| = h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This shows that ξ ∈ Γh, and so ∅ ̸= B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γh ⊆ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg because h ≥ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, let u ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r)∩Γh and put r′ := min{δ, λ (r−∥f −u∥p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let v ∈ B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We define the function γ : Ω → C by γ(x) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 v(x), if x ∈ D � |v(x)| + β|u(x) − v(x)| � θ(x), if x ∈ Ω \\ D where θ(x) := \uf8f1 \uf8f2 \uf8f3 v(x) |v(x)|, if v(x) ̸= 0 1, if v(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Therefore, for each x ∈ Ω \\ D we have |γ(x) − v(x)| = β |u(x) − v(x)| and |γ(x)| ≥ β |u(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Easily, ∥γ − v∥p p = ∥(γ − v) χD∥p p + ∥(γ − v) χΩ\\D∥p p = ∥(γ − v) χΩ\\D∥p p = βp ∥(u − v) χΩ\\D∥p p ≤ βp ∥u − v∥p p < λp ∥u − v∥p p, DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 5 and hence, γ ∈ B � v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' λ ∥u − v∥p � ⊆ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In addition, |γ(x)| ≥ β |u(x)| ≥ β h(x) = g(x) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' x ∈ (Ω \\ D) ∩ A and |γ(x)| = |v(x)| ≥ |u(x)| − δ ≥ g(x) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' x ∈ D ∩ A, because ∥u − v∥p ≤ δ and also |u| ≥ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Therefore, B � v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' λ ∥u − v∥p � ∩ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' and this competes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Note that, in general, the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1) in the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3 does not implies that Ω is a discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In particular, if in the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1) we set A := Ω, then it implies that Lp(Ω, µ) ⊆ L∞(Ω, µ), and this inclusion is equivalent to α := inf{µ(E) : µ(E) > 0} > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3) and equivalently, for each q > p, Lp(Ω, µ) ⊆ Lq(Ω, µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' see [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If in addition, suppµ = Ω, then the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3) implies that for each x ∈ Ω, µ({x}) = inf{µ(F) : F is a compact neighborhood of x} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Specially, if Ω is a locally compact group (or hypergroup) and µ is a left Haar measure of it, then the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1) implies that Ω is a discrete topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The next result is a direct conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Ω be a discrete topological space and ϕ := (ϕj)j∈Ω ⊆ [1, ∞) such that for each j, ϕj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Put µϕ := � j∈Ω ϕj δj, where δj is the point-mass measure at j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, for each g ∈ Lp(Ω, µϕ), the set Γg := � f ∈ Lp(Ω, µϕ) : |f| ≥ |g| � is not σ-porous in Lp(Ω, µϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Just note that for each k ∈ Ω and f ∈ Lp(Ω, µϕ), ∥f∥p p = � j∈Ω |f(j)|p µϕ({j}) ≥ |f(k)| ϕk ≥ |f(k)|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ In particular, if a set is endowed with the counting measure, we get the fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1 and A be a non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, for each g ∈ ℓp(A), the set Γg := � f ∈ ℓp(A) : |f| ≥ |g| � is not σ-porous in ℓp(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The situation for L∞-spaces is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Ω be a locally compact Hausdorff space and µ be a non- negative Radon measure on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, for each g ∈ L∞(Ω, µ), the set Γg := � f ∈ L∞(Ω, µ) : |f| ≥ |g| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' � is not σ-porous in L∞(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Same as the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3 fix 0 < λ ≤ 1 2, and set F := � Γg : g ∈ L∞(Ω, µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This collection satisfies the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Trivially, Γg is a closed subset of L∞(Ω, µ) for all g ∈ L∞(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Assume that 0 ≤ g ∈ L∞(Ω, µ), and let f ∈ L∞(Ω, µ) and r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg ̸= ∅, we choose some k ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg and we find some ε ∈ (0, 1) such that ∥k − f∥∞ < εr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Pick some δ ∈ (0, (1 − ε)r), and set h := g + δ and ξ := (|k| + δ)η, where η is as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, we get ∥ξ − f∥∞ ≤ ∥ξ − k∥∞ + ∥k − f∥∞ ≤ δ + εr < (1 − ε)r + εr = r, so ξ ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r), and ξ ∈ Γh since |k| ≥ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Next, let u ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γh, and set r′ := min{δ, λ (r − ∥f − u∥∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Pick some v ∈ B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, ∥u − v∥∞ < r′ ≤ δ, so |v| ≥ |u| − δ ≥ h − δ = g a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=', so v ∈ Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Thus, v ∈ B(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' λ∥u − v∥∞) ∩ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The main Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3 is valid also for the sequence space c0, because the sequences with finitely many non-zero coefficients approximate sequences in c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' At the end of this section, we give a class of non-σ-porous subsets of the Lp-space on real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In the proof of this result, which is also based on Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2, we apply some functions defined in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, and τ be the Lebesgue measure on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For each g ∈ Lp(R, τ) put Θg := � f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥gχ[m,m+1]∥p for all m ∈ Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, Θg is not σ-porous in Lp(R, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let 0 < λ ≤ 1 2, and 0 < β < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Denote F := � Θg : g ∈ Lp(R, τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We prove that the collection F satisfies the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let 0 ≤ g ∈ Lp(R, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, easily Θg ̸= ∅ and it is closed in Lp(R, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, assume that f ∈ Lp(R, τ) and r > 0 with B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r)∩Θg ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, there exists DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 7 a large enough number N ∈ N, some 0 < ǫ < 1 and a function k ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r)∩Θg such that ∥k − f∥p < ǫ 1 p r and � [−N,N]c � |f| + β−1g �p dτ < (1 − ǫ) rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Pick some α with ∥k − f∥p < α < ǫ 1 p r, and denote δ := ǫ 1 p r−α 2(2N) 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Put A1 := {m ∈ [N] : g = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' on [m, m + 1]}, A2 := [N] \\ A1 and B1 := {m ∈ [N] : k = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' on [m, m + 1]}, B2 := [N] \\ B1, where [N] := {−N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' , N − 1}, and then define ρ := � m∈A1 χ[m,m+1] + � m∈A2 gχ[m,m+1] ∥gχ[m,m+1]∥p , and η := � m∈B1 χ[m,m+1] + � m∈B2 kχ[m,m+1] ∥kχ[m,m+1]∥p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, we define h, ξ : R → C by h := gχ[−N,N] + δρ + β−1g χ[−N,N]c and ξ := |k| χ[−N,N] + δη + hχ[−N,N]c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Clearly, h ∈ Lp(R, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For each x ∈ [−N, N] we have |k(x) − ξ(x)| = δ |η(x)|, and so ∥(k − ξ)χ[−N,N]∥p p = δp ∥ηχ[−N,N]∥p p = δp � m∈[N] ∥ηχ[m,m+1]∥p p = δp 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Hence, ∥(k − ξ)χ[−N,N]∥p = δ (2N) 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, similar to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3 we have ξ ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Moreover, ∥ξχ[m,m+1]∥p = ∥kχ[m,m+1]∥p + δ ≥ ∥gχ[m,m+1]∥p + δ = ∥hχ[m,m+1]∥p for all m ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' And also for each m /∈ [N], ∥ξχ[m,m+1]∥p = ∥hχ[m,m+1]∥p ≥ ∥gχ[m,m+1]∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' So, ξ ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Θh ⊆ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Θg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE Now, let u ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Θh and put r′ := min{δ, λ (r − ∥f − u∥p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let v ∈ B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We define the function γ : R → C by γ(x) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 v(x), if x ∈ [−N, N] � |v(x)| + β|u(x) − v(x)| � θ(x), if x ∈ [−N, N]c where θ(x) := \uf8f1 \uf8f2 \uf8f3 v(x) |v(x)|, if v(x) ̸= 0 1, if v(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Similar to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3, we have γ ∈ B � v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' λ ∥u − v∥p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, for each m /∈ [N], |γ|χ(m,m+1) = (|v| + β|u − v|)χ(m,m+1) ≥ β|u|χ(m,m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Hence, ∥γχ[m,m+1]∥p ≥ β ∥uχ[m,m+1]∥p ≥ β ∥hχ[m,m+1]∥p since u ∈ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Θh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' However, in this case we have (m, m + 1) ∈ [−N, N]c, so hχ(m,m+1) = β−1gχ(m,m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Thus, β∥hχ[m,m+1]∥p = ∥gχ[m,m+1]∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If m ∈ [N], we have γχ[m,m+1] = vχ[m,m+1] because γχ[−N,N] = vχ[−N,N] and [m, m+ 1] ⊆ [−N, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We get �� ∥uχ[m,m+1]∥p − ∥vχ[m,m+1]∥p �� ≤ ||(u − v)χ[m,m+1]∥p ≤ ∥u − v∥p < δ because v ∈ B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r′), hence ∥γχ[m,m+1]∥p = ∥vχ[m,m+1]∥p ≥ ∥uχ[m,m+1]∥p − δ ≥ ∥hχ[m,m+1]∥p − δ = ∥gχ[m,m+1]∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Therefore, γ ∈ B � v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' λ ∥u − v∥p � ∩ B(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' r) ∩ Θg, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Applications In this section, we will apply the results of the previous section, to prove that the set of all non-hypercyclic vectors of some sequences of weighted translation operators is non-σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' A sequence (Tn)n∈N0 of operators in B(X) is called hypercyclic if there is an element x ∈ X (called hypercyclic vector) such that the orbit {Tn(x) : n ∈ N0} is dense in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The set of all hypercyclic vectors of a sequence (Tn)n∈N0 is denoted by HC((Tn)n∈N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' An operator T ∈ B(X) is called hypercyclic if the sequence (T n)n∈N0 is hyper- cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 9 Let G be a locally compact group and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, for each function f : G → C we define Laf : G → C by Laf(x) := f(a−1x) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Note that if p ≥ 1, then the left translation operator La : Lp(G) → Lp(G), f �→ Laf is not hypercyclic because ∥La∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Hypercyclicity of weigted translation operators on Lp(G) and regarding an aperiodic element a was studied in [5] (an element a ∈ G is called aperiodic if the closed subgroup of G generated by a is not compact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let G be a locally compact group with a left Haar measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Fix p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We denote Lp(G) := Lp(G, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Assume that w : G → (0, ∞) is a bounded measurable function (called a weight) and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the weighted translation operator Ta,w,p : Lp(G) → Lp(G) is defined by Ta,w,p(f) := w Laf, (f ∈ Lp(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For each n ∈ N we denote ϕn := w Law .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Lan−1w, where a0 := e, the identity element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, G be a discrete group and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let µ be a left Haar measure on G with µ({e}) ≥ 1 and (γn)n be an unbounded sequence of non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let w : G → (0, ∞) be a bounded function such that for some finite nonempty set F ⊆ G and some N > 0 we have aγnF ∩ F = ∅ (n ≥ N), and β := inf � γn � k=1 w(akt) : n ≥ N, t ∈ F � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set Λ := � f ∈ Lp(G, µ) : ∥T γn a,w,pf − χF ∥p ≥ µ(F) 1 p for all n ≥ N � is non-σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Γ := {f ∈ Lp(G, µ) : |f| ≥ 1 βχF }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, Γ is not σ-porous in Lp(G, µ) thanks to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Also, for each f ∈ Γ and n ≥ N we have ∥T γn a,w,pf − χF∥p p = � G | n � k=1 w(a−γn+kx) f(a−γnx) − χF (x)|p dµ(x) = � G | γn � k=1 w(akx) f(x) − χF(aγnx)|p dµ(x) = � G | γn � k=1 w(akx) f(x) − χa−γnF (x)|p dµ(x) ≥ � F | γn � k=1 w(akx) f(x) − χa−γnF (x)|p dµ(x) = � F | γn � k=1 w(akx) f(x)|p dµ(x) ≥ � F |β 1 β |p dµ(x) = µ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let G be the additive group Z with the counting measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let F be a finite non-empty subset of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Put N := max{|j| : j ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If w := (wn)n∈Z ⊆ (0, ∞) is a bounded sequence with wn ≥ 1 for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then the required conditions in the previous theorem hold with respect to F and a := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The following fact is a direct conclusion of the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, G be a discrete group and a ∈ G with infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let µ be the counting measure on G and (γn)n be an unbounded sequence of non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let w : G → (0, ∞) be a bounded function such that for some t ∈ G, inf � γn � k=1 w(akt) : n ∈ N � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set � f ∈ Lp(G, µ) : ∥T γn a,wf − χ{t}∥p ≥ 1 for all n � is non-σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, G be a discrete group, and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let µ be a left Haar measure on G with µ({e}) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let (γn)n be an unbounded sequence of non-negative integers and let w : G → (0, ∞) be a bounded function such that inf n∈N γn � k=1 w(ak) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 11 Then, the set Γ := � f ∈ Lp(G, µ) : |f(e)| inf n∈N γn � k=1 w(ak) ≥ 1 � is non-σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In particular, setting Tn := T γn a,w,p for all n, the set of all non-hypercyclic vectors of the sequence (Tn)n is not σ-porous in Lp(G, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Since µ({e}) ≥ 1, applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='3 the set Γ is non-σ-porous, because [ inf n∈N γn � k=1 w(ak)]−1 χ{e} ∈ Lp(G, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let f ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If n is a nonnegative integer, then for every x in G we have ∥Tnf∥p ≥ ��ϕγn(x) Laγn f(x) ��, and so setting x = aγn we have ∥Tnf∥p ≥ ��ϕn(aγn) Laγn f(aγn) �� = � γn � k=1 w(ak) � |f(e)| ≥ |f(e)| inf m∈N γm � k=1 w(ak) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This implies that the set {Tnf : n ∈ N} is not dense in Lp(G, µ), and so Γ is a subset of the set of all non-hypercyclic vectors of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Now, we recall the definition of hypergroups which are generalizations of locally compact groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' see the monograph [4] and the basic paper [12] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In locally compact hypergroups the convolution of two Dirac measures is not necessarily a Dirac measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let K be a locally compact Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We denote by M(K) the space of all regular complex Borel measures on K, and by δx the Dirac measure at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The support of a measure µ ∈ M(K) is denoted by supp(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Suppose that K is a locally compact Hausdorff space, (µ, ν) �→ µ∗ν is a bilinear positive-continuous mapping from M(K)×M(K) into M(K) (called convolution), and x �→ x− is an involutive homeomorphism on K (called involution) with the following properties: (i) M(K) with ∗ is a complex associative algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (ii) if x, y ∈ K, then δx∗δy is a probability measure with compact support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (iii) the mapping (x, y) �→ supp(δx ∗ δy) from K × K into C(K) is contin- uous, where C(K) is the set of all non-empty compact subsets of K equipped with Michael topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (iv) there exists a (necessarily unique) element e ∈ K (called identity) such that for all x ∈ K, δx ∗ δe = δe ∗ δx = δx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE (v) for all x, y ∈ K, e ∈ supp(δx ∗ δy) if and only if x = y−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, K ≡ (K, ∗,− , e) is called a locally compact hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' A nonzero nonnegative regular Borel measure m on K is called the (left) Haar measure if for each x ∈ K, δx∗m = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For each x, y ∈ K and measurable function f : K → C we denote f(x ∗ y) := � K f d(δx ∗ δy), while this integral exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Suppose that a := (an)n∈N0 is a sequence in a hypergroup K, and w is a weight function on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For each n ∈ N0 we define the bounded linear operator Λn+1 on Lp(K) by Λn+1f(x) := w(a0 ∗ x) w(a1 ∗ x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an ∗ x) f(an+1 ∗ x) (f ∈ Lp(K)) for all x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Also, we assume that Λ0 is the identity operator on Lp(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Some linear dynamical properties of this sequence of operators were studied in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' The sequence {Λn}n is a generalization of the usual powers of a single weighted translation operator on Lp(G), where G is a locally compact group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In fact, any locally compact group G with the mapping µ ∗ ν �→ � G � G δxydµ(x)dν(y) (µ, ν ∈ M(G)) as convolution, and x �→ x−1 from G onto G as involution is a locally compact hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let η := (an)n∈N0 be a sequence in G, and w be a weight on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then for each f ∈ Lp(G), n ∈ N0 and x ∈ G, we have Λn+1f(x) = w(a0x) w(a1x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(anx) f(an+1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In particular, let a ∈ G and for each n ∈ N0, put an := a−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, Λn = T n a,w,p for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In this case, the operator Ta,w,p is hypercyclic if and only if the sequence (Λn)n is hypercyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let K be a discrete hypergroup with the convolution ∗ between Radon measures of K and the involution ·− : K → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, by [12, Theorem7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1A], the measure µ on K given by µ({x}) := 1 δx ∗ δx−({e}), (x ∈ K) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1) is a left Haar measure on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let K be a discrete hypergroup, µ be the Haar measure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1), and p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then for each g ∈ Lp(K, µ), the set � f ∈ Lp(K, µ) : |f| ≥ |g| � is not σ-porous in Lp(K, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Just note that for each x ∈ K we have µ({x}) ≥ 1 because 1 = δx ∗ δx−(K) ≥ δx ∗ δx−({e}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Hence, the measure space (K, µ) satisfies the condition of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Let a := (an)n∈N be a sequence in a discrete hypergroup K such that an ̸= am for each m ̸= n, and let w : K → (0, ∞) be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' We define ha,w : K → C by ha,w := � n∈N0 1 w(a0)w(a1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an)χ{an+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, and K be a discrete hypergroup endowed with the left Haar measure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let a := (an)n∈N0 ⊆ K with distinct terms, and w be a weight on K such that ha,w ∈ Lp(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set of all non-hypercyclic vectors of the sequence (Λn)n is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' First, thanks to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='9, the set E := � f ∈ Lp(K) : |f(an+1)| ≥ 1 w(a0)w(a1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an) for all n � is not σ-porous because it equals to the set � f ∈ Lp(K) : |f| ≥ ha,w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, for each f ∈ E, ∥Λn+1f∥p ≥ sup x∈K w(a0 ∗ x) w(a1 ∗ x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an ∗ x) |f(an+1 ∗ x)| ≥ w(a0) w(a1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an) |f(an+1)| ≥ 1 for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This implies that 0 does not belong to the closure of {Λnf : n ∈ N} in Lp(K), and so E ⊆ [HC((Λn)n)]c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Since any group is a hypergroup, we can give the fact below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, and G be a discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let a ∈ G be of infinite order, (γn)n∈N0 ⊆ N be with distinct terms and w : G → (0, ∞) be a weight such that � 1 w(aγ0)w(aγ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(aγn) � n ∈ ℓp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set of all non-hypercyclic vectors of the sequence (T γn a,w,p)n is not σ-porous in ℓp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, we can write the next corollary which is a generalization of [1, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let p ≥ 1, (γn)n ⊆ N be strictly increasing and (wn)n∈Z be a bounded sequence in (0, ∞) such that � 1 wγ0wγ1wγ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' wγn � n ∈ ℓp(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE Then, the set of all non-hypercyclic vectors of the sequence (Tn)n is not σ- porous, where (Tn+1a)k := wγ0wγ1wγ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' wγnak+γn+1 (k ∈ N0) for all a := (aj)j ∈ ℓp(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='7 we can speak regarding some more general situation in the case of p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Ω be a locally compact Hausdorff space endowed with a nonnegative Radon measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let w : Ω → (0, ∞) be a bounded measurable function, and α : Ω → Ω be a bi-measurable mapping such that ∥f ◦ α±1∥∞ = ∥f∥∞ for all f ∈ L∞(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, we define Tα,w,∞ : L∞(Ω, µ) → L∞(Ω, µ) by Tα,w,∞(f) := w (f ◦ α) (f ∈ L∞(Ω, µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If Ω be a locally compact group and a ∈ Ω, setting αa(x) := ax for all x ∈ Ω, we denote Ta,w,∞ := Tαa,w,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Note that α−1 means the inverse function of α, and for each k ∈ N, α−k := (α−1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Tα,w,∞ be the weighted composition operator defined as above and let {γn}n ⊆ N be a fixed unbounded sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Suppose that there exists a sequence {An}n of disjoint subsets of Ω with µ(An) > 0 for all n such that jα,w := � n∈N 1 (w ◦ α−γn) (w ◦ α−γn+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (w ◦ α−1)χAn ∈ L∞(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set {f ∈ L∞(Ω, µ) : ∥T γn α,w,∞(f)∥∞ ≥ 1 for all n} is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In particular, the set of all non-hypercyclic vectors of the sequence {T γn α,w,∞}n is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let E := {f ∈ L∞(Ω, µ) : |f| ≥ jα,w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, E is not σ-porous thanks to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For each f ∈ E and n ∈ N we have ∥T γn α,w,∞(f)∥∞ = ∥ γn � k=1 (w ◦ αγn−k) (f ◦ αγn)∥∞ = ∥ γn � k=1 (w ◦ α−k) f∥∞ ≥ ∥ γn � k=1 (w ◦ α−k) χAn f∥∞ ≥ ∥ γn � k=1 (w ◦ α−k) χAn jα,w∥∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 15 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let G be a locally compact group and µ be a left Haar mea- sure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let a ∈ G and w : G → (0, ∞) be a bounded measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, if � 1 w(a)w(a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' w(an) � n ∈ L∞(G, µ), then the set of all non-hypercyclic vectors of the operator Ta,w,∞ on L∞(G, µ) is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If (wn)n∈Z is a bounded sequence such that � 1 w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' wn � n ∈ ℓ∞, then the set of all non-hypercyclic vectors of the sequence (Tγn,w)n is not σ- porous in ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Tα,w,∞ be the weighted composition operator on L∞(Ω, µ) and let F ⊆ Ω be a Borel set with 0 < µ(F) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let there exists a constant N > 0 such that for all n ≥ N, αn(F) ∩ F = ∅, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2) and β := inf{ n � k=1 (w ◦ α−k)(t) : n ≥ N, t ∈ F} ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set {f ∈ L∞(Ω, µ) : ∥T n α,w,∞f − χF∥∞ ≥ 1 for all n ≥ N} is not σ-porous in L∞(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Γ := {f ∈ L∞(Ω, µ) : |f| ≥ 1 βχF }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='7, Γ is not σ-porous in L∞(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Also, for each f ∈ Γ we have ∥T n α,w,∞f − χF ∥∞ = ∥ n � k=1 (w ◦ αn−k) (f ◦ αn) − χF∥∞ = ∥ n � k=1 (w ◦ α−k) f − χF ◦ αn∥∞ = ∥ n � k=1 (w ◦ α−k) f − χαn(F )∥∞ ≥ ∥ n � k=1 (w ◦ α−k) fχF − χαn(F )χF∥∞ = ∥ n � k=1 (w ◦ α−k) fχF∥∞ ≥ β∥fχF∥∞ ≥ β 1 β ∥χF∥∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Ω := R and µ be the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Put α(t) := t−1 for all t ∈ R and F := [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' If w ∈ Cb(R) such that |w(t)| ≥ 1 for all t ≥ k > 0 and inf{|w(t)| : t ∈ [0, 1]} > 0, then the required conditions in the previous theorem hold with respect to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' With some similar proof, one can prove the next fact without the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Let Tα,w,∞ be the weighted composition operator on L∞(Ω, µ) and let F ⊆ Ω be a Borel set with 0 < µ(F) < ∞ such that inf{ n � k=1 (w ◦ α−k)(t) : n ≥ N, t ∈ F} ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set {f ∈ L∞(Ω, µ) : ∥T n α,w,∞f∥∞ ≥ 1 for all n ≥ N} is not σ-porous in L∞(Ω, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In particular, the set of all non-hypercyclic vec- tors of the operator Tα,w,∞ is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' In sequel, we find some application for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='9 regarding hypercyclicity of shift operators on Lp(R, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Consider the weighted translation operator Tα,w on Lp(R, τ) given by Tα,wf := w · (f ◦α), where 0 < w, w−1 ∈ Cb(R) and α(t) = t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For each n ∈ N put An := [n, n + 1] = αn([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Set yα,w := � n∈N 1 inft∈An �n k=1(w ◦ α−k)(t)χAn and assume that yα,w ∈ Lp(R, τ) (in particular inft∈An �n k=1(w ◦ α−k)(t) > 0 for all n ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set {f ∈ Lp(R, τ) : ∥T n α,w(f)∥p ≥ 1 for all n ∈ N} is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='9, the set E := {f ∈ Lp(R, τ) : ∥fχAn∥p ≥ ∥yα,wχAn∥p for all n ∈ N} is not σ-porous, because it equals to {f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥yα,wχ[m,m+1]∥p for all m ∈ Z}, DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 17 as yα,wχ[m,m+1] = 0 for all m ∈ Z with m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Now, note that for each f ∈ E and n ∈ N, ∥T n α,w(f)∥p p = � R � n � k=1 (w ◦ αn−k)(t) �p |(f ◦ αn)(t)|p dτ = � R � n � k=1 (w ◦ α−k)(t) �p |f(t)|p dτ ≥ � An � n � k=1 (w ◦ α−k)(t) �p |f(t)|p dτ ≥ inf t∈An � n � k=1 (w ◦ α−k)(t) �p ∥yα,wχAn∥p p = inf t∈An � n � k=1 (w ◦ α−k)(t) �p 1 inft∈An [�n k=1(w ◦ α−k)(t)]p τ(An) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' □ Assume now that there exists some l ∈ Z such that β := inf{ n � k=1 (w ◦ α−k)(t) : t ∈ [l, l + 1], n ∈ N} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Put F := {f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥ 1 β χ[l,l+1]χ[m,m+1]∥p for all m ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' So by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='9, F is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' For every f ∈ F we have ∥T n α,w(f)∥p p = � R � n � k=1 (w ◦ αn−k)(t) �p |(f ◦ αn)(t)|p dτ = � R � n � k=1 (w ◦ α−k)(t) �p |f(t)|p dτ ≥ � [l,l+1] � n � k=1 (w ◦ α−k)(t) �p |f(t)|p dτ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Hence, the set {f ∈ Lp(R, τ) : ∥T n α,w(f)∥p ≥ 1 for all n ∈ N} is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Next, suppose that α is an aperiodic function on R (this means that for each compact set C ⊂ R, there exists a constant N > 0 such that αn(C) ∩ C = ∅ for all n ≥ N) and β > 0, where β is as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Then, the set {f ∈ Lp(R, τ) : ∥T n α,w(f) − χ[l,l+1]∥p ≥ 1 for all n ≥ N} 18 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' IVKOVI´C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨OZTOP, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' TABATABAIE is not σ-porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Indeed, for all f ∈ F, and n ≥ N we have ∥T n α,w(f) − χ[l,l+1]∥p ≥ ∥ n � k=1 (w ◦ α−k) fχ[l,l+1]∥p by the similar calculations as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' However, ∥ n � k=1 (w ◦ α−k) fχ[l,l+1]∥p ≥ β ∥fχ[l,l+1]∥p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Bayart, Porosity and hypercyclic operators, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 133(11) (2005) 3309-3316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Bayart and ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 179, Cambridge University Press, Cambridge, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Belna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Evans and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='Humke, Symmetric and ordinary differentiation, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 72(2) (1978) 261-267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Bloom and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Kruyter, Berlin, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' C-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Chen and C-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Chu, Hypercyclic weighted translations on groups, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 139 (2011) 2839-2846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' C-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' ¨Oztop and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Tabatabaie, Disjoint dynamics on weighted Orlicz spaces, Disjoint dynamics on weighted Orlicz spaces, Complex Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Theory, 14(72) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1007/s11785-020-01034-x 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Chen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Tabatabaie, Chaotic operators on hypergroups, Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 12(1) (2018) 143-156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Dolˇzenko, Boundary properties of arbitrary functions, Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Nauk SSSR Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 31 (1967) 3-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Folland, Real Analysis, Modern Techniques and Their Applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Second Edition, John Wiley and Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' New York (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' K-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 139 (2000) 47-68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' K-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Grosse-Erdmann and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Peris, Linear Chaos, Universitext, Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Jewett, Spaces with an abstract convolution of measures, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=', 18 (1975) 1-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Kumar and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Tabatabaie, Hypercyclic sequences of weighted translations on hy- pergroups, Semigroup Forum, 103 (2021) 916–934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Preiss and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Zaj´ıˇcek, Fr´echet differentiation of convex functions in a Banach space with a separable dual, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 91(2) (1984) 202–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Salas, Hypercyclic weighted shifts, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 347 (1995) 993-1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Sawano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Tabatabaie and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Shahhoseini, Disjoint Dynamics of Weighted Translations on Solid Spaces, Topology Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 298, 107709, 14 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' (2021) DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='1016/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='TOPOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='107709 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Tabatabaie and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Ivkovic, Linear dynamics of cosine operator functions on solid Banach function spaces, Positivity, 25 (2021) 1437–1448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Villani, Another note on the inclusion Lp(µ) ⊂ Lq(µ), Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Monthly, 92 (1985) 485–487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Z´ajiˇcek, Porosity and σ-porous, Real Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Exchange 13 (1987/1988) 314-350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Zaj´ıˇcek, Small non-σ-porous sets in topologically complete metric spaces, Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 77(2) (1998) 293-304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Z´ajiˇcek, On σ-porous sets in abstract spaces, Abstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 5 (2005) 509-534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS 19 Mathematical Institute of the Serbian Academy of Sciences and Arts, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' 367, Kneza Mihaila 36, 11000 Beograd, Serbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Email address: stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='iv10@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='com Department of Mathematics, Faculty of Science, Istanbul University, Istan- bul, Turkey Email address: oztops@istanbul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='tr Department of Mathematics, University of Qom, Qom, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content=' Email address: sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='tabatabaie@qom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} +page_content='ir' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfGAMz/content/2301.02908v1.pdf'} diff --git a/IdE2T4oBgHgl3EQfowif/content/tmp_files/2301.04022v1.pdf.txt b/IdE2T4oBgHgl3EQfowif/content/tmp_files/2301.04022v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0307b798a9ebc725fe44836283ebd1c51675beca --- /dev/null +++ b/IdE2T4oBgHgl3EQfowif/content/tmp_files/2301.04022v1.pdf.txt @@ -0,0 +1,2070 @@ +Distributed Sparse Linear Regression +under Communication Constraints ∗ +Rodney Fonseca and Boaz Nadler +Department of Computer Science and Applied Mathematics, +Weizmann Institute of Science, Rehovot, Israel +e-mail: rodney.fonseca@weizmann.ac.il; boaz.nadler@weizmann.ac.il +Abstract: In multiple domains, statistical tasks are performed in dis- +tributed settings, with data split among several end machines that are con- +nected to a fusion center. In various applications, the end machines have +limited bandwidth and power, and thus a tight communication budget. +In this work we focus on distributed learning of a sparse linear regression +model, under severe communication constraints. We propose several two +round distributed schemes, whose communication per machine is sublinear +in the data dimension. In our schemes, individual machines compute debi- +ased lasso estimators, but send to the fusion center only very few values. On +the theoretical front, we analyze one of these schemes and prove that with +high probability it achieves exact support recovery at low signal to noise +ratios, where individual machines fail to recover the support. We show in +simulations that our scheme works as well as, and in some cases better, +than more communication intensive approaches. +MSC2020 subject classifications: Primary 62J07, 62J05; secondary +68W15. +Keywords and phrases: Divide and conquer, communication-efficient, +debiasing, high-dimensional. +1. Introduction +In various applications, datasets are stored in a distributed manner among sev- +eral sites or machines (Fan et al., 2020, chap. 1.2). Often, due to communication +constraints as well as privacy restrictions, the raw data cannot be shared be- +tween the various machines. Such settings have motivated the development of +methods and supporting theory for distributed learning and inference. See, e.g., +the reviews by Huo and Cao (2019), Gao et al. (2022) and references therein. +∗This research was supported by a grant from the Council for Higher Education Compet- +itive Program for Data Science Research Centers. RF acknowledges support provided by the +Mor´a Miriam Rozen Gerber Fellowship for Brazilian postdocs. +1 +arXiv:2301.04022v1 [cs.LG] 9 Jan 2023 + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +2 +In this paper we consider distributed learning of a sparse linear regression +model. Specifically, we assume that the response y ∈ R and the vector X ∈ Rd +of explanatory variables are linearly related via +y = X⊤θ∗ + w, +(1) +where w ∼ N(0, σ2), σ > 0 is the noise level, and θ∗ ∈ Rd is an unknown +vector of coefficients. We further assume X ∈ Rd is random with mean zero +and covariance matrix Σ. We focus on a high-dimensional setting d ≫ 1, and +assume that θ∗ is sparse with only K ≪ d nonzero coefficients. The support set +of θ∗ ∈ Rd is denoted by S = {i ∈ [d] | |θ∗ +i | > 0}. +Given N samples {(Xi, yi)}N +i=1 from the model (1), common tasks are to +estimate the vector θ∗ and its support set S. Motivated by contemporary ap- +plications, we consider these tasks in a distributed setting where the data are +randomly split among M machines. Specifically, we consider a star topology +network, whereby the end machines communicate only with a fusion center. +As reviewed in Section 2, estimating θ∗ and its support in the above or simi- +lar distributed settings were studied by several authors, see for example Mateos, +Bazerque and Giannakis (2010); Chen and Xie (2014); Lee et al. (2017); Battey +et al. (2018); Chen et al. (2020); Liu et al. (2021); Barghi, Najafi and Mota- +hari (2021) and references therein. Most prior works on distributed regression +required communication of at least O(d) bits per machine, as in their schemes +each machine sends to the fusion center its full d-dimensional estimate of the +unknown vector θ∗. Some works in the literature denote this as communication +efficient, in the sense that for a machine holding n samples, an O(d) communi- +cation is still significantly less than the size O(n · d) of its data. +The design and analysis of communication efficient distributed schemes is +important, as in various distributed settings the communication channel is the +critical bottleneck. Moreover, in some practical cases, such as mobile devices and +sensor networks, the end machines may have very limited bandwidth. Thus, in +high dimensional settings with d ≫ 1, it may not even be feasible for each ma- +chine to send messages of length O(d). In this work, we study such a restricted +communication setting, assuming that each machine is allowed to send to the fu- +sion center only a limited number of bits, significantly lower than the dimension +d. Our goals are to develop low communication distributed schemes to estimate +θ∗ and its support and to theoretically analyze their performance. +We make the following contributions. On the methodology side, in Section 4 +we present several two round distributed schemes. The schemes vary slightly by + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +3 +the messages sent, but in all of them, the fusion center estimates the support set +of θ∗ in the first round and the vector θ∗ in the second round. In our schemes, +each machine computes its own debiased lasso estimate. However, it sends to +the center only the indices of its top few largest values, possibly along with their +signs. Hence, the communication per machine is significantly less than d bits. +In the simplest variant, the fusion center estimates the support of θ∗ by voting, +selecting the few indices that were sent by the largest number of machines. +Next, on the theoretical side, we prove in Section 5 that under suitable condi- +tions, with high probability the first round of our scheme achieves exact support +recovery with communication per machine sublinear in d. Specifically, we present +support guarantees under two different parameter regimes. Theorem 2 consid- +ers a case with a relatively large number of machines. Here, each machine sends +a short message of O(K ln d) bits. Next, Theorem 3 considers a setting with +relatively few machines, M = O(ln d). Here, to achieve exact support recovery +each machine sends a much longer message, of length O(dα) for some suitable +α < 1. This is still sublinear in d, and much less than the communication re- +quired if a machine were to send its full d-dimensional estimated vector. The +proofs of our theorems rely on recent results regarding the distribution of debi- +ased lasso estimators, combined with sharp bounds on tails of binomial random +variables. Exact support recovery follows by showing that with high probability, +all non-support indices receive fewer votes than support indices. +In Section 6 we present simulations comparing our schemes to previously +proposed methods. These illustrate that with our algorithms, the fusion center +correctly detects the support of θ∗ and consequently accurately estimates θ∗, +even at low signal to noise ratios where each machine is unable to do so. Fur- +thermore, this is achieved with very little communication per machine compared +to the dimension d. One insight from both the simulations and our theoretical +analysis is that for the fusion center to detect the correct support, it is not +necessary to require M/2 votes as suggested in Barghi, Najafi and Motahari +(2021) and Chen and Xie (2014). Instead, as few as O(ln d) votes suffice to dis- +tinguish support from non-support indices. Interestingly, under a broad range +of parameter values, our schemes work as well as, and in some cases better than +more communication intensive approaches. Our simulations also highlight the +importance and advantages of a second round of communication. Specifically, +even though a single-round scheme based on averaging debiased lasso estimates, +as proposed by Lee et al. (2017), is minimax rate optimal and finds the correct +support, it nonetheless may output an estimate with a larger mean squared er- + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +4 +ror than that of our scheme. We conclude with a summary and discussion in +Section 7. Proofs appear in the Appendix. +Notation +For an integer k ≥ 1, we denote [k] = {1, 2, . . . , k}. The indicator +function is denoted as I(A), which equals one if condition A holds and zero +otherwise. The ℓq norm of a vector Y ∈ Rn for q ≥ 1 is ∥Y ∥q = (�n +i=1 |Yi|q)1/q, +whereas ∥Y ∥0 = �n +i=1 I(Yi ̸= 0) is its number of nonzero entries. We denote +by |Y | the vector whose entries are (|Y1|, |Y2|, . . . , |Yn|). For a d × d matrix +A = {aij}d +i,j=1, we denote ∥A∥∞ = max1≤i≤d +�d +j=1 |aij|. We further denote by +σmin(A) and σmax(A) its smallest and largest singular values, respectively. For +a subset J ⊂ [d], AJ is the d×|J| matrix whose columns are those in the subset +J. Similarly, AJ,J is the |J|×|J| submatrix whose rows and columns correspond +to the indices in J. The cumulative distribution function (CDF) of a standard +Gaussian is denoted by Φ(·) whereas Φc(·) = 1−Φ(·). We write an ≳ bn for two +sequences {an}n≥1 and {bn}n≥1 if there are positive constants C and n0 such +that an ≥ Cbn for all n > n0. +2. Previous works +Distributed linear regression schemes under various settings, not necessarily +involving sparsity, have been proposed and theoretically studied in multiple +fields, including sensor networks, statistics and machine learning, see for example +(Guestrin et al., 2004; Predd, Kulkarni and Poor, 2006; Boyd et al., 2011; Zhang, +Duchi and Wainwright, 2013; Heinze et al., 2014; Rosenblatt and Nadler, 2016; +Jordan, Lee and Yang, 2019; Chen et al., 2020; Dobriban and Sheng, 2020; Zhu, +Li and Wang, 2021; Dobriban and Sheng, 2021). +Mateos, Bazerque and Giannakis (2010) were among the first to study dis- +tributed sparse linear regression in a general setting without a fusion center, +where machines are connected and communicate with each other. They devised +a multi-round scheme whereby all the machines reach a consensus and jointly +approximate the centralized solution, that would have been computed if all data +were available at a single machine. Several later works focused on the setting +which we also consider in this paper, where machines are connected in a star +topology to a fusion center, and only one or two communication rounds are +made. In a broader context of generalized sparse linear models, Chen and Xie +(2014) proposed a divide-and-conquer approach where each machine estimates +θ∗ by minimizing a penalized objective with a sparsity inducing penalty, such + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +5 +as ∥θ∥1. Each machine sends its sparse estimate to the fusion center, which es- +timates the support by voting over the indices of the individual estimates of +the M machines. Finally, the center estimates θ∗ by a weighted average of these +M estimates. For sparse linear regression, with each machine computing a lasso +estimate of θ∗, their method suffers from the well known bias of the lasso, which +is not reduced by averaging. +To overcome the bias of the lasso, in recent years several debiased lasso es- +timators were derived and theoretically studied, see Zhang and Zhang (2014); +van de Geer et al. (2014); Javanmard and Montanari (2018). For distributed +learning, debiased estimators have been applied in various settings, including +hypothesis testing, quantile regression and more, see for example Lee et al. +(2017); Battey et al. (2018); Liu et al. (2021); Lv and Lian (2022). +In particular, Lee et al. (2017) proposed a single round scheme whereby each +machine computes its own debiased lasso estimator, and sends it to the fusion +center. The center averages these debiased estimators and thresholds the result +to estimate θ∗ and recover its support. Lee et al. (2017) proved that the re- +sulting estimator achieves the same error rate as the centralized solution, and +is minimax rate optimal. However, their scheme requires a communication of +O(d) bits per machine and is thus not applicable in the restricted communica- +tion setting considered in this manuscript. Moreover, as we demonstrate in the +simulation section, unless the signal strength is very low, our two round scheme +in fact achieves a smaller mean squared error, with a much lower communica- +tion. This highlights the potential sub-optimality of lasso and debiased lasso in +sparse regression problems with sufficiently strong signals. +Most related to our paper is the recent work by Barghi, Najafi and Motahari +(2021). In their method, each machine computes a debiased lasso estimator +ˆθ, but sends to the fusion center only the indices i for which |ˆθi| is above a +certain threshold. The support set estimated by the fusion center consists of all +indices that were sent by at least half of the machines, i.e., indices that received +at least M/2 votes. Focusing on the consistency of feature selection, Barghi, +Najafi and Motahari (2021) derive bounds on the type-I and type-II errors +of the estimated support set. Their results, however, are given as rates with +unspecified multiplicative constants. As we show in this work, both theoretically +and empirically, consistent support estimation is possible with a much lower +voting threshold. Furthermore, requiring at least M/2 votes implies that their +scheme achieves exact support recovery only for much stronger signals. +We remark that voting is a natural approach for distributed support esti- + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +6 +mation under communication constraints. Amiraz, Krauthgamer and Nadler +(2022) analyzed voting-based distributed schemes in the context of a simpler +problem of sparse Gaussian mean estimation. They proved that even at low +signal strengths, their schemes achieve exact support recovery with high prob- +ability using communication sublinear in the dimension. Their setting can be +viewed as a particular case of sparse linear regression but with a unitary design +matrix. Their proofs, which rely on this property, do not extend to our setting. +3. The lasso and debiased lasso estimators +For our paper to be self contained, we briefly review the lasso and debiased lasso +and some of their theoretical properties. The lasso (Tibshirani, 1996) is perhaps +the most popular method to fit high-dimensional sparse linear models. Given +a regularization parameter λ > 0 and n samples (Xi, yi), stacked in a design +matrix X ∈ Rn×d and a response vector Y ∈ Rn, the lasso estimator is given by +˜θ = ˜θ(X, Y, λ) = arg min +θ∈Rd +� 1 +2n∥Y − Xθ∥2 +2 + λ∥θ∥1 +� +. +(2) +The lasso has two desirable properties. First, computationally Eq. (2) is a convex +problem for which there are fast solvers. Second, from a theoretical standpoint, +it enjoys strong recovery guarantees, assuming the data follows the model (1) +with an exact or approximately sparse θ∗, see for example (Candes and Tao, +2005; Bunea, Tsybakov and Wegkamp, 2007; van de Geer and B¨uhlmann, 2009; +Hastie, Tibshirani and Wainwright, 2015). However, the lasso has two major +drawbacks: it may output significantly biased estimates and it does not have +a simple asymptotic distribution. The latter is needed for confidence intervals +and hypothesis testing. To overcome these limitations, and in particular derive +confidence intervals for high-dimensional sparse linear models, several authors +developed debiased lasso estimators (Zhang and Zhang, 2014; van de Geer et al., +2014; Javanmard and Montanari, 2014a,b, 2018). +For random X with a known population covariance matrix Σ, Javanmard and +Montanari (2014a) proposed 1 +nΣ−1X⊤(Y − X˜θ) as a debiasing term. . As Σ is +often unknown, both van de Geer et al. (2014) and Javanmard and Montanari +(2014b) developed methods to estimate its inverse Ω = Σ−1. In our work, we +estimate Ω using the approach of van de Geer et al. (2014), who assume that Ω +is sparse. In their method, presented in Algorithm 1, ˆΩ is constructed by fitting +a lasso regression with regularization λΩ > 0 to each column of X against all +the other columns. Hence, it requires solving d separate lasso problems. + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +7 +Given the lasso estimate ˜θ of Eq. (2) and the matrix ˆΩ, the debiased lasso is +ˆθ = ˆθ(Y, X, λ, λΩ) = ˜θ + 1 +n +ˆΩX⊤(Y − X˜θ). +(3) +An appealing property of ˆθ is that, under some conditions, it is asymptotically +unbiased with a Gaussian distribution. For our analysis, we shall use the follow- +ing result (Javanmard and Montanari, 2018, Theorem 3.13). +Theorem 1. Consider the linear model Y = Xθ∗+W, where W ∼ N(0, σ2In×n) +and X ∈ Rn×d has independent Gaussian rows with zero mean and covariance +matrix Σ ∈ Rd×d. Suppose that Σ satisfies the following conditions: +i. For all i ∈ [d], Σii ≤ 1. +ii. For some constants Cmax, Cmin > 0, +0 < Cmin < σmin(Σ) ≤ σmax(Σ) < Cmax. +(4) +iii. For C0 = (32Cmax/Cmin) + 1, and a constant ρ > 0, +max +J⊆[d], |J|≤C0K ∥Σ−1 +J,J∥∞ ≤ ρ. +Let KΩ be the maximum row-wise sparsity of Ω = Σ−1, that is, +KΩ = max +i∈[d] |{j ∈ [d]; Ωij ̸= 0, j ̸= i}| . +Let ˜θ be the lasso estimator computed using λ = κσ +� +(ln d)/n for κ ∈ [8, κmax], +and let ˆθ be the debiased lasso estimator in Eq. (3) with ˆΩ computed by Algorithm +1 with λΩ = κΩ +� +(ln d)/n for some suitable large κΩ > 0. Let ˆΣ = X⊤X/n de- +note the empirical covariance matrix. Then there exist constants c, c∗, C depend- +ing solely on Cmin, Cmax, κmax and κΩ such that, for n ≥ c max{K, KΩ} ln d, +the following holds: +√n(ˆθ − θ∗) = Z + R, +Z|X ∼ N(0, σ2 ˆΩˆΣˆΩ⊤), +(5) +where Z = n−1/2 ˆΩX⊤W and R = √n +� +ˆΩˆΣ − I +� +(θ∗ − ˜θ), and with probability +at least 1 − 2de−c∗n/K − de−cn − 6d−2, +∥R∥∞ ≤ Cσ ln d +√n +� +ρ +√ +K + min{K, KΩ} +� +. +(6) +Assumptions (i) and (ii) in this theorem are common in the literature. As- +sumption (iii) is satisfied, for example, by circulant matrices Σij = ς|i−j|, + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +8 +Algorithm 1 Computation of a precision matrix estimate ˆΩ +Input: design matrix X ∈ Rn×d, regularization parameter λΩ > 0. +Output: precision matrix estimate ˆΩ ∈ Rd×d. +xi ∈ Rn denotes the i-th column of X. +X−i ∈ Rn×(d−1) denotes the design matrix with the i-th column removed. +1: for i = 1, . . . , d do +2: +Fit a lasso with response xi, design matrix X−i and regularization parameter λΩ. +3: +Let ˜γi = {˜γi,j}d +j=1,j̸=i ∈ Rd−1 be the estimated regression coefficients of step 2. +4: +Compute ˜τ 2 +i = (2n)−1∥xi − X−i˜γi∥2 +2 + λΩ∥˜γi∥1, i ∈ [d]. +5: end for +6: Construct d × d matrix +˜C = +� +���� +1 +−˜γ1,2 +· · · +−˜γ1,d +−˜γ2,1 +1 +· · · +−˜γ2,d +... +... +... +... +−˜γd,1 +−˜γd,2 +· · · +1 +� +���� . +7: return ˆΩ = diag{˜τ −2 +1 +, . . . , ˜τ −2 +d +} ˜C. +ς ∈ (0, 1). The quantity R in Eq. (5) can be viewed as a bias term. By Theorem +1, this bias is small if the sample size and dimension are suitably large, which +in turn implies that ˆθi is approximately Gaussian. The following lemma, proven +in the Appendix, bounds the error of this approximation. It will be used in +analyzing the probability of exact support recovery of our distributed scheme. +Lemma 1. Under the assumptions of Theorem 1, for any τ > 0, +���Pr +� √n(ˆθi−θ∗ +i ) +σ√cii +≤ τ +� +− Φ (τ) +��� ≤ +δR +σ√cii +φ (τ) + 2de−c∗n/K + de−cn + 6 +d2 , +(7) +where φ(·) denotes the Gaussian density function, cii = (ˆΩˆΣˆΩ⊤)ii, and δR is +the upper bound on the bias term in Eq. (6), namely +δR = Cσ ln d +√n +� +ρ +√ +K + min{K, KΩ} +� +. +(8) +4. Distributed sparse regression with restricted communication +As described in Section 1, we consider a distributed setting with M machines +connected in a star topology to a fusion center. For simplicity, we assume that +each machine m has a sample (Xm, Y m) of n = N/M i.i.d. observations from +the model (1), where Y m ∈ Rn and Xm ∈ Rn×d. In describing our schemes, +we further assume that the noise level σ is known. If σ is unknown, it may + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +9 +Algorithm 2 Distributed voting based scheme for support estimation +Input: Data (Xm, Y m) ∈ Rn×(d+1), threshold τ and regularization parameters λΩ and λ. +Output: Support estimate ˆS. +At each local machine m = 1, . . . , M +1: Compute a lasso estimator ˜θm via Eq. (2) with regularization parameter λ. +2: Compute a precision matrix estimate ˆΩm ∈ Rd×d by Algorithm 1 with Xm and λΩ. +3: Compute a debiased lasso estimate ˆθm ∈ Rd, Eq. (3), with data (Xm, Y m), λ and ˆΩm. +4: Calculate the empirical covariance matrix ˆΣm = n−1(Xm)⊤Xm. +5: Use ˆΩm and ˆΣm to compute the standardized estimator ˆξm ∈ Rd, Eq. (9). +6: Set Sm = {i; |ˆξm +i | > τ} and send it to the fusion center. +At the fusion center +7: For each i ∈ [d], compute Vi = �M +m=1 I(i ∈ Sm). +8: Sort Vj1 ≥ Vj2 ≥ · · · ≥ Vjd. +9: return ˆS = {j1, . . . , jK}. +be consistently estimated, for example, by the scaled lasso of Sun and Zhang +(2012), see also (Javanmard and Montanari, 2018, Corollary 3.10). +We present several two round distributed schemes to estimate the sparse +vector θ∗ of Eq. (1) under the constraint of limited communication between the +M machines and the fusion center. Here we present the simplest scheme and +discuss other variants in section 4.1. In all variants, the fusion center estimates +the support of θ∗ in the first round, and θ∗ itself in the second round. +The first round of our scheme is described in Algorithm 2, whereas the full two +round scheme is outlined in Algorithm 3. In the first round, each machine m ∈ +[M] computes the following quantities using its own data (Xm, Y m): (i) a lasso +estimate ˜θm by Eq. (2); (ii) a matrix ˆΩm by Algorithm 1; and (iii) a debiased +lasso ˆθm by Eq. (3). Up to this point, this is identical to Lee et al. (2017). The +main difference is that in their scheme, each machine sends to the center its +debiased lasso estimate ˆθm ∈ Rd, incurring O(d) bits of communication. +In contrast, in our scheme each machine sends only a few indices. Towards +this end and in light of Eq. (7) of Lemma 1, each machine computes a normalized +vector ˆξm whose coordinates are given by +ˆξm +k = +√nˆθm +k +σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2 +kk +, +∀k ∈ [d]. +(9) +In the simplest variant, each machine sends to the center only indices k such +that |ˆξm +k | > τ for some suitable threshold τ > 0. +Given the messages sent by the M machines, the fusion center counts the +number of votes received by each index. If the sparsity level K is known, its + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +10 +Algorithm 3 Two round distributed scheme to estimate θ∗ +Input: Data (Xm, Y m) ∈ Rn×(d+1), sparsity K, threshold τ and regularizations λΩ and λ. +Output: A two-round estimate ˆθ ∈ Rd of θ∗. +First round +1: The fusion center estimates ˆS with Algorithm 2. +Second round +2: The fusion center sends ˆS to all M machines. +At each local machine m = 1, . . . , M +3: Let Xm +ˆ +S ∈ Rn×K be the K columns of Xm corresponding to indices in ˆS. +4: Compute ˆβm = arg minβ ∥Xm +ˆ +S β − Y m∥2 +2. +5: Send ˆβm to the fusion center. +At the fusion center +6: Given ˆβ1, . . . , ˆβM, compute the estimate ˆθ according to Eq. (11). +7: return ˆθ. +estimated support set ˆS consists of the K indices with the largest number of +votes. Otherwise, as discussed in Remark 4.5 below, the center may estimate the +support set by the indices whose number of votes exceed a suitable threshold. +Next, we describe the second round. At its start, the fusion center sends +the estimated support ˆS to all M machines. Next, each machine computes the +standard least squares regression solution, restricted to the set ˆS, namely +ˆβm = arg min +β ∥Xm +ˆ +S β − Y m∥2 +2 +(10) +where Xm +ˆ +S ∈ Rn×| ˆ +S| consists of the columns of Xm corresponding to the indices +in ˆS. Each machine then sends its vector ˆβm to the fusion center. Finally, the +fusion center estimates θ∗ by averaging these M vectors, +ˆθi = +� +1 +M +�M +m=1 ˆβm +i +i ∈ ˆS +0 +otherwise +(11) +In the next section we present several variants of this basic two round scheme. +Before that we make a few remarks and observations. +Remark 4.1. The communication of the first round (Algorithm 2) depends on +the threshold τ. A high threshold leads to only few sent indices. However, at +low signal strengths, the signal coordinates may not have the highest values |ˆξm +k | +and thus may not be sent. Hence, for successful support recovery by the fusion +center, a lower threshold leading to many more sent coordinates is required. Since +the maxima of d standard Gaussian variables scales as +√ +2 ln d, to comply with + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +11 +the communication constraints, the threshold τ should also scale as O( +√ +ln d). In +Section 5, we present suitable thresholds and sufficient conditions on the number +of machines and on the signal strength, which guarantee support recovery by +Algorithm 2, with high probability and little communication per machine. +Remark 4.2. With known K, the communication per machine of the second +round is O(K ln d) bits. For suitable choices of the threshold τ in the first round, +this is negligible or at most comparable to the communication of the first round. +Remark 4.3. A two round scheme, whereby given an estimated set ˆS, the +second round is identical to ours was discussed by Battey et al. (2018) in Eq. +(A.2) of their supplementary. The difference is that in their first round, similar +to Lee et al. (2017), each machine sends its full debiased lasso vector, with a +communication of O(d) bits. Battey et al. (2018) showed that, under certain +conditions, their two-round estimator attains an optimal rate. In Section 5, we +prove that for a sufficiently high SNR, our method achieves the same rate, but +using much less communication. +Remark 4.4. With a higher communication per machine in the second round, +it is possible for the fusion center to compute the exact centralized least squares +solution corresponding to the set ˆS, denoted ˆθLS. Specifically, suppose that each +machine sends to the center both the vector (Xm +ˆ +S )⊤Y m of length | ˆS|, and the +| ˆS| × | ˆS| matrix (Xm +ˆ +S )⊤Xm +ˆ +S . The center may then compute ˆθLS as follows +ˆθ +LS = +� M +� +m=1 +(Xm +ˆ +S )⊤Xm +ˆ +S +�−1 +M +� +m=1 +(Xm +ˆ +S )⊤Y m. +(12) +With K known and | ˆS| = K, such a second round has a communication of +O(K2) bits. If the sparsity K is non-negligible, this is much higher than the +O(K) bits of our original scheme. In particular, if K = O(d1/2), the resulting +communication is comparable to that of sending the full debiased lasso vector. +Remark 4.5. In practice, the sparsity K is often unknown. Instead of step 9 +in Algorithm 2, one alternative is to estimate S by thresholding the number of +votes. For some threshold τvotes > 0, ˆS could be set as all indices i such that +Vi > τvotes. Lemma 3 in Appendix A shows that, under suitable conditions, +non-support indices have a small probability of receiving more than 2 ln d votes. +Hence, τvotes = 2 ln d is a reasonable choice for such a threshold value. + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +12 +4.1. Variations of Algorithm 2 +Various adaptations of Algorithm 2 are possible and may offer better perfor- +mance. One example is a top L algorithm where each machine sends to the cen- +ter the indices of the L largest entries of |ˆξm|, for some parameter K ≤ L ≪ d. +A similar approach was proposed in Amiraz, Krauthgamer and Nadler (2022) +for the simpler problem of sparse normal means estimation. One advantage of +this variant is that its communication per machine is fixed and known a priori +O(L ln d). This is in contrast to the above thresholding based scheme, whose +communication per machine is random. +A different variant is to use sums of signs to estimate the support. Here +machines send both the indices corresponding to the largest entries in |ˆξm| and +their signs. Hence, in step 5 of Algorithm 2 the message sent by machine m is +Sm = +�� +i, sign(ˆξm +i ) +� +; |ˆξm +i | > τ +� +. +Next, the fusion center computes for each index i ∈ [d] its corresponding sum +of received signs, i.e., +V sign +i += +M +� +m=1 +sign(ˆξm +i )I +�� +i, sign(ˆξm +i ) +� +∈ Sm� +. +(13) +For known K, the estimated support set are the K indices with largest values +of |V sign +i +|. This algorithm uses a few more bits than a voting scheme. How- +ever, sums of signs are expected to better distinguish between support and +non-support coefficients when the number of machines is large. The reason is +that at non-support indices j ̸∈ S, the random variable V sign +j +has approximately +zero mean, unlike sums of votes Vj, whereas at support indices |V sign +i +| ≈ Vi since +support indices are unlikely to be sent to the fusion center with the opposite +sign of θ∗ +i . In the simulation section we illustrate the improved performance of +a sign-based over a votes-based distributed scheme. +5. Theoretical results +In this section, we present a theoretical analysis for one of our schemes. Specif- +ically, both Theorems 2 and 3 show that under suitable conditions, with high +probability Algorithm 2 achieves exact support recovery with little communi- +cation per machine. In Theorem 2, the number of machines is relatively large, +and the communication per machine is linear in the sparsity K. In Theorem 3 + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +13 +the number of machines M is logarithmic in d, in which case the communica- +tion per machine is much higher, though still sublinear in d. Both theorems are +based on the Gaussian approximation in Lemma 1 and on probability bounds +for binomial random variables. Their proofs appear in Appendix A. +To put our theorems in context, let us briefly review previous results on exact +support recovery in the simpler (non-distributed) sparse linear regression set- +ting. A key quantity characterizing the ability of exact support recovery is the +signal strength, defined as θmin = mini∈S |θ∗ +i |. As proven by Wainwright (2009), +under suitable conditions on the design matrix, the lasso estimator based on +n samples and an appropriately chosen regularization parameter λn, achieves +exact support recovery with high probability, provided that θmin ≳ +� +(ln d)/n. +The same rate θmin ≳ +� +(ln d)/n is also sufficient for support recovery using +a debiased lasso estimator (see, e.g., section 2.2 of Javanmard and Montanari +(2014b)). In a distributed setting, Lee et al. (2017) proved that with high proba- +bility, their scheme achieves exact support recovery when θmin ≳ +� +(ln d)/(nM). +While this result matches the centralized setting, their scheme requires each ma- +chine to send to the center its d-dimensional debiased lasso estimate, incurring +O(d) communication per machine. Hence, an interesting range for the signal +strength, for the study of support recovery under communication constraints, +is +� +ln d +nM ≲ θmin ≲ +� +ln d +n . In this range, individual machines may be unable to +exactly recover the support using the lasso or debiased lasso estimators. +To derive support recovery guarantees, we assume the smallest nonzero co- +efficient of θ∗ is sufficiently large, namely |θ∗ +i | ≥ θmin for all i ∈ S and some +suitable θmin > 0. For our analysis below, conditional on the design matrices +X1, . . . , XM at the M machines, it will be convenient to make the following +change of variables from θmin to the (data-dependent) SNR parameter r, +θmin = θmin(d, σ, r, n, cΩ) = σ +� +2cΩ +n r ln d, +(14) +where cΩ is defined as +cΩ = +max +i∈[d],m∈[M] +� +ˆΩm ˆΣm(ˆΩm)⊤� +ii . +(15) +Recall from Eq. (9) that by Theorem 1, σ2 � +n−1 ˆΩm ˆΣm(ˆΩm)⊤� +ii is the asymp- +totic variance of ˆθm +i . Hence, σ2cΩ/n is the largest variance of all d coordinates +of the M debiased estimators computed by the M machines. In terms of the +SNR parameter r, the range of interest is thus +1 +M < r < 1. + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +14 +Recall that our scheme is based on thresholding the normalized debiased lasso +estimators ˆξm +k of Eq. (9). We denote the corresponding normalized signal by +ϑm +k = +√nθ∗ +k +σ +� +ˆΩm ˆΣm(ˆΩm)⊤ +�1/2 +kk +, +∀k ∈ [d]. +(16) +Lemma 1 states that under suitable conditions, ˆξm +k − ϑm +k has approximately a +standard Gaussian distribution. This property plays an important role in our +theoretical results. Eq. (7) of Lemma 1 provides a bound on the error between +the CDF of ˆξm +k −ϑm +k and that of a standard Gaussian. For a threshold τ, let ϵ(τ) +be the largest of these error bounds over all d coordinates in all M machines, +ϵ(τ) = +max +k∈[d],m∈[M] +� +� +� +� +� +δRφ (τ − ϑm +k ) +σ +� +ˆΩm ˆΣm(ˆΩm)⊤ +�1/2 +kk ++ 2de−c∗n/K + de−cn + 6 +d2 +� +� +� +� +� +. (17) +Recall that δR, defined in Eq. (8), is an upper bound on the bias ˆθm +k − θ∗ +k. +By Lemma 5.4 of van de Geer et al. (2014), if the row sparsity of Ω satisfies +KΩ = o(n/ ln d) and ˆΩm is computed with regularization λΩ ∝ +� +(ln d)/n, then +� +ˆΩm ˆΣm(ˆΩm)⊤� +kk ≥ Ωkk + oP (1) ≥ C−1 +max + oP (1) when ln d +n → 0. Hence, when +n and d are large, all terms on the right hand side of Eq. (17) are small, and +the Gaussian approximation is accurate. +To prove that our scheme recovers S with high probability, we assume that: +(C1) The n samples in each of the M machines are i.i.d. from the model (1) +and the conditions of Theorem 1 all hold. Additionally, all machines use +the same regularization parameters λ and λΩ to compute the lasso (2) and +debiased lasso (3) estimators, respectively. +(C2) |θ∗ +i | ≥ θmin(d, σ, r, n, cΩ) for all i ∈ S, where θmin and cΩ are defined in +Eqs. (14) and (15), respectively. +The following theorem provides a recovery guarantee for Algorithm 2, where +the sparsity K is assumed to be known to the fusion center. +Theorem 2. Suppose Algorithm 2 is run with threshold τ = +√ +2 ln d. Assume +that d is sufficiently large and that the SNR in Eq. (14) satisfies +1 +4 +ln2(48√π ln3/2 d) +ln2(d) +< r < 1. +Additionally, assume conditions C1 and C2 hold and the approximation error + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +15 +in Eq. (17) satisfies ϵ(τ) ≤ 1/d. Then, if the number of machines satisfies +8 ln d +� +2(1−√r) +2 ln d +√ +2π +� +2(1−√r) +2 ln d+1 +� − ϵ(τ)d(1−√r) +2 d(1−√r) +2 +≤ M ≤ d +3, +(18) +with probability at least 1 − K+1 +d , Algorithm 2 achieves exact support recovery. +Additionally, the expected communication per machine is O (K ln d) bits. +Let us make a few remarks regarding this theorem. The upper bound M < +d/3 is rather artificial and stems from the fact that in our proof we assume +M < d. It is possible to derive support guarantees also for the case M > d, +though this setting seems to be unlikely in practice. The lower bound on the +number of machines is required to guarantee that with high probability, all +support indices receive more votes than any non-support coordinate. The lower +bound on the SNR r ensures that the lower bound on the number of machines +in Eq. (18) is indeed smaller than d/3, so the range of possible values for M is +not empty. A similar lower bound on r appeared in Amiraz, Krauthgamer and +Nadler (2022) after their Theorem 1.B. +Another important remark is that the threshold τ = +√ +2 ln d in Theorem 2 +is relatively high, so each machine sends only few indices to the center. How- +ever, to guarantee support recovery, this requires a relatively large number of +machines M = polylog(d) · d(1−√r)2. In Theorem 3, we give sufficient conditions +to still achieve a high probability of exact support recovery when the number +of machines is much smaller, of order only logarithmic in d. The price to pay is +a higher communication per machine, which nonetheless is still sub-linear in d, +namely much lower than the communication required to send the whole debi- +ased lasso vector. For the next theorem, we assume that a lower bound on the +SNR is known to all machines, which set a threshold that depends on it. +Theorem 3. Suppose Algorithm 2 is run with threshold τ = +√ +2r ln d. Assume +that d is sufficiently large and that the SNR in Eq. (14) satisfies +ln(16 ln d) +ln d +< r < 1. +Additionally, assume conditions C1 and C2 hold and the approximation error +in Eq. (17) satisfies ϵ(τ) < 1/(4dr). If the number of machines satisfies +16 ln d +1 − 2ϵ(τ) ≤ M ≤ dr, +(19) + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +16 +then with probability at least 1 − K+1 +d , Algorithm 2 achieves exact support re- +covery, with expected communication per machine O +� +d1−r ln d +� +bits. +Beyond support recovery, another quantity of interest is the accuracy of the +distributed estimator ˆθ of Eq. (11). The following corollary, proven in Appendix +A, shows that once S is precisely recovered, ˆθ is close to the oracle least squares +estimator ˆθLS computed with all data in a single machine and with knowledge +of the true support. Consequently, ˆθ is also close to the true vector θ∗. +Corollary 1. Assume the conditions of Theorem 2 hold. Let N = nM denote +the total sample size in all M machines. If M = O +� +NK +(max{K,ln N})2 +� +, then +∥ˆθ − ˆθ +LS∥2 = OP +�√ +M max{K, ln N} +N +� +and +∥ˆθ − θ∗∥2 = OP +�� +K +N +� +, +as d, N → ∞ and +ln d +N/M → 0, where ˆθ is defined in Eq. (11) and ˆθLS is the +least squares solution using all N samples and with a known S, as in Eq. (12), +appended by zeros at all coordinates j /∈ S. +Corollary 1 shows that in a high dimensional sparse setting, for a sufficiently +strong signal, Algorithm 3 with a threshold τ = +√ +2 ln d achieves the same error +rate as the oracle estimator. Let us put this result in a broader context. If the +support S were known, then each machine could have computed its least squares +solution restricted to S and send it to the center for averaging. As discussed +in Rosenblatt and Nadler (2016), in a general setting of M-estimators, if the +number of machines is not too large, averaging is optimal and to leading order +coincides with the centralized solution. Yet, while being rate optimal, we note +that averaging does lead to a loss of accuracy and is not as efficient as the oracle +estimator, see Dobriban and Sheng (2021). +As mentioned in Remark 4.3, Battey et al. (2018) also proposed a two-round +estimator that attains the optimal rate in Corollary 1, but requires each machine +to send at least d values to the fusion center. In contrast, ˆθ is computed using +a much lower communication cost. Similar results can also be established for +Algorithm 3 under the conditions of Theorem 3. +5.1. Comparison to other works +Theorems 2 and 3 can be viewed as analogous to Theorems 2.A and 2.B of +Amiraz, Krauthgamer and Nadler (2022), who studied distributed estimation + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +17 +of a sparse vector under communication constraints. A key difference is that +in their setting, the d coordinates of the estimated vector at each machine are +all independent and unbiased. This allowed them to analyze a top-L scheme +since the probability of sending any non-support index was the same and could +be easily bounded by L/d. As such, their proofs do not directly apply to our +setting. In our case, the debiased lasso ˆθm still has a small bias term and its +d coordinates are in general correlated. This implies that the probabilities of +sending non-support indices are not all identical. In our analysis, we bypass +this issue by analyzing instead a thresholding approach at step 5 of Algorithm +2, where each ˆθm +i +for i ∈ [d] is compared separately to a fixed threshold. This +way, we do not need to account for the complex dependence among different +coordinates of the debiased lasso. +Barghi, Najafi and Motahari (2021) considered a similar distributed scheme +to estimate the support of θ∗. They did not normalize the debiased lasso estimate +at each machine, and more importantly the estimated support set consists only +of those indices that received at least M/2 votes. The authors performed a +theoretical analysis of their scheme, though various quantities are described only +up to unspecified multiplicative constants. We remark that both theoretically as +well as empirically, the SNR must be quite high for support indices to obtain at +least M/2 votes. In our work, we present explicit expressions for the minimum +SNR in Eq. (14), sufficient for exact support recovery, requiring far fewer votes +at the fusion center. The simulations in the next section illustrate the advantages +of our scheme as compared to that of Barghi, Najafi and Motahari (2021). +6. Simulations +We present simulations that illustrate the performance of our proposed methods +in comparison to other distributed schemes. We focus on methods based on +debiased lasso estimates, and specifically consider the following five distributed +schemes to estimate θ∗ and its support. +• thresh-votes: Algorithm 2 with a threshold of τ = +√ +2 ln d. +• top-L-votes: The top L algorithm presented in section 4.1. Each machine +sends the indices of its top L values of |ˆξm +i | to the fusion center. +• top-L-signs: Each machine sends both the indices and signs of the top +L values of |ˆξm +i |. The center forms ˆS using sums of signs as in Eq. (13). +• BNM21: The algorithm proposed by Barghi, Najafi and Motahari (2021) + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +18 +with a threshold τ = +√ +2 ln d. It is similar to Algorithm 2, the difference +is that ˆS consists of the indices i ∈ [d] with Vi ≥ M/2. +• AvgDebLasso: Based on Lee et al. (2017), each machine sends its debiased +lasso estimate ˆθm. The center computes ˆθavg = +1 +M +�M +m=1 ˆθm and estimates +the support as the indices with the K largest values |ˆθavg +i +|, i ∈ [d]. +In all these algorithms, each machine computes a debiased lasso estimator using +its own data. The methods differ by the content and length of the messages sent +to the fusion center and hence by the manner in which the fusion center estimates +S and θ∗. For a fair comparison, we run all methods with the same regularization +parameters λΩ = 2 +� +(ln d)/n and λ = 8 +� +(ln d)/n in each machine to compute +the precision matrix ˆΩm and the lasso estimator ˜θm, respectively. +We performed simulations with both known sparsity, where the methods +were run as described above, as well as unknown sparsity. In the latter case, +for thresh-votes, top-L-votes and top-L-signs, the center computes ˆS as +the indices i such that Vi or |V sign +i +| is larger than 2 ln d, see Remark 4.5 in +Section 4; For AvgDebLasso, we set ˆS as the indices i such that |ˆθavg +i +| > 11 ln d +n . +This scaling is motivated by Theorem 16 of Lee et al. (2017), whereby in our +simulation setting this term is larger than the term O +�� +ln d/(nM) +� +in their +bound for ∥ˆθavg − θ∗∥∞. The factor 11 was manually tuned for good results. +BNM21 is unchanged, as it does not require knowledge of K. +We evaluate the accuracy of an estimated support set ˆS by the F-measure, +F-measure = 2 · precision · recall +precision + recall , +where precision = |S ∩ ˆS|/| ˆS| and recall = |S ∩ ˆS|/K. An F-measure equal to +one indicates that exact support recovery was achieved. +Given a support estimate ˆS, the vector θ∗ is estimated as follows. For all +four methods excluding AvgDebLasso, we perform a second round and compute +the estimator ˆθ given by Eq. (11). AvgDebLasso is a single round scheme. Its +estimate of θ∗ consists of ˆθavg restricted to the indices i ∈ ˆS. The error of +an estimate ˆθ is measured by its ℓ2-norm ∥ˆθ − θ∗∥2. As a benchmark for the +achievable accuracy, we also computed the oracle centralized estimator ˆθLS that +knows the support S and estimates θ∗ by least squares on the whole data. +We generated data as follows: The design matrix Xm ∈ Rn×d in machine +m has n rows i.i.d. N(0, Σ), with Σi,j = 0.5|i−j|. We then computed for each +machine its matrix ˆΩm, and the quantity cΩ in Eq. (15). Next, we generated a K +sparse vector θ∗ ∈ Rd, whose nonzero indices are sampled uniformly at random + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +19 +(a) +(b) +Fig 1: Results for known sparsity, averaged over 500 realizations, as a function +of SNR in the range r ∈ [1/M, 1]. (a) F-measure; (b) ℓ2 error, on a log scale. +from [d]. Its nonzero coefficients have random ±1 signs and their magnitudes +are chosen from K equally spaced values {θmin, . . . , 2θmin}, where +θmin = +� +2cΩ ln d +n +. +These matrices and the vector θ∗ are then kept fixed. For a simulation with +SNR parameter r, we set σ = 1/√r. Finally, in each realization we generated +the response Y m ∈ Rn according to the model (1). +Our first simulation compared the performance of various schemes as a func- +tion of the SNR, with a known sparsity K = 5. We fixed the dimension d = 5000, +the sample size in each machine n = 250, and the number of machines M = 100. +The top L method was run with L = K (see Sec. 4.1). Figure 1a displays the +F-measure of each method, averaged over 500 realizations. As expected, at low +SNR values, AvgDebLasso achieved the best performance in terms of support +recovery. However, for stronger signals with r > 0.4, both top-K-votes and +thresh-votes achieved an F-measure of one, in accordance with our theoret- +ical results regarding exact recovery. In particular, at sufficiently high SNR, +our methods estimate the support as accurately as AvgDebLasso, but with 2-3 +orders of magnitude less communication. The scheme of BNM21 achieves good + +n=250.d=5000.K=5.M=100 +1.0 +0.8 +measure +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +r +AvgDebLasso +top-K-votes +BNM21 +oracle +thresh-votesn=250,d=5000,K=5,M=100 +0.00 +-0.25 +error +-0.50 +-0.75 +) +1.00 +-1.25 +1.50 +-1.75 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +r +AvgDebLasso +top-K-votes +BNM21 +oracle +thresh-votesR. Fonseca and B. Nadler/Distributed Sparse Linear Regression +20 +(a) +(b) +Fig 2: Results for unknown sparsity, averaged over 500 realizations, as a function +of SNR in the range r ∈ [1/M, 1]. (a) F-measure; (b) ℓ2 error on a log scale. +performance only at higher SNR. +Figure 1b shows the errors ∥ˆθ − θ∗∥2, averaged over 500 realizations. At low +SNR, 1/M < r < 0.1, AvgDebLasso has the smallest error. However, for r > 0.4, +thresh-votes and top-K-votes yield more accurate estimates. Thus, Figure +1b shows the benefits of an accurate support estimate followed by a distributed +least squares in a second round. Indeed, at these SNR levels, our methods exactly +recover the support. Consequently, the second round reduces to a distributed +ordinary least squares restricted to the correct support set S. In accordance +with Corollary 1, Algorithm 2 then has the same error rate as the oracle. +Next we present simulation results for unknown sparsity, as a function of the +SNR in the range r ∈ [ 1 +M , 1]. As seen in Figure 2, throughout this SNR range, +AvgDebLasso with a threshold of 11(ln d)/n achieves an F-measure close to one +and ℓ2 errors close to those of the oracle. In contrast, thresh-votes achieves +accurate estimates only for r > 0.6. These results illustrate that even when +sparsity is unknown, our schemes can accurately estimate the vector θ∗ and its +support, albeit with a higher SNR as compared to the case of known sparsity. +Finally, we present simulation results as a function of number of samples +n ∈ [nmin, nmax] = [100, 400] with M = 100 machines, and as a function of +number of machines M ∈ [Mmin, Mmax] = [40, 160] with n = 250 samples + +n=250,d=5000,K=5,M=100 +1.0 +0.8 +measure +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +r +AvgDebLasso +thresh-votes +BNM21 +oraclen=250,d=5000,K=5,M=100 +0.00 +-0.25 +error) +-0.50 +-0.75 +log1o(l2 +-1.00 +-1.25 +-1.50 +-1.75 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +r +AvgDebLasso +thresh-votes +BNM21 +oracleR. Fonseca and B. Nadler/Distributed Sparse Linear Regression +21 +(a) +(b) +Fig 3: ℓ2 error vs. sample size n (a) and vs. number of machines M (b), both on +a log-log scale, at an SNR of r = 0.4. Values are averaged over 1000 realizations. +per machine. Initially, for each machine m ∈ [Mmax], we generated its full data +matrix Xm with nmax number of samples. We then computed the corresponding +matrix ˆΩm and the quantity cΩ of Eq. (15). We then generated the sparse +vector θ∗ as described above. To save on run-time, in simulations with a smaller +number of samples n < nmax we nonetheless used the decorrelation matrices +Ωm that correspond to nmax samples. This can be viewed as a semi-supervised +setting, as also mentioned in Javanmard and Montanari (2018). We fixed r = 0.4 +and d = 5000, and the sparsity K = 5 was known to the center. Figure 3 +shows the resulting ℓ2 errors averaged over 1000 simulations. In this simulation, +top-K-votes and thresh-votes are close to the centralized least squares oracle +since their support estimates are accurate, as can be seen in Figure 1 for r = 0.4. +Both plots in Figure 3 show a linear dependence on a log-log scale with a slope of +approximately −1/2, namely that the resulting errors decay as 1/√n and 1/ +√ +M, +respectively. This is in agreement with our theoretical result in Corollary 1. +6.1. The advantages of sending signs +We now illustrate the advantages of using sums of signs instead of sums of votes, +in terms of both support recovery and parameter estimation. In this simulation, +we fixed n = 250, d = 5000, M = 100 and K = 25. The results in Figure +4 show that using sums of signs is more accurate than sums of votes for low + +d=5000,K=5,r=0.40,M=100 +-1.0 +log1o(l2 error) +-1.2 +-1.4 +-1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +log1o(n) +AvgDebLasso +top-K-votes +thresh-votes +oracled=5000,K=5,r=0.40,n=250 +-1.0 +log1o(l2 error) +-1.2 +-1.4 +-1.6 +-1.8 +1.6 +1.8 +2.0 +2.2 +log1o(M) +AvgDebLasso +top-K-votes +thresh-votes +oracleR. Fonseca and B. Nadler/Distributed Sparse Linear Regression +22 +(a) +(b) +Fig 4: Results for schemes using sums of signs and sums of votes, averaged over +500 realizations, as a function of r∈[ 1 +M , 0.1]. (a) F-measure; (b) log10 of ℓ2 error. +SNR values in the range r ∈ [1/M, 0.1]. Figure 4 also shows that using L = 5K +instead of L = K significantly improves the accuracy of the support estimator, +at the expense of increasing the communication. This illustrates the potential +trade-offs between the accuracy of support estimation and communication. +7. Summary and Discussion +The development and analysis of distributed statistical inference schemes having +low communication are important contemporary problems. Given its simplicity +and ubiquity, the sparse linear regression model has attracted significant atten- +tion in the literature. Most previous inference schemes for this model require +communication per machine of at least O(d) bits. In this work we proved the- +oretically and showed via simulations that, under suitable conditions, accurate +distributed inference for sparse linear regression is possible with a much lower +communication per machine. +Over the past years, several authors studied distributed statistical infer- +ence under communication constraints. Specifically, for sparse linear regression, +Braverman et al. (2016) proved that without a lower bound on the SNR, to ob- + +n=250.d=5000.K=25.M=100 +1.0 +0.8 +measure +0.6 +0.4 +0.2 +0.0 +0.02 +0.04 +0.06 +0.08 +0.10 +r +AvgDebLasso +top-5K-votes +thresh-votes +top-5K-signs +top-K-votes +oracle +top-K-signsn=250,d=5000,K=25,M=100 +0.4 +0.2 +error) +0.0 +0.2 +)0 +0.4 +0.6 +-0.8 +-1.0 +0.02 +0.04 +0.06 +0.08 +0.10 +r +AvgDebLasso +top-5K-votes +thresh-votes +top-5K-signs +top-K-votes +oracle +top-K-signsR. Fonseca and B. Nadler/Distributed Sparse Linear Regression +23 +tain a risk comparable to that of the minimax lower bound, a communication of +at least Ω(M min(n, d)/ log d) bits is required. Acharya et al. (2019) proved that, +under certain conditions, rate optimal estimates of a linear regression model can +be computed using total communication sublinear in the dimension. However, +as they mention in their appendix B.3, a precise characterization of the ability +to recover the support with sublinear communication in d and its dependency on +other parameters such as SNR and the number of machines is still an open prob- +lem. In our theoretical results, we presented explicit expressions for the minimal +SNR at which our scheme is guaranteed to achieve exact recovery with high +probability and with sublinear communication. While we did not address the +open problem of tight lower bounds, our results highlight the potential tradeoffs +between SNR, communication and number of machines. +We believe that using more refined techniques, our theoretical analysis can be +extended and improved. For example, since the d coordinates of a debiased lasso +estimator are correlated, sharp concentration bounds for dependent variables, +like those of Lopes and Yao (2022), could improve our analysis and extend it to +other schemes such as top-L. In our analysis, we focused on a setting where both +the noise and covariates have Gaussian distribution. Lee et al. (2017) and Battey +et al. (2018), for example, considered sub-Gaussian distributions for these terms. +Our results can be adapted for this case, but a careful control of the various +constants in probability bounds is needed to derive explicit expressions. +Finally, our low-communication schemes could also be applied to other prob- +lems, such as sparse M-estimators, sparse covariance estimation and distributed +estimation of jointly sparse signals. We leave these for future research. +Appendix A: Proofs +A.1. Proof of Lemma 1 +Proof. Consider the debiased lasso estimator ˆθi given in Eq. (3). Making the +change of variables t = σ√ciiτ/√n, gives that +Pr +�√n(ˆθi − θ∗ +i ) +σ√cii +≤ τ +� += Pr(ˆθi − θ∗ +i ≤ t). +(20) + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +24 +It follows from Eq. (5) that Pr(ˆθi − θ∗ +i ≤ t) = Pr (Zi ≤ √nt − Ri). By the law +of total probability, +Pr +� +Zi ≤ √nt − Ri +� += +Pr +� +{Zi ≤ √nt − Ri} ∩ {|Ri| ≤ δR} +� ++ Pr +� +{Zi ≤ √nt − Ri} ∩ {|Ri| > δR} +� +≤ +Pr +� +Zi ≤ √nt + δR +� ++ Pr (|Ri| > δR) . +(21) +From Eq. (5) it follows that Pr (Zi ≤ √nt + δR) = Φ +� √nt+δR +σ√cii +� +. Hence, from +Eqs. (20) and (21) we get that +Pr +�√n(ˆθi − θ∗ +i ) +σ√cii +≤ τ +� +≤ Φ +�√nt + δR +σ√cii +� ++ Pr (|Ri| > δR) . +(22) +The second term on the right-hand side of Eq. (22) can be bounded by Eq. (6). +This gives the last three terms on the right hand side in Eq. (7). +Let us analyze Φ +� √nt+δR +σ√cii +� +. For any fixed x and δ > 0, by the mean value +theorem, |Φ(x + δ) − Φ(x)| ≤ δφ(x∗), where x∗ ∈ (x, x + δ). Since φ(x) is a +decreasing function for x > 0, we have φ(x∗) ≤ φ(x). Thus, +|Φ(x + δ) − Φ(x)| ≤ δφ(x). +Applying this result with x = +√nt +σ√cii and δ = +δR +σ√cii , gives +����Φ +�√nt + δR +σ√cii +� +− Φ +� √nt +σ√cii +����� ≤ +δR +σ√cii +φ +� √nt +σ√cii +� +. +Combining the above with Eq. (22), and replacing t = σ√ciiτ +√n +proves Eq. (7). +A.2. Proofs of Theorem 2 and Theorem 3 +Let us first provide an overview of the proofs. Recall that we consider a dis- +tributed setting with M machines each with its own data, and each machine +sends an independent message containing a few indices to the fusion center. For +any index i ∈ [d] and machine m, let pm +i +denote the probability that index i +is sent by machine m, namely that |ˆξm +i | > τ. Since data at different machines +are statistically independent, the total number of votes Vi received at the fusion +center for index i is distributed as Vi ∼ �M +i=1 Ber(pm +i ). Our proof strategy is as +follows: we compute an upper bound for pm +j for non-support indices j ̸∈ S, and +a lower bound for pm +i for support indices i ∈ S. Next, we employ tail bounds for + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +25 +binomial random variables. Combining these implies that under suitable condi- +tions on the SNR and number of machines, with high probability, the number +of votes Vi can perfectly distinguish between support and non-support indices. +To carry out this proof outline, we now introduce some auxiliary lemmas and +results. The following are standard Gaussian tail bounds, +t +√ +2π(t2 + 1)e−t2/2 ≤ 1 − Φ(t) ≤ +1 +√ +2πte−t2/2, +∀t > 0. +(23) +We also use the following inequality for a binomial variable V ∼ Bin(M, p) +(Boucheron, Lugosi and Massart, 2013, exercise 2.11). For any 0 < p ≤ a < 1, +Pr (V > Ma) ≤ +��p +a +�a �1 − p +1 − a +�1−a�M += eMF (a,p) +(24) +where +F(a, p) = a ln +�p +a +� ++ (1 − a) ln +�1 − p +1 − a +� +. +(25) +The following result appeared in (Amiraz, Krauthgamer and Nadler, 2022, +Lemma A.3). It is used in our proof to show that with high probability, support +indices receive a relatively large number of votes. +Lemma 2. Assume that mini∈S |θ∗ +i | is sufficiently large so that for some suitable +pmin > 0, for all i ∈ S, and m ∈ [M], pm +i ≥ pmin. If pmin ≥ 8 ln d +M , then +Pr +� +min +i∈S Vi < 4 ln d +� +≤ K +d . +The next lemma shows that under suitable conditions, non-support indices +receive relatively few total number of votes. +Lemma 3. Assume that d ≥ 4 and M > 2 ln d. In addition, assume that +pm +j ≤ +1 +M for all non-support indices j ̸∈ S and all machines m ∈ [M]. Then +Pr +� +max +j̸∈S Vj > 2 ln d +� +≤ 1 +d. +(26) +Proof. Recall that the number of votes received by an index j ̸∈ S at the fusion +center is distributed as Vj ∼ �M +m=1 Ber(M, pm +j ). Since pm +j ≤ +1 +M for all j ̸∈ S, +then Vj is stochastically dominated by +V ∼ Bin(M, p), +where +p = 1/M. +(27) + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +26 +Thus, by a union bound, +Pr +� +max +j̸∈S Vj > t +� +≤ (d − K) · Pr(V > t) ≤ d · Pr +� +V > M t +M +� +. +We now apply Eq. (24) with t ≥ 1, so that the value of a = t/M = tp indeed +satisfies that a ≥ p. With F(a, p) defined in Eq. (25), this gives +Pr +� +max +j̸∈S Vj > t +� +≤ d · eMF (tp,p). +(28) +Next, we upper bound F(tp, p). Since ln(1 + x) ≤ x holds for all x ≥ 0, +F(tp, p) = −tp ln(t) + (1 − tp) ln +� +1 + tp − p +1 − tp +� +≤ −tp ln(t) + tp − p +< −tp ln(t) + tp = −tp ln(t/e). +Inserting this into Eq. (28) with t = 2 ln d, p = 1/M and M > 2 ln d gives +Pr +� +max +j̸∈S Vj > 2 ln d +� +≤ deMF (tp,p) ≤ de−2 ln(d) ln(2 ln(d)/e). +For Eq. (26) to hold, we thus require that +2 ln(d) ln +�2 ln d +e +� +≥ ln +� +d2� +. +(29) +This holds for d ≥ exp{e/2} ≈ 3.89. +Proof of Theorem 2. Our goal is to show that the event { ˆS = S} occurs with +high probability. Recall that ˆS is determined at the fusion center as the K +indices with the largest number of votes. Further recall that pm +i +denotes the +probability that machine m sends index i. Our strategy is to show that these +probabilities are sufficiently large for support indices and sufficiently small for +non-support indices. This shall allow us to apply Lemmas 2 and 3 to prove the +required result. To derive bounds on pm +i +we employ Lemma 1. +First, we prove that the condition of Lemma 2 holds, i.e. that +pm +i ≥ 8 ln d +M +for all i ∈ S, +(30) +where pm +i += Pr(|ˆξm +i | > τ), and ˆξm +i += +√nˆθm +i +σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2 +ii +is the standardized +debiased lasso estimator, defined in Eq. (9). Without loss of generality, assume +that θ∗ +i > 0. Otherwise, we could do the same calculations for −ˆξm +i . Clearly, +pm +i = Pr(|ˆξm +i | > τ) ≥ Pr(ˆξm +i +> τ) = Pr +� +√n +ˆθm +i − θ∗ +i +σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2 +ii +> τ − ϑm +i +� +, + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +27 +where ϑm +i +is defined in Eq. (16). Since condition C1 holds, applying Eq. (7) of +Lemma 1 gives that +pm +i ≥ Φc(τ − ϑm +i ) − ϵ(τ), +(31) +where ϵ(τ) is the error defined in Eq. (17). +Next, by condition C2, with the definition of cΩ in Eq. (15) and Eq. (14) +ϑm +i = +√nθ∗ +i +� +ˆΩm ˆΣm(ˆΩm)⊤ +�1/2 +ii +≥ +√nθmin +√cΩ += +√ +2r ln d. +At a threshold τ = +√ +2 ln d we thus obtain +pm +i ≥ Φc � +(1 − √r) +√ +2 ln d +� +− ϵ(τ). +By the Gaussian tail bound (23), +Φc +�� +2(1 − √r)2 ln d +� +≥ C(r, d)d−(1−√r)2, +where +C(r, d) = +� +2(1 − √r)2 ln(d) +√ +2π{2(1 − √r)2 ln(d) + 1}. +Therefore, for the condition (30) to hold, it suffices that +M ≥ +8 ln d +C(r, d)d−(1−√r)2 − ϵ(τ) = +8 ln d +C(r, d) − ϵ(τ)d(1−√r)2 d(1−√r)2, +which is precisely condition (18) of the theorem. Notice that the requirement +ϵ(τ) < 1/d guarantees that the denominator in the fraction above is positive. +The lower bound on r in the theorem, guarantees that the range of possible +values for M is non-empty. +Next, we prove that the conditions of Lemma 3 hold. The condition M > +2 ln d is satisfied given the requirement of Eq. (18). The next condition to verify +is pm +j ≤ 1/M for all j ̸∈ S. Since pm +j = Pr(|ˆξm +j | > τ) and ϑm +j = 0 for j ̸∈ S, then +pm +j − 2Φc(τ) = +� +Pr(ˆξm +j > τ) − Φc(τ) +� ++ +� +Pr(ˆξm +j < −τ) − Φc(τ) +� +. +According to Eq. (5) of Theorem 1, apart from a bias term, ξm +j +and −ξm +j +have +the same distribution because ϑm +j = 0. Hence, applying Eq. (7) in Lemma 1 to +each of the above bracketed terms separately gives that +pm +j ≤ 2Φc(τ) + 2ϵ(τ), + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +28 +where ϵ(τ) was defined in Eq. (17). By Eq. (23), for τ = +√ +2 ln d we have 2Φc(τ) ≤ +2 +√ +2π +√ +2 ln d +1 +d. Since ϵ(τ) < 1/d, we obtain that for all j ̸∈ S +pm +j ≤ +1 +√ +π ln d +1 +d + 2 +d ≤ 3 +d ≤ 1 +M , +(32) +where the last inequality follows from the assumption that M ≤ d/3 in Eq. (18). +Therefore, all conditions of Lemma 3 are satisfied. +Since Lemmas 2 and 3 hold, we apply their results with a union bound to +derive a lower bound on the probability of exact support recovery, as follows: +Pr( ˆS = S) ≥ Pr +�� +max +j̸∈S Vj ≤ 4 ln d +� +∩ +� +min +i∈S Vi ≥ 4 ln d +�� +≥ 1 − Pr +� +max +j̸∈S Vj ≥ 4 ln d +� +− Pr +� +min +i∈S Vi ≤ 4 ln d +� +≥ 1 − (K + 1) +d +. +We remark that Lemma 3 provides a bound on Pr +� +maxj /∈S Vj > 2 ln d +� +. Clearly, +the probability for a higher threshold 4 ln d above is much smaller. +Finally, we analyze the communication per machine. Let Bm denote the num- +ber of bits sent by machine m. Note that Bm is a sum of Bernoulli random +variables Bm +k ∼ Ber(pm +k ) times some factor ∝ ln d corresponding to the num- +ber of bits necessary to represent indices in [d]. The random variable Bm +k is an +indicator whether machine m sends index k to the center. Then +E(Bm) = O +� d +� +k=1 +E(Bm +k ) ln d +� += O +� +� +� +� +� +� +i∈S +pm +i + +� +j̸∈S +pm +j +� +� +� ln d +� +� . +(33) +Since pm +j ≤ 3/d for all j ̸∈ S, then � +j̸∈S pm +j ≤ 3. Additionally, � +i∈S pm +i ≤ K. +Therefore, E(Bm) = O (K ln d). +Proof of Theorem 3. The proof is similar to that of Theorem 2. We first show +that the conditions of Lemmas 2 and 3 hold. Then, we derive a lower bound +for the probability of Algorithm 2 achieving exact support recovery. Recall that +here the threshold is τ = +√ +2r ln d, where r is the SNR introduced in Eq. (14). +Let us start by proving that the condition of Lemma 2 holds. We need to +show that pm +i ≥ 8 ln d +M +for all i ∈ S. As in Eq. (31) in the proof of Theorem 2, +pm +i ≥ Φc(τ − ϑm +i ) − ϵ(τ) +where ϑm +i +is given by Eq. (16). + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +29 +By condition C2, ϑm +i ≥ √nθmin/√cΩ = +√ +2r ln d, where θmin and cΩ are given +by Eqs. (14) and (15), respectively. Since τ = +√ +2r ln d, then τ − ϑm +i ≤ 0 and +pm +i ≥ Φc(0) − ϵ(τ) ≥ 1 +2 − ϵ(τ). +The inequality pm +i +≥ 8 ln d +M +follows directly from the requirement on M in Eq. +(19), i.e., M ≥ +8 ln d +1/2−ϵ(τ). +Next, we verify the conditions of Lemma 3. Its first condition that M > 4 ln d +holds given Eq. (19). For the second condition, we need to show that pm +j ≤ 1/M +for all j ̸∈ S. As in the proof of Theorem 2, +pm +j ≤ 2Φc(τ) + 2ϵ(τ). +Plugging the value τ = +√ +2r ln d into the Gaussian tail bound (23) gives +pm +j ≤ +1 +√ +πr ln d +1 +dr + 2ϵ(τ) +(i) +≤ 1 +dr , +(34) +where inequality (i) follows from the assumptions that ϵ(τ) < 1/(4dr) and +r > ln(16 ln d)/(ln d). This latter assumption implies that +1 +√ +πr ln(d) ≤ +1 +2 for +sufficienly large d. Since we assume that M ≤ dr in Eq. (19), then pm +j ≤ 1/dr ≤ +1/M for all j ̸∈ S. Hence, both conditions of Lemma 3 are satisfied. +Applying a union bound and the result of Lemmas 2 and 3, it follows that +Pr( ˆS = S) ≥ Pr +�� +max +j̸∈S Vj ≤ 4 ln d +� +∩ +� +min +i∈S Vi ≥ 4 ln d +�� +≥ 1 − (K + 1) +d +. +Finally, let us analyze the average communication per machine. Let B de- +note the number of bits sent by a single machine. Following the same steps +used to compute Eq. (33), the expectation of B may be bounded as E(B) ≤ +O +�� +K + d−K +dr +� +ln d +� +, where the factor 1/dr is due to Eq. (34). Hence, the ex- +pected communication of a single machine is O +� +d1−r ln d +� +bits. +A.3. Proof of Corollary 1 +Proof. We proceed similar to Battey et al. (2018) in the proof of their Corollary +A.3. By the law of total probability, for any constant C′ > 0, +Pr +� +∥ˆθ − ˆθ +LS∥2 > C′ +√ +M max{K, ln N} +N +� +≤ Pr +�� +∥ˆθ − ˆθ +LS∥2 > C′ +√ +M max{K, ln N} +N +� +∩ { ˆS = S} +� ++ Pr( ˆS ̸= S). + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +30 +By Theorem 2, Pr( ˆS ̸= S) ≤ (K + 1)/d. In addition, when ˆS = S, then ˆθj = +ˆθLS +j += 0 for all j ̸∈ S. Consequently, ∥ˆθ − ˆθLS∥2 = ∥ˆθS − ˆθLS +S ∥2. Furthermore, the +first term on the right hand side above may be bounded by +Pr +� +∥ˆθS − ˆθ +LS +S ∥2 > C′ +√ +M max{K, ln N} +N +� +. +The next step is to apply a result from the proof of Theorem A.1 of Battey +et al. (2018), which appeared in the last line of their proof. For clarity we state +it here as a lemma. +Lemma 4. Consider the linear model in dimension K, +y = X⊤β∗ + σw, +where w ∼ N(0, 1), X ∼ N(0, Σ), and Σ ∈ RK×K satisfies 0 < Cmin ≤ +σmin(Σ) ≤ σmax(Σ) ≤ Cmax < ∞. Suppose N i.i.d. samples from this model +are uniformly distributed to M machines, with n > K. Denote by ˆβm the least +squares solution at the m-th machine and ˆβLS the centralized least squares solu- +tion. If the number of machines satisfies M = O +� +NK +(max{K,ln N})2 +� +, then +Pr +���� 1 +M +� +m +ˆβm − ˆβ +LS��� +2 > C′ +√ +M max{K,ln N} +N +� +≤ cMe− max{K ln N} + Me−c N +M , +where c, C′ > 0 are constants that do not depend on K or N. +Applying this lemma to our case gives +Pr +� +∥ˆθS − ˆθ +LS +S ∥2 > C′ +√ +M max{K, ln N} +N +� +≤ cMe− max{K ln N} + Me−c N +M . +Since M = O +� +NK +(max{K,ln N})2 +� +, it follows that ∥ˆθ − ˆθLS∥2 = OP +�� +K +N +� +. As the +oracle estimator has rate ∥ˆθLS − θ∗∥2 = OP +�� +K +N +� +, by the triangle inequality +∥ˆθ − θ∗∥2 = OP +�� +K +N +� +as well. +References +Acharya, J., De Sa, C., Foster, D. J. and Sridharan, K. (2019). +Distributed +Learning +with +Sublinear +Communication. +arXiv +preprint +arXiv:1902.11259. + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +31 +Amiraz, C., Krauthgamer, R. and Nadler, B. (2022). Distributed sparse +normal means estimation with sublinear communication. Information and In- +ference: A Journal of the IMA iaab030. +Barghi, H., Najafi, A. and Motahari, S. A. (2021). Distributed sparse +feature selection in communication-restricted networks. arXiv preprint +arXiv:2111.02802. +Battey, H., Fan, J., Liu, H., Lu, J. and Zhu, Z. (2018). Distributed testing +and estimation under sparse high dimensional models. Annals of Statistics 46 +1352–1382. +Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequal- +ities: A Nonasymptotic Theory of Independence. Oxford University Press, Ox- +ford. +Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011). +Distributed optimization and statistical learning via the alternating direction +method of multipliers. Foundations and Trends in Machine learning 3 1–122. +Braverman, M., Garg, A., Ma, T., Nguyen, H. L. and Woodruff, D. P. +(2016). Communication lower bounds for statistical estimation problems via +a distributed data processing inequality. In Proceedings of the forty-eighth +annual ACM symposium on Theory of Computing 1011–1020. +Bunea, F., Tsybakov, A. and Wegkamp, M. (2007). Sparsity oracle inequal- +ities for the Lasso. Electronic Journal of Statistics 1 169–194. +Candes, E. J. and Tao, T. (2005). Decoding by linear programming. IEEE +Transactions on Information Theory 51 4203–4215. +Chen, X. and Xie, M. (2014). A split-and-conquer approach for analysis of +extraordinarily large data. Statistica Sinica 24 1655–1684. +Chen, X., Liu, W., Mao, X. and Yang, Z. (2020). Distributed high- +dimensional regression under a quantile loss function. Journal of Machine +Learning Research 21 1–43. +Dobriban, E. and Sheng, Y. (2020). WONDER: Weighted one-shot dis- +tributed ridge regression in high dimensions. Journal of Machine Learning +Research 21 1–52. +Dobriban, E. and Sheng, Y. (2021). Distributed linear regression by averag- +ing. Annals of Statistics 49 918–943. +Fan, J., Li, R., Zhang, C.-H. and Zou, H. (2020). Statistical Foundations of +Data Science. CRC Press, Boca Raton. +Gao, Y., Liu, W., Wang, H., Wang, X., Yan, Y. and Zhang, R. (2022). +A review of distributed statistical inference. Statistical Theory and Related + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +32 +Fields 6 89–99. +Guestrin, C., Bodik, P., Thibaux, R., Paskin, M. and Madden, S. (2004). +Distributed regression: an efficient framework for modeling sensor network +data. In Third International Symposium on Information Processing in Sensor +Networks 1–10. +Hastie, T., Tibshirani, R. and Wainwright, M. (2015). Statistical Learning +with Sparsity: The Lasso and Generalizations. CRC Press, Boca Raton. +Heinze, C., McWilliams, B., Meinshausen, N. and Krummenacher, G. +(2014). LOCO: Distributing ridge regression with random projections. arXiv +preprint arXiv:1406.3469. +Huo, X. and Cao, S. (2019). Aggregated inference. Wiley Interdisciplinary +Reviews: Computational Statistics 11 e1451. +Javanmard, A. and Montanari, A. (2014a). Hypothesis testing in high- +dimensional regression under the Gaussian random design model: Asymptotic +theory. IEEE Transactions on Information Theory 60 6522–6554. +Javanmard, A. and Montanari, A. (2014b). Confidence intervals and hy- +pothesis testing for high-dimensional regression. The Journal of Machine +Learning Research 15 2869–2909. +Javanmard, A. and Montanari, A. (2018). Debiasing the lasso: Optimal +sample size for Gaussian designs. The Annals of Statistics 46 2593–2622. +Jordan, M. I., Lee, J. D. and Yang, Y. (2019). Communication-efficient dis- +tributed statistical inference. Journal of the American Statistical Association +114 668–681. +Lee, J. D., Liu, Q., Sun, Y. and Taylor, J. E. (2017). Communication- +efficient sparse regression. The Journal of Machine Learning Research 18 1– +30. +Liu, M., Xia, Y., Cho, K. and Cai, T. (2021). Integrative high dimensional +multiple testing with heterogeneity under data sharing constraints. Journal +of Machine Learning Research 22 1–26. +Lopes, M. E. and Yao, J. (2022). A sharp lower-tail bound for Gaussian +maxima with application to bootstrap methods in high dimensions. Electronic +Journal of Statistics 16 58–83. +Lv, S. and Lian, H. (2022). Debiased distributed learning for sparse partial +linear models in high dimensions. Journal of Machine Learning Research 23 +1–32. +Mateos, G., Bazerque, J. A. and Giannakis, G. B. (2010). Distributed +sparse linear regression. IEEE Transactions on Signal Processing 58 5262– + +R. Fonseca and B. Nadler/Distributed Sparse Linear Regression +33 +5276. +Predd, J. B., Kulkarni, S. B. and Poor, H. V. (2006). Distributed learning +in wireless sensor networks. IEEE Signal Processing Magazine 23 56–69. +Rosenblatt, J. D. and Nadler, B. (2016). On the optimality of averaging in +distributed statistical learning. Information and Inference: A Journal of the +IMA 5 379–404. +Sun, T. and Zhang, C.-H. (2012). Scaled sparse linear regression. Biometrika +99 879–898. +Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. Jour- +nal of the Royal Statistical Society: Series B 58 267–288. +van de Geer, S. A. and B¨uhlmann, P. (2009). On the conditions used to +prove oracle results for the Lasso. Electronic Journal of Statistics 3 1360– +1392. +van de Geer, S., B¨uhlmann, P., Ritov, Y. and Dezeure, R. (2014). +On asymptotically optimal confidence regions and tests for high-dimensional +models. The Annals of Statistics 42 1166–1202. +Wainwright, M. J. (2009). Sharp thresholds for high-dimensional and noisy +sparsity recovery using ℓ1 - constrained quadratic programming (Lasso). IEEE +Transactions on Information Theory 55 2183–2202. +Zhang, Y., Duchi, J. C. and Wainwright, M. J. (2013). Communication- +efficient algorithms for statistical optimization. Journal of Machine Learning +Research 14 3321–3363. +Zhang, C. H. and Zhang, S. S. (2014). Confidence intervals for low dimen- +sional parameters in high dimensional linear models. Journal of the Royal +Statistical Society: Series B 76 217–242. +Zhu, X., Li, F. and Wang, H. (2021). Least-square approximation for a dis- +tributed system. Journal of Computational and Graphical Statistics 30 1004– +1018. + diff --git a/IdE2T4oBgHgl3EQfowif/content/tmp_files/load_file.txt b/IdE2T4oBgHgl3EQfowif/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..146955531825c1fd2990585ad0dd227a27e529b6 --- /dev/null +++ b/IdE2T4oBgHgl3EQfowif/content/tmp_files/load_file.txt @@ -0,0 +1,1220 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf,len=1219 +page_content='Distributed Sparse Linear Regression under Communication Constraints ∗ Rodney Fonseca and Boaz Nadler Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel e-mail: rodney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='fonseca@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='il;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' boaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='nadler@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='il Abstract: In multiple domains, statistical tasks are performed in dis- tributed settings, with data split among several end machines that are con- nected to a fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In various applications, the end machines have limited bandwidth and power, and thus a tight communication budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In this work we focus on distributed learning of a sparse linear regression model, under severe communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We propose several two round distributed schemes, whose communication per machine is sublinear in the data dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our schemes, individual machines compute debi- ased lasso estimators, but send to the fusion center only very few values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' On the theoretical front, we analyze one of these schemes and prove that with high probability it achieves exact support recovery at low signal to noise ratios, where individual machines fail to recover the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We show in simulations that our scheme works as well as, and in some cases better, than more communication intensive approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' MSC2020 subject classifications: Primary 62J07, 62J05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' secondary 68W15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Keywords and phrases: Divide and conquer, communication-efficient, debiasing, high-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Introduction In various applications, datasets are stored in a distributed manner among sev- eral sites or machines (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', 2020, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Often, due to communication constraints as well as privacy restrictions, the raw data cannot be shared be- tween the various machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Such settings have motivated the development of methods and supporting theory for distributed learning and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', the reviews by Huo and Cao (2019), Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2022) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' ∗This research was supported by a grant from the Council for Higher Education Compet- itive Program for Data Science Research Centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' RF acknowledges support provided by the Mor´a Miriam Rozen Gerber Fellowship for Brazilian postdocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='04022v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='LG] 9 Jan 2023 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 2 In this paper we consider distributed learning of a sparse linear regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Specifically, we assume that the response y ∈ R and the vector X ∈ Rd of explanatory variables are linearly related via y = X⊤θ∗ + w, (1) where w ∼ N(0, σ2), σ > 0 is the noise level, and θ∗ ∈ Rd is an unknown vector of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We further assume X ∈ Rd is random with mean zero and covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We focus on a high-dimensional setting d ≫ 1, and assume that θ∗ is sparse with only K ≪ d nonzero coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The support set of θ∗ ∈ Rd is denoted by S = {i ∈ [d] | |θ∗ i | > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Given N samples {(Xi, yi)}N i=1 from the model (1), common tasks are to estimate the vector θ∗ and its support set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Motivated by contemporary ap- plications, we consider these tasks in a distributed setting where the data are randomly split among M machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Specifically, we consider a star topology network, whereby the end machines communicate only with a fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As reviewed in Section 2, estimating θ∗ and its support in the above or simi- lar distributed settings were studied by several authors, see for example Mateos, Bazerque and Giannakis (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Chen and Xie (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Barghi, Najafi and Mota- hari (2021) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Most prior works on distributed regression required communication of at least O(d) bits per machine, as in their schemes each machine sends to the fusion center its full d-dimensional estimate of the unknown vector θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Some works in the literature denote this as communication efficient, in the sense that for a machine holding n samples, an O(d) communi- cation is still significantly less than the size O(n · d) of its data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The design and analysis of communication efficient distributed schemes is important, as in various distributed settings the communication channel is the critical bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Moreover, in some practical cases, such as mobile devices and sensor networks, the end machines may have very limited bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Thus, in high dimensional settings with d ≫ 1, it may not even be feasible for each ma- chine to send messages of length O(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In this work, we study such a restricted communication setting, assuming that each machine is allowed to send to the fu- sion center only a limited number of bits, significantly lower than the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Our goals are to develop low communication distributed schemes to estimate θ∗ and its support and to theoretically analyze their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We make the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' On the methodology side, in Section 4 we present several two round distributed schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The schemes vary slightly by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 3 the messages sent, but in all of them, the fusion center estimates the support set of θ∗ in the first round and the vector θ∗ in the second round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our schemes, each machine computes its own debiased lasso estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' However, it sends to the center only the indices of its top few largest values, possibly along with their signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, the communication per machine is significantly less than d bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In the simplest variant, the fusion center estimates the support of θ∗ by voting, selecting the few indices that were sent by the largest number of machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, on the theoretical side, we prove in Section 5 that under suitable condi- tions, with high probability the first round of our scheme achieves exact support recovery with communication per machine sublinear in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Specifically, we present support guarantees under two different parameter regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Theorem 2 consid- ers a case with a relatively large number of machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Here, each machine sends a short message of O(K ln d) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, Theorem 3 considers a setting with relatively few machines, M = O(ln d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Here, to achieve exact support recovery each machine sends a much longer message, of length O(dα) for some suitable α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This is still sublinear in d, and much less than the communication re- quired if a machine were to send its full d-dimensional estimated vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The proofs of our theorems rely on recent results regarding the distribution of debi- ased lasso estimators, combined with sharp bounds on tails of binomial random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Exact support recovery follows by showing that with high probability, all non-support indices receive fewer votes than support indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Section 6 we present simulations comparing our schemes to previously proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' These illustrate that with our algorithms, the fusion center correctly detects the support of θ∗ and consequently accurately estimates θ∗, even at low signal to noise ratios where each machine is unable to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fur- thermore, this is achieved with very little communication per machine compared to the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' One insight from both the simulations and our theoretical analysis is that for the fusion center to detect the correct support, it is not necessary to require M/2 votes as suggested in Barghi, Najafi and Motahari (2021) and Chen and Xie (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Instead, as few as O(ln d) votes suffice to dis- tinguish support from non-support indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Interestingly, under a broad range of parameter values, our schemes work as well as, and in some cases better than more communication intensive approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Our simulations also highlight the importance and advantages of a second round of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Specifically, even though a single-round scheme based on averaging debiased lasso estimates, as proposed by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017), is minimax rate optimal and finds the correct support, it nonetheless may output an estimate with a larger mean squared er- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 4 ror than that of our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We conclude with a summary and discussion in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Proofs appear in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Notation For an integer k ≥ 1, we denote [k] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The indicator function is denoted as I(A), which equals one if condition A holds and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The ℓq norm of a vector Y ∈ Rn for q ≥ 1 is ∥Y ∥q = (�n i=1 |Yi|q)1/q, whereas ∥Y ∥0 = �n i=1 I(Yi ̸= 0) is its number of nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We denote by |Y | the vector whose entries are (|Y1|, |Y2|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , |Yn|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For a d × d matrix A = {aij}d i,j=1, we denote ∥A∥∞ = max1≤i≤d �d j=1 |aij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We further denote by σmin(A) and σmax(A) its smallest and largest singular values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For a subset J ⊂ [d], AJ is the d×|J| matrix whose columns are those in the subset J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Similarly, AJ,J is the |J|×|J| submatrix whose rows and columns correspond to the indices in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The cumulative distribution function (CDF) of a standard Gaussian is denoted by Φ(·) whereas Φc(·) = 1−Φ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We write an ≳ bn for two sequences {an}n≥1 and {bn}n≥1 if there are positive constants C and n0 such that an ≥ Cbn for all n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Previous works Distributed linear regression schemes under various settings, not necessarily involving sparsity, have been proposed and theoretically studied in multiple fields, including sensor networks, statistics and machine learning, see for example (Guestrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Predd, Kulkarni and Poor, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Zhang, Duchi and Wainwright, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Heinze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Rosenblatt and Nadler, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Jordan, Lee and Yang, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Dobriban and Sheng, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Zhu, Li and Wang, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Dobriban and Sheng, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Mateos, Bazerque and Giannakis (2010) were among the first to study dis- tributed sparse linear regression in a general setting without a fusion center, where machines are connected and communicate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' They devised a multi-round scheme whereby all the machines reach a consensus and jointly approximate the centralized solution, that would have been computed if all data were available at a single machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Several later works focused on the setting which we also consider in this paper, where machines are connected in a star topology to a fusion center, and only one or two communication rounds are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In a broader context of generalized sparse linear models, Chen and Xie (2014) proposed a divide-and-conquer approach where each machine estimates θ∗ by minimizing a penalized objective with a sparsity inducing penalty, such R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 5 as ∥θ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Each machine sends its sparse estimate to the fusion center, which es- timates the support by voting over the indices of the individual estimates of the M machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Finally, the center estimates θ∗ by a weighted average of these M estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For sparse linear regression, with each machine computing a lasso estimate of θ∗, their method suffers from the well known bias of the lasso, which is not reduced by averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To overcome the bias of the lasso, in recent years several debiased lasso es- timators were derived and theoretically studied, see Zhang and Zhang (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Javanmard and Montanari (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For distributed learning, debiased estimators have been applied in various settings, including hypothesis testing, quantile regression and more, see for example Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lv and Lian (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In particular, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017) proposed a single round scheme whereby each machine computes its own debiased lasso estimator, and sends it to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The center averages these debiased estimators and thresholds the result to estimate θ∗ and recover its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017) proved that the re- sulting estimator achieves the same error rate as the centralized solution, and is minimax rate optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' However, their scheme requires a communication of O(d) bits per machine and is thus not applicable in the restricted communica- tion setting considered in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Moreover, as we demonstrate in the simulation section, unless the signal strength is very low, our two round scheme in fact achieves a smaller mean squared error, with a much lower communica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This highlights the potential sub-optimality of lasso and debiased lasso in sparse regression problems with sufficiently strong signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Most related to our paper is the recent work by Barghi, Najafi and Motahari (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In their method, each machine computes a debiased lasso estimator ˆθ, but sends to the fusion center only the indices i for which |ˆθi| is above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The support set estimated by the fusion center consists of all indices that were sent by at least half of the machines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', indices that received at least M/2 votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Focusing on the consistency of feature selection, Barghi, Najafi and Motahari (2021) derive bounds on the type-I and type-II errors of the estimated support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Their results, however, are given as rates with unspecified multiplicative constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As we show in this work, both theoretically and empirically, consistent support estimation is possible with a much lower voting threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Furthermore, requiring at least M/2 votes implies that their scheme achieves exact support recovery only for much stronger signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We remark that voting is a natural approach for distributed support esti- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 6 mation under communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Amiraz, Krauthgamer and Nadler (2022) analyzed voting-based distributed schemes in the context of a simpler problem of sparse Gaussian mean estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' They proved that even at low signal strengths, their schemes achieve exact support recovery with high prob- ability using communication sublinear in the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Their setting can be viewed as a particular case of sparse linear regression but with a unitary design matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Their proofs, which rely on this property, do not extend to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The lasso and debiased lasso estimators For our paper to be self contained, we briefly review the lasso and debiased lasso and some of their theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The lasso (Tibshirani, 1996) is perhaps the most popular method to fit high-dimensional sparse linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Given a regularization parameter λ > 0 and n samples (Xi, yi), stacked in a design matrix X ∈ Rn×d and a response vector Y ∈ Rn, the lasso estimator is given by ˜θ = ˜θ(X, Y, λ) = arg min θ∈Rd � 1 2n∥Y − Xθ∥2 2 + λ∥θ∥1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2) The lasso has two desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' First, computationally Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2) is a convex problem for which there are fast solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Second, from a theoretical standpoint, it enjoys strong recovery guarantees, assuming the data follows the model (1) with an exact or approximately sparse θ∗, see for example (Candes and Tao, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Bunea, Tsybakov and Wegkamp, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' van de Geer and B¨uhlmann, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hastie, Tibshirani and Wainwright, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' However, the lasso has two major drawbacks: it may output significantly biased estimates and it does not have a simple asymptotic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The latter is needed for confidence intervals and hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To overcome these limitations, and in particular derive confidence intervals for high-dimensional sparse linear models, several authors developed debiased lasso estimators (Zhang and Zhang, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Javanmard and Montanari, 2014a,b, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For random X with a known population covariance matrix Σ, Javanmard and Montanari (2014a) proposed 1 nΣ−1X⊤(Y − X˜θ) as a debiasing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As Σ is often unknown, both van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014) and Javanmard and Montanari (2014b) developed methods to estimate its inverse Ω = Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our work, we estimate Ω using the approach of van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014), who assume that Ω is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In their method, presented in Algorithm 1, ˆΩ is constructed by fitting a lasso regression with regularization λΩ > 0 to each column of X against all the other columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, it requires solving d separate lasso problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 7 Given the lasso estimate ˜θ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2) and the matrix ˆΩ, the debiased lasso is ˆθ = ˆθ(Y, X, λ, λΩ) = ˜θ + 1 n ˆΩX⊤(Y − X˜θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (3) An appealing property of ˆθ is that, under some conditions, it is asymptotically unbiased with a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For our analysis, we shall use the follow- ing result (Javanmard and Montanari, 2018, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Consider the linear model Y = Xθ∗+W, where W ∼ N(0, σ2In×n) and X ∈ Rn×d has independent Gaussian rows with zero mean and covariance matrix Σ ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Suppose that Σ satisfies the following conditions: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For all i ∈ [d], Σii ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For some constants Cmax, Cmin > 0, 0 < Cmin < σmin(Σ) ≤ σmax(Σ) < Cmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (4) iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For C0 = (32Cmax/Cmin) + 1, and a constant ρ > 0, max J⊆[d], |J|≤C0K ∥Σ−1 J,J∥∞ ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let KΩ be the maximum row-wise sparsity of Ω = Σ−1, that is, KΩ = max i∈[d] |{j ∈ [d];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Ωij ̸= 0, j ̸= i}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let ˜θ be the lasso estimator computed using λ = κσ � (ln d)/n for κ ∈ [8, κmax], and let ˆθ be the debiased lasso estimator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (3) with ˆΩ computed by Algorithm 1 with λΩ = κΩ � (ln d)/n for some suitable large κΩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let ˆΣ = X⊤X/n de- note the empirical covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Then there exist constants c, c∗, C depend- ing solely on Cmin, Cmax, κmax and κΩ such that, for n ≥ c max{K, KΩ} ln d, the following holds: √n(ˆθ − θ∗) = Z + R, Z|X ∼ N(0, σ2 ˆΩˆΣˆΩ⊤), (5) where Z = n−1/2 ˆΩX⊤W and R = √n � ˆΩˆΣ − I � (θ∗ − ˜θ), and with probability at least 1 − 2de−c∗n/K − de−cn − 6d−2, ∥R∥∞ ≤ Cσ ln d √n � ρ √ K + min{K, KΩ} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (6) Assumptions (i) and (ii) in this theorem are common in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As- sumption (iii) is satisfied, for example, by circulant matrices Σij = ς|i−j|, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 8 Algorithm 1 Computation of a precision matrix estimate ˆΩ Input: design matrix X ∈ Rn×d, regularization parameter λΩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Output: precision matrix estimate ˆΩ ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' xi ∈ Rn denotes the i-th column of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' X−i ∈ Rn×(d−1) denotes the design matrix with the i-th column removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 1: for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , d do 2: Fit a lasso with response xi, design matrix X−i and regularization parameter λΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 3: Let ˜γi = {˜γi,j}d j=1,j̸=i ∈ Rd−1 be the estimated regression coefficients of step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 4: Compute ˜τ 2 i = (2n)−1∥xi − X−i˜γi∥2 2 + λΩ∥˜γi∥1, i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 5: end for 6: Construct d × d matrix ˜C = � ���� 1 −˜γ1,2 · · −˜γ1,d −˜γ2,1 1 · · −˜γ2,d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' −˜γd,1 −˜γd,2 · · 1 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 7: return ˆΩ = diag{˜τ −2 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , ˜τ −2 d } ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' ς ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The quantity R in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (5) can be viewed as a bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' By Theorem 1, this bias is small if the sample size and dimension are suitably large, which in turn implies that ˆθi is approximately Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The following lemma, proven in the Appendix, bounds the error of this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' It will be used in analyzing the probability of exact support recovery of our distributed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Under the assumptions of Theorem 1, for any τ > 0, ���Pr � √n(ˆθi−θ∗ i ) σ√cii ≤ τ � − Φ (τ) ��� ≤ δR σ√cii φ (τ) + 2de−c∗n/K + de−cn + 6 d2 , (7) where φ(·) denotes the Gaussian density function, cii = (ˆΩˆΣˆΩ⊤)ii, and δR is the upper bound on the bias term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (6), namely δR = Cσ ln d √n � ρ √ K + min{K, KΩ} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (8) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed sparse regression with restricted communication As described in Section 1, we consider a distributed setting with M machines connected in a star topology to a fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For simplicity, we assume that each machine m has a sample (Xm, Y m) of n = N/M i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' observations from the model (1), where Y m ∈ Rn and Xm ∈ Rn×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In describing our schemes, we further assume that the noise level σ is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If σ is unknown, it may R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 9 Algorithm 2 Distributed voting based scheme for support estimation Input: Data (Xm, Y m) ∈ Rn×(d+1), threshold τ and regularization parameters λΩ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Output: Support estimate ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' At each local machine m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , M 1: Compute a lasso estimator ˜θm via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2) with regularization parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 2: Compute a precision matrix estimate ˆΩm ∈ Rd×d by Algorithm 1 with Xm and λΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 3: Compute a debiased lasso estimate ˆθm ∈ Rd, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (3), with data (Xm, Y m), λ and ˆΩm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 4: Calculate the empirical covariance matrix ˆΣm = n−1(Xm)⊤Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 5: Use ˆΩm and ˆΣm to compute the standardized estimator ˆξm ∈ Rd, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 6: Set Sm = {i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' |ˆξm i | > τ} and send it to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' At the fusion center 7: For each i ∈ [d], compute Vi = �M m=1 I(i ∈ Sm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 8: Sort Vj1 ≥ Vj2 ≥ · · · ≥ Vjd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 9: return ˆS = {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , jK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' be consistently estimated, for example, by the scaled lasso of Sun and Zhang (2012), see also (Javanmard and Montanari, 2018, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We present several two round distributed schemes to estimate the sparse vector θ∗ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (1) under the constraint of limited communication between the M machines and the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Here we present the simplest scheme and discuss other variants in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In all variants, the fusion center estimates the support of θ∗ in the first round, and θ∗ itself in the second round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The first round of our scheme is described in Algorithm 2, whereas the full two round scheme is outlined in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In the first round, each machine m ∈ [M] computes the following quantities using its own data (Xm, Y m): (i) a lasso estimate ˜θm by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (ii) a matrix ˆΩm by Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and (iii) a debiased lasso ˆθm by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Up to this point, this is identical to Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The main difference is that in their scheme, each machine sends to the center its debiased lasso estimate ˆθm ∈ Rd, incurring O(d) bits of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In contrast, in our scheme each machine sends only a few indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Towards this end and in light of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (7) of Lemma 1, each machine computes a normalized vector ˆξm whose coordinates are given by ˆξm k = √nˆθm k σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2 kk , ∀k ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (9) In the simplest variant, each machine sends to the center only indices k such that |ˆξm k | > τ for some suitable threshold τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Given the messages sent by the M machines, the fusion center counts the number of votes received by each index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If the sparsity level K is known, its R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 10 Algorithm 3 Two round distributed scheme to estimate θ∗ Input: Data (Xm, Y m) ∈ Rn×(d+1), sparsity K, threshold τ and regularizations λΩ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Output: A two-round estimate ˆθ ∈ Rd of θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' First round 1: The fusion center estimates ˆS with Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Second round 2: The fusion center sends ˆS to all M machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' At each local machine m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , M 3: Let Xm ˆ S ∈ Rn×K be the K columns of Xm corresponding to indices in ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 4: Compute ˆβm = arg minβ ∥Xm ˆ S β − Y m∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 5: Send ˆβm to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' At the fusion center 6: Given ˆβ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , ˆβM, compute the estimate ˆθ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 7: return ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' estimated support set ˆS consists of the K indices with the largest number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Otherwise, as discussed in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='5 below, the center may estimate the support set by the indices whose number of votes exceed a suitable threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, we describe the second round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' At its start, the fusion center sends the estimated support ˆS to all M machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, each machine computes the standard least squares regression solution, restricted to the set ˆS, namely ˆβm = arg min β ∥Xm ˆ S β − Y m∥2 2 (10) where Xm ˆ S ∈ Rn×| ˆ S| consists of the columns of Xm corresponding to the indices in ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Each machine then sends its vector ˆβm to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Finally, the fusion center estimates θ∗ by averaging these M vectors, ˆθi = � 1 M �M m=1 ˆβm i i ∈ ˆS 0 otherwise (11) In the next section we present several variants of this basic two round scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Before that we make a few remarks and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The communication of the first round (Algorithm 2) depends on the threshold τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A high threshold leads to only few sent indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' However, at low signal strengths, the signal coordinates may not have the highest values |ˆξm k | and thus may not be sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, for successful support recovery by the fusion center, a lower threshold leading to many more sent coordinates is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since the maxima of d standard Gaussian variables scales as √ 2 ln d, to comply with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 11 the communication constraints, the threshold τ should also scale as O( √ ln d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Section 5, we present suitable thresholds and sufficient conditions on the number of machines and on the signal strength, which guarantee support recovery by Algorithm 2, with high probability and little communication per machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' With known K, the communication per machine of the second round is O(K ln d) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For suitable choices of the threshold τ in the first round, this is negligible or at most comparable to the communication of the first round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A two round scheme, whereby given an estimated set ˆS, the second round is identical to ours was discussed by Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2) of their supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The difference is that in their first round, similar to Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017), each machine sends its full debiased lasso vector, with a communication of O(d) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018) showed that, under certain conditions, their two-round estimator attains an optimal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Section 5, we prove that for a sufficiently high SNR, our method achieves the same rate, but using much less communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' With a higher communication per machine in the second round, it is possible for the fusion center to compute the exact centralized least squares solution corresponding to the set ˆS, denoted ˆθLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Specifically, suppose that each machine sends to the center both the vector (Xm ˆ S )⊤Y m of length | ˆS|, and the | ˆS| × | ˆS| matrix (Xm ˆ S )⊤Xm ˆ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The center may then compute ˆθLS as follows ˆθ LS = � M � m=1 (Xm ˆ S )⊤Xm ˆ S �−1 M � m=1 (Xm ˆ S )⊤Y m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (12) With K known and | ˆS| = K, such a second round has a communication of O(K2) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If the sparsity K is non-negligible, this is much higher than the O(K) bits of our original scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In particular, if K = O(d1/2), the resulting communication is comparable to that of sending the full debiased lasso vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In practice, the sparsity K is often unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Instead of step 9 in Algorithm 2, one alternative is to estimate S by thresholding the number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For some threshold τvotes > 0, ˆS could be set as all indices i such that Vi > τvotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lemma 3 in Appendix A shows that, under suitable conditions, non-support indices have a small probability of receiving more than 2 ln d votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, τvotes = 2 ln d is a reasonable choice for such a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Variations of Algorithm 2 Various adaptations of Algorithm 2 are possible and may offer better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' One example is a top L algorithm where each machine sends to the cen- ter the indices of the L largest entries of |ˆξm|, for some parameter K ≤ L ≪ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A similar approach was proposed in Amiraz, Krauthgamer and Nadler (2022) for the simpler problem of sparse normal means estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' One advantage of this variant is that its communication per machine is fixed and known a priori O(L ln d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This is in contrast to the above thresholding based scheme, whose communication per machine is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A different variant is to use sums of signs to estimate the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Here machines send both the indices corresponding to the largest entries in |ˆξm| and their signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, in step 5 of Algorithm 2 the message sent by machine m is Sm = �� i, sign(ˆξm i ) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' |ˆξm i | > τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, the fusion center computes for each index i ∈ [d] its corresponding sum of received signs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', V sign i = M � m=1 sign(ˆξm i )I �� i, sign(ˆξm i ) � ∈ Sm� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (13) For known K, the estimated support set are the K indices with largest values of |V sign i |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This algorithm uses a few more bits than a voting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' How- ever, sums of signs are expected to better distinguish between support and non-support coefficients when the number of machines is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The reason is that at non-support indices j ̸∈ S, the random variable V sign j has approximately zero mean, unlike sums of votes Vj, whereas at support indices |V sign i | ≈ Vi since support indices are unlikely to be sent to the fusion center with the opposite sign of θ∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In the simulation section we illustrate the improved performance of a sign-based over a votes-based distributed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Theoretical results In this section, we present a theoretical analysis for one of our schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Specif- ically, both Theorems 2 and 3 show that under suitable conditions, with high probability Algorithm 2 achieves exact support recovery with little communi- cation per machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Theorem 2, the number of machines is relatively large, and the communication per machine is linear in the sparsity K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Theorem 3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 13 the number of machines M is logarithmic in d, in which case the communica- tion per machine is much higher, though still sublinear in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Both theorems are based on the Gaussian approximation in Lemma 1 and on probability bounds for binomial random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Their proofs appear in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To put our theorems in context, let us briefly review previous results on exact support recovery in the simpler (non-distributed) sparse linear regression set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A key quantity characterizing the ability of exact support recovery is the signal strength, defined as θmin = mini∈S |θ∗ i |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As proven by Wainwright (2009), under suitable conditions on the design matrix, the lasso estimator based on n samples and an appropriately chosen regularization parameter λn, achieves exact support recovery with high probability, provided that θmin ≳ � (ln d)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The same rate θmin ≳ � (ln d)/n is also sufficient for support recovery using a debiased lasso estimator (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 of Javanmard and Montanari (2014b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In a distributed setting, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017) proved that with high proba- bility, their scheme achieves exact support recovery when θmin ≳ � (ln d)/(nM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' While this result matches the centralized setting, their scheme requires each ma- chine to send to the center its d-dimensional debiased lasso estimate, incurring O(d) communication per machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, an interesting range for the signal strength, for the study of support recovery under communication constraints, is � ln d nM ≲ θmin ≲ � ln d n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In this range, individual machines may be unable to exactly recover the support using the lasso or debiased lasso estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To derive support recovery guarantees, we assume the smallest nonzero co- efficient of θ∗ is sufficiently large, namely |θ∗ i | ≥ θmin for all i ∈ S and some suitable θmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For our analysis below, conditional on the design matrices X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , XM at the M machines, it will be convenient to make the following change of variables from θmin to the (data-dependent) SNR parameter r, θmin = θmin(d, σ, r, n, cΩ) = σ � 2cΩ n r ln d, (14) where cΩ is defined as cΩ = max i∈[d],m∈[M] � ˆΩm ˆΣm(ˆΩm)⊤� ii .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (15) Recall from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (9) that by Theorem 1, σ2 � n−1 ˆΩm ˆΣm(ˆΩm)⊤� ii is the asymp- totic variance of ˆθm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, σ2cΩ/n is the largest variance of all d coordinates of the M debiased estimators computed by the M machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In terms of the SNR parameter r, the range of interest is thus 1 M < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 14 Recall that our scheme is based on thresholding the normalized debiased lasso estimators ˆξm k of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We denote the corresponding normalized signal by ϑm k = √nθ∗ k σ � ˆΩm ˆΣm(ˆΩm)⊤ �1/2 kk , ∀k ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (16) Lemma 1 states that under suitable conditions, ˆξm k − ϑm k has approximately a standard Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This property plays an important role in our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (7) of Lemma 1 provides a bound on the error between the CDF of ˆξm k −ϑm k and that of a standard Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For a threshold τ, let ϵ(τ) be the largest of these error bounds over all d coordinates in all M machines, ϵ(τ) = max k∈[d],m∈[M] � � � � � δRφ (τ − ϑm k ) σ � ˆΩm ˆΣm(ˆΩm)⊤ �1/2 kk + 2de−c∗n/K + de−cn + 6 d2 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (17) Recall that δR, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (8), is an upper bound on the bias ˆθm k − θ∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 of van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014), if the row sparsity of Ω satisfies KΩ = o(n/ ln d) and ˆΩm is computed with regularization λΩ ∝ � (ln d)/n, then � ˆΩm ˆΣm(ˆΩm)⊤� kk ≥ Ωkk + oP (1) ≥ C−1 max + oP (1) when ln d n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, when n and d are large, all terms on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (17) are small, and the Gaussian approximation is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To prove that our scheme recovers S with high probability, we assume that: (C1) The n samples in each of the M machines are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' from the model (1) and the conditions of Theorem 1 all hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Additionally, all machines use the same regularization parameters λ and λΩ to compute the lasso (2) and debiased lasso (3) estimators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (C2) |θ∗ i | ≥ θmin(d, σ, r, n, cΩ) for all i ∈ S, where θmin and cΩ are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (14) and (15), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The following theorem provides a recovery guarantee for Algorithm 2, where the sparsity K is assumed to be known to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Suppose Algorithm 2 is run with threshold τ = √ 2 ln d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Assume that d is sufficiently large and that the SNR in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (14) satisfies 1 4 ln2(48√π ln3/2 d) ln2(d) < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Additionally, assume conditions C1 and C2 hold and the approximation error R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 15 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (17) satisfies ϵ(τ) ≤ 1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Then, if the number of machines satisfies 8 ln d � 2(1−√r) 2 ln d √ 2π � 2(1−√r) 2 ln d+1 � − ϵ(τ)d(1−√r) 2 d(1−√r) 2 ≤ M ≤ d 3, (18) with probability at least 1 − K+1 d , Algorithm 2 achieves exact support recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Additionally, the expected communication per machine is O (K ln d) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let us make a few remarks regarding this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The upper bound M < d/3 is rather artificial and stems from the fact that in our proof we assume M < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' It is possible to derive support guarantees also for the case M > d, though this setting seems to be unlikely in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The lower bound on the number of machines is required to guarantee that with high probability, all support indices receive more votes than any non-support coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The lower bound on the SNR r ensures that the lower bound on the number of machines in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (18) is indeed smaller than d/3, so the range of possible values for M is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A similar lower bound on r appeared in Amiraz, Krauthgamer and Nadler (2022) after their Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Another important remark is that the threshold τ = √ 2 ln d in Theorem 2 is relatively high, so each machine sends only few indices to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' How- ever, to guarantee support recovery, this requires a relatively large number of machines M = polylog(d) · d(1−√r)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Theorem 3, we give sufficient conditions to still achieve a high probability of exact support recovery when the number of machines is much smaller, of order only logarithmic in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The price to pay is a higher communication per machine, which nonetheless is still sub-linear in d, namely much lower than the communication required to send the whole debi- ased lasso vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For the next theorem, we assume that a lower bound on the SNR is known to all machines, which set a threshold that depends on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Suppose Algorithm 2 is run with threshold τ = √ 2r ln d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Assume that d is sufficiently large and that the SNR in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (14) satisfies ln(16 ln d) ln d < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Additionally, assume conditions C1 and C2 hold and the approximation error in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (17) satisfies ϵ(τ) < 1/(4dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If the number of machines satisfies 16 ln d 1 − 2ϵ(τ) ≤ M ≤ dr, (19) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 16 then with probability at least 1 − K+1 d , Algorithm 2 achieves exact support re- covery, with expected communication per machine O � d1−r ln d � bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Beyond support recovery, another quantity of interest is the accuracy of the distributed estimator ˆθ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The following corollary, proven in Appendix A, shows that once S is precisely recovered, ˆθ is close to the oracle least squares estimator ˆθLS computed with all data in a single machine and with knowledge of the true support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Consequently, ˆθ is also close to the true vector θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Assume the conditions of Theorem 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let N = nM denote the total sample size in all M machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If M = O � NK (max{K,ln N})2 � , then ∥ˆθ − ˆθ LS∥2 = OP �√ M max{K, ln N} N � and ∥ˆθ − θ∗∥2 = OP �� K N � , as d, N → ∞ and ln d N/M → 0, where ˆθ is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (11) and ˆθLS is the least squares solution using all N samples and with a known S, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (12), appended by zeros at all coordinates j /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Corollary 1 shows that in a high dimensional sparse setting, for a sufficiently strong signal, Algorithm 3 with a threshold τ = √ 2 ln d achieves the same error rate as the oracle estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let us put this result in a broader context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If the support S were known, then each machine could have computed its least squares solution restricted to S and send it to the center for averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As discussed in Rosenblatt and Nadler (2016), in a general setting of M-estimators, if the number of machines is not too large, averaging is optimal and to leading order coincides with the centralized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Yet, while being rate optimal, we note that averaging does lead to a loss of accuracy and is not as efficient as the oracle estimator, see Dobriban and Sheng (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As mentioned in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='3, Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018) also proposed a two-round estimator that attains the optimal rate in Corollary 1, but requires each machine to send at least d values to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In contrast, ˆθ is computed using a much lower communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Similar results can also be established for Algorithm 3 under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Comparison to other works Theorems 2 and 3 can be viewed as analogous to Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='A and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='B of Amiraz, Krauthgamer and Nadler (2022), who studied distributed estimation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 17 of a sparse vector under communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A key difference is that in their setting, the d coordinates of the estimated vector at each machine are all independent and unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This allowed them to analyze a top-L scheme since the probability of sending any non-support index was the same and could be easily bounded by L/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As such, their proofs do not directly apply to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our case, the debiased lasso ˆθm still has a small bias term and its d coordinates are in general correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This implies that the probabilities of sending non-support indices are not all identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our analysis, we bypass this issue by analyzing instead a thresholding approach at step 5 of Algorithm 2, where each ˆθm i for i ∈ [d] is compared separately to a fixed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This way, we do not need to account for the complex dependence among different coordinates of the debiased lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Barghi, Najafi and Motahari (2021) considered a similar distributed scheme to estimate the support of θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' They did not normalize the debiased lasso estimate at each machine, and more importantly the estimated support set consists only of those indices that received at least M/2 votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The authors performed a theoretical analysis of their scheme, though various quantities are described only up to unspecified multiplicative constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We remark that both theoretically as well as empirically, the SNR must be quite high for support indices to obtain at least M/2 votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our work, we present explicit expressions for the minimum SNR in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (14), sufficient for exact support recovery, requiring far fewer votes at the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The simulations in the next section illustrate the advantages of our scheme as compared to that of Barghi, Najafi and Motahari (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Simulations We present simulations that illustrate the performance of our proposed methods in comparison to other distributed schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We focus on methods based on debiased lasso estimates, and specifically consider the following five distributed schemes to estimate θ∗ and its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' thresh-votes: Algorithm 2 with a threshold of τ = √ 2 ln d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' top-L-votes: The top L algorithm presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Each machine sends the indices of its top L values of |ˆξm i | to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' top-L-signs: Each machine sends both the indices and signs of the top L values of |ˆξm i |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The center forms ˆS using sums of signs as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' BNM21: The algorithm proposed by Barghi, Najafi and Motahari (2021) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 18 with a threshold τ = √ 2 ln d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' It is similar to Algorithm 2, the difference is that ˆS consists of the indices i ∈ [d] with Vi ≥ M/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' AvgDebLasso: Based on Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017), each machine sends its debiased lasso estimate ˆθm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The center computes ˆθavg = 1 M �M m=1 ˆθm and estimates the support as the indices with the K largest values |ˆθavg i |, i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In all these algorithms, each machine computes a debiased lasso estimator using its own data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The methods differ by the content and length of the messages sent to the fusion center and hence by the manner in which the fusion center estimates S and θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For a fair comparison, we run all methods with the same regularization parameters λΩ = 2 � (ln d)/n and λ = 8 � (ln d)/n in each machine to compute the precision matrix ˆΩm and the lasso estimator ˜θm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We performed simulations with both known sparsity, where the methods were run as described above, as well as unknown sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In the latter case, for thresh-votes, top-L-votes and top-L-signs, the center computes ˆS as the indices i such that Vi or |V sign i | is larger than 2 ln d, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='5 in Section 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For AvgDebLasso, we set ˆS as the indices i such that |ˆθavg i | > 11 ln d n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This scaling is motivated by Theorem 16 of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017), whereby in our simulation setting this term is larger than the term O �� ln d/(nM) � in their bound for ∥ˆθavg − θ∗∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The factor 11 was manually tuned for good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' BNM21 is unchanged, as it does not require knowledge of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We evaluate the accuracy of an estimated support set ˆS by the F-measure, F-measure = 2 · precision · recall precision + recall , where precision = |S ∩ ˆS|/| ˆS| and recall = |S ∩ ˆS|/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' An F-measure equal to one indicates that exact support recovery was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Given a support estimate ˆS, the vector θ∗ is estimated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For all four methods excluding AvgDebLasso, we perform a second round and compute the estimator ˆθ given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' AvgDebLasso is a single round scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Its estimate of θ∗ consists of ˆθavg restricted to the indices i ∈ ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The error of an estimate ˆθ is measured by its ℓ2-norm ∥ˆθ − θ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As a benchmark for the achievable accuracy, we also computed the oracle centralized estimator ˆθLS that knows the support S and estimates θ∗ by least squares on the whole data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We generated data as follows: The design matrix Xm ∈ Rn×d in machine m has n rows i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' N(0, Σ), with Σi,j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='5|i−j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We then computed for each machine its matrix ˆΩm, and the quantity cΩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, we generated a K sparse vector θ∗ ∈ Rd, whose nonzero indices are sampled uniformly at random R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 19 (a) (b) Fig 1: Results for known sparsity, averaged over 500 realizations, as a function of SNR in the range r ∈ [1/M, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (a) F-measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (b) ℓ2 error, on a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' from [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Its nonzero coefficients have random ±1 signs and their magnitudes are chosen from K equally spaced values {θmin, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' , 2θmin}, where θmin = � 2cΩ ln d n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' These matrices and the vector θ∗ are then kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For a simulation with SNR parameter r, we set σ = 1/√r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Finally, in each realization we generated the response Y m ∈ Rn according to the model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Our first simulation compared the performance of various schemes as a func- tion of the SNR, with a known sparsity K = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We fixed the dimension d = 5000, the sample size in each machine n = 250, and the number of machines M = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The top L method was run with L = K (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Figure 1a displays the F-measure of each method, averaged over 500 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As expected, at low SNR values, AvgDebLasso achieved the best performance in terms of support recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' However, for stronger signals with r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4, both top-K-votes and thresh-votes achieved an F-measure of one, in accordance with our theoret- ical results regarding exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In particular, at sufficiently high SNR, our methods estimate the support as accurately as AvgDebLasso, but with 2-3 orders of magnitude less communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The scheme of BNM21 achieves good n=250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='d=5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='K=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='M=100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 measure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 r AvgDebLasso top-K-votes BNM21 oracle thresh-votesn=250,d=5000,K=5,M=100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='25 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='75 ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 r AvgDebLasso top-K-votes BNM21 oracle thresh-votesR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 20 (a) (b) Fig 2: Results for unknown sparsity, averaged over 500 realizations, as a function of SNR in the range r ∈ [1/M, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (a) F-measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (b) ℓ2 error on a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' performance only at higher SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Figure 1b shows the errors ∥ˆθ − θ∗∥2, averaged over 500 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' At low SNR, 1/M < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1, AvgDebLasso has the smallest error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' However, for r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4, thresh-votes and top-K-votes yield more accurate estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Thus, Figure 1b shows the benefits of an accurate support estimate followed by a distributed least squares in a second round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Indeed, at these SNR levels, our methods exactly recover the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Consequently, the second round reduces to a distributed ordinary least squares restricted to the correct support set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In accordance with Corollary 1, Algorithm 2 then has the same error rate as the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next we present simulation results for unknown sparsity, as a function of the SNR in the range r ∈ [ 1 M , 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As seen in Figure 2, throughout this SNR range, AvgDebLasso with a threshold of 11(ln d)/n achieves an F-measure close to one and ℓ2 errors close to those of the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In contrast, thresh-votes achieves accurate estimates only for r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' These results illustrate that even when sparsity is unknown, our schemes can accurately estimate the vector θ∗ and its support, albeit with a higher SNR as compared to the case of known sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Finally, we present simulation results as a function of number of samples n ∈ [nmin, nmax] = [100, 400] with M = 100 machines, and as a function of number of machines M ∈ [Mmin, Mmax] = [40, 160] with n = 250 samples n=250,d=5000,K=5,M=100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 measure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 r AvgDebLasso thresh-votes BNM21 oraclen=250,d=5000,K=5,M=100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='25 error) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='75 log1o(l2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 r AvgDebLasso thresh-votes BNM21 oracleR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 21 (a) (b) Fig 3: ℓ2 error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' sample size n (a) and vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' number of machines M (b), both on a log-log scale, at an SNR of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Values are averaged over 1000 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' per machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Initially, for each machine m ∈ [Mmax], we generated its full data matrix Xm with nmax number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We then computed the corresponding matrix ˆΩm and the quantity cΩ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We then generated the sparse vector θ∗ as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To save on run-time, in simulations with a smaller number of samples n < nmax we nonetheless used the decorrelation matrices Ωm that correspond to nmax samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This can be viewed as a semi-supervised setting, as also mentioned in Javanmard and Montanari (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We fixed r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 and d = 5000, and the sparsity K = 5 was known to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Figure 3 shows the resulting ℓ2 errors averaged over 1000 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In this simulation, top-K-votes and thresh-votes are close to the centralized least squares oracle since their support estimates are accurate, as can be seen in Figure 1 for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Both plots in Figure 3 show a linear dependence on a log-log scale with a slope of approximately −1/2, namely that the resulting errors decay as 1/√n and 1/ √ M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This is in agreement with our theoretical result in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The advantages of sending signs We now illustrate the advantages of using sums of signs instead of sums of votes, in terms of both support recovery and parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In this simulation, we fixed n = 250, d = 5000, M = 100 and K = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The results in Figure 4 show that using sums of signs is more accurate than sums of votes for low d=5000,K=5,r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='40,M=100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 log1o(l2 error) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 log1o(n) AvgDebLasso top-K-votes thresh-votes oracled=5000,K=5,r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='40,n=250 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 log1o(l2 error) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 log1o(M) AvgDebLasso top-K-votes thresh-votes oracleR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 22 (a) (b) Fig 4: Results for schemes using sums of signs and sums of votes, averaged over 500 realizations, as a function of r∈[ 1 M , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (a) F-measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (b) log10 of ℓ2 error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' SNR values in the range r ∈ [1/M, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Figure 4 also shows that using L = 5K instead of L = K significantly improves the accuracy of the support estimator, at the expense of increasing the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This illustrates the potential trade-offs between the accuracy of support estimation and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Summary and Discussion The development and analysis of distributed statistical inference schemes having low communication are important contemporary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Given its simplicity and ubiquity, the sparse linear regression model has attracted significant atten- tion in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Most previous inference schemes for this model require communication per machine of at least O(d) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In this work we proved the- oretically and showed via simulations that, under suitable conditions, accurate distributed inference for sparse linear regression is possible with a much lower communication per machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Over the past years, several authors studied distributed statistical infer- ence under communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Specifically, for sparse linear regression, Braverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2016) proved that without a lower bound on the SNR, to ob- n=250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='d=5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='K=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='M=100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 measure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='10 r AvgDebLasso top-5K-votes thresh-votes top-5K-signs top-K-votes oracle top-K-signsn=250,d=5000,K=25,M=100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 error) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2 )0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='10 r AvgDebLasso top-5K-votes thresh-votes top-5K-signs top-K-votes oracle top-K-signsR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 23 tain a risk comparable to that of the minimax lower bound, a communication of at least Ω(M min(n, d)/ log d) bits is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2019) proved that, under certain conditions, rate optimal estimates of a linear regression model can be computed using total communication sublinear in the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' However, as they mention in their appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='3, a precise characterization of the ability to recover the support with sublinear communication in d and its dependency on other parameters such as SNR and the number of machines is still an open prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our theoretical results, we presented explicit expressions for the minimal SNR at which our scheme is guaranteed to achieve exact recovery with high probability and with sublinear communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' While we did not address the open problem of tight lower bounds, our results highlight the potential tradeoffs between SNR, communication and number of machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We believe that using more refined techniques, our theoretical analysis can be extended and improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For example, since the d coordinates of a debiased lasso estimator are correlated, sharp concentration bounds for dependent variables, like those of Lopes and Yao (2022), could improve our analysis and extend it to other schemes such as top-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In our analysis, we focused on a setting where both the noise and covariates have Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017) and Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018), for example, considered sub-Gaussian distributions for these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Our results can be adapted for this case, but a careful control of the various constants in probability bounds is needed to derive explicit expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Finally, our low-communication schemes could also be applied to other prob- lems, such as sparse M-estimators, sparse covariance estimation and distributed estimation of jointly sparse signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We leave these for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Appendix A: Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Proof of Lemma 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Consider the debiased lasso estimator ˆθi given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Making the change of variables t = σ√ciiτ/√n, gives that Pr �√n(ˆθi − θ∗ i ) σ√cii ≤ τ � = Pr(ˆθi − θ∗ i ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (20) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 24 It follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (5) that Pr(ˆθi − θ∗ i ≤ t) = Pr (Zi ≤ √nt − Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' By the law of total probability, Pr � Zi ≤ √nt − Ri � = Pr � {Zi ≤ √nt − Ri} ∩ {|Ri| ≤ δR} � + Pr � {Zi ≤ √nt − Ri} ∩ {|Ri| > δR} � ≤ Pr � Zi ≤ √nt + δR � + Pr (|Ri| > δR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (21) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (5) it follows that Pr (Zi ≤ √nt + δR) = Φ � √nt+δR σ√cii � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (20) and (21) we get that Pr �√n(ˆθi − θ∗ i ) σ√cii ≤ τ � ≤ Φ �√nt + δR σ√cii � + Pr (|Ri| > δR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (22) The second term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (22) can be bounded by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This gives the last three terms on the right hand side in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let us analyze Φ � √nt+δR σ√cii � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For any fixed x and δ > 0, by the mean value theorem, |Φ(x + δ) − Φ(x)| ≤ δφ(x∗), where x∗ ∈ (x, x + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since φ(x) is a decreasing function for x > 0, we have φ(x∗) ≤ φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Thus, |Φ(x + δ) − Φ(x)| ≤ δφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Applying this result with x = √nt σ√cii and δ = δR σ√cii , gives ����Φ �√nt + δR σ√cii � − Φ � √nt σ√cii ����� ≤ δR σ√cii φ � √nt σ√cii � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Combining the above with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (22), and replacing t = σ√ciiτ √n proves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Proofs of Theorem 2 and Theorem 3 Let us first provide an overview of the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Recall that we consider a dis- tributed setting with M machines each with its own data, and each machine sends an independent message containing a few indices to the fusion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For any index i ∈ [d] and machine m, let pm i denote the probability that index i is sent by machine m, namely that |ˆξm i | > τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since data at different machines are statistically independent, the total number of votes Vi received at the fusion center for index i is distributed as Vi ∼ �M i=1 Ber(pm i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Our proof strategy is as follows: we compute an upper bound for pm j for non-support indices j ̸∈ S, and a lower bound for pm i for support indices i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, we employ tail bounds for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 25 binomial random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Combining these implies that under suitable condi- tions on the SNR and number of machines, with high probability, the number of votes Vi can perfectly distinguish between support and non-support indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To carry out this proof outline, we now introduce some auxiliary lemmas and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The following are standard Gaussian tail bounds, t √ 2π(t2 + 1)e−t2/2 ≤ 1 − Φ(t) ≤ 1 √ 2πte−t2/2, ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (23) We also use the following inequality for a binomial variable V ∼ Bin(M, p) (Boucheron, Lugosi and Massart, 2013, exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For any 0 < p ≤ a < 1, Pr (V > Ma) ≤ ��p a �a �1 − p 1 − a �1−a�M = eMF (a,p) (24) where F(a, p) = a ln �p a � + (1 − a) ln �1 − p 1 − a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (25) The following result appeared in (Amiraz, Krauthgamer and Nadler, 2022, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' It is used in our proof to show that with high probability, support indices receive a relatively large number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Assume that mini∈S |θ∗ i | is sufficiently large so that for some suitable pmin > 0, for all i ∈ S, and m ∈ [M], pm i ≥ pmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If pmin ≥ 8 ln d M , then Pr � min i∈S Vi < 4 ln d � ≤ K d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The next lemma shows that under suitable conditions, non-support indices receive relatively few total number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Assume that d ≥ 4 and M > 2 ln d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In addition, assume that pm j ≤ 1 M for all non-support indices j ̸∈ S and all machines m ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Then Pr � max j̸∈S Vj > 2 ln d � ≤ 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Recall that the number of votes received by an index j ̸∈ S at the fusion center is distributed as Vj ∼ �M m=1 Ber(M, pm j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since pm j ≤ 1 M for all j ̸∈ S, then Vj is stochastically dominated by V ∼ Bin(M, p), where p = 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (27) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 26 Thus, by a union bound, Pr � max j̸∈S Vj > t � ≤ (d − K) · Pr(V > t) ≤ d · Pr � V > M t M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We now apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (24) with t ≥ 1, so that the value of a = t/M = tp indeed satisfies that a ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' With F(a, p) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (25), this gives Pr � max j̸∈S Vj > t � ≤ d · eMF (tp,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (28) Next, we upper bound F(tp, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since ln(1 + x) ≤ x holds for all x ≥ 0, F(tp, p) = −tp ln(t) + (1 − tp) ln � 1 + tp − p 1 − tp � ≤ −tp ln(t) + tp − p < −tp ln(t) + tp = −tp ln(t/e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Inserting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (28) with t = 2 ln d, p = 1/M and M > 2 ln d gives Pr � max j̸∈S Vj > 2 ln d � ≤ deMF (tp,p) ≤ de−2 ln(d) ln(2 ln(d)/e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (26) to hold, we thus require that 2 ln(d) ln �2 ln d e � ≥ ln � d2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (29) This holds for d ≥ exp{e/2} ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Our goal is to show that the event { ˆS = S} occurs with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Recall that ˆS is determined at the fusion center as the K indices with the largest number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Further recall that pm i denotes the probability that machine m sends index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Our strategy is to show that these probabilities are sufficiently large for support indices and sufficiently small for non-support indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This shall allow us to apply Lemmas 2 and 3 to prove the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' To derive bounds on pm i we employ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' First, we prove that the condition of Lemma 2 holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' that pm i ≥ 8 ln d M for all i ∈ S, (30) where pm i = Pr(|ˆξm i | > τ), and ˆξm i = √nˆθm i σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2 ii is the standardized debiased lasso estimator, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Without loss of generality, assume that θ∗ i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Otherwise, we could do the same calculations for −ˆξm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Clearly, pm i = Pr(|ˆξm i | > τ) ≥ Pr(ˆξm i > τ) = Pr � √n ˆθm i − θ∗ i σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2 ii > τ − ϑm i � , R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 27 where ϑm i is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since condition C1 holds, applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (7) of Lemma 1 gives that pm i ≥ Φc(τ − ϑm i ) − ϵ(τ), (31) where ϵ(τ) is the error defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, by condition C2, with the definition of cΩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (15) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (14) ϑm i = √nθ∗ i � ˆΩm ˆΣm(ˆΩm)⊤ �1/2 ii ≥ √nθmin √cΩ = √ 2r ln d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' At a threshold τ = √ 2 ln d we thus obtain pm i ≥ Φc � (1 − √r) √ 2 ln d � − ϵ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' By the Gaussian tail bound (23), Φc �� 2(1 − √r)2 ln d � ≥ C(r, d)d−(1−√r)2, where C(r, d) = � 2(1 − √r)2 ln(d) √ 2π{2(1 − √r)2 ln(d) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Therefore, for the condition (30) to hold, it suffices that M ≥ 8 ln d C(r, d)d−(1−√r)2 − ϵ(τ) = 8 ln d C(r, d) − ϵ(τ)d(1−√r)2 d(1−√r)2, which is precisely condition (18) of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Notice that the requirement ϵ(τ) < 1/d guarantees that the denominator in the fraction above is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The lower bound on r in the theorem, guarantees that the range of possible values for M is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, we prove that the conditions of Lemma 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The condition M > 2 ln d is satisfied given the requirement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The next condition to verify is pm j ≤ 1/M for all j ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since pm j = Pr(|ˆξm j | > τ) and ϑm j = 0 for j ̸∈ S, then pm j − 2Φc(τ) = � Pr(ˆξm j > τ) − Φc(τ) � + � Pr(ˆξm j < −τ) − Φc(τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (5) of Theorem 1, apart from a bias term, ξm j and −ξm j have the same distribution because ϑm j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (7) in Lemma 1 to each of the above bracketed terms separately gives that pm j ≤ 2Φc(τ) + 2ϵ(τ), R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 28 where ϵ(τ) was defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (23), for τ = √ 2 ln d we have 2Φc(τ) ≤ 2 √ 2π √ 2 ln d 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since ϵ(τ) < 1/d, we obtain that for all j ̸∈ S pm j ≤ 1 √ π ln d 1 d + 2 d ≤ 3 d ≤ 1 M , (32) where the last inequality follows from the assumption that M ≤ d/3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Therefore, all conditions of Lemma 3 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since Lemmas 2 and 3 hold, we apply their results with a union bound to derive a lower bound on the probability of exact support recovery, as follows: Pr( ˆS = S) ≥ Pr �� max j̸∈S Vj ≤ 4 ln d � ∩ � min i∈S Vi ≥ 4 ln d �� ≥ 1 − Pr � max j̸∈S Vj ≥ 4 ln d � − Pr � min i∈S Vi ≤ 4 ln d � ≥ 1 − (K + 1) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We remark that Lemma 3 provides a bound on Pr � maxj /∈S Vj > 2 ln d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Clearly, the probability for a higher threshold 4 ln d above is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Finally, we analyze the communication per machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let Bm denote the num- ber of bits sent by machine m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Note that Bm is a sum of Bernoulli random variables Bm k ∼ Ber(pm k ) times some factor ∝ ln d corresponding to the num- ber of bits necessary to represent indices in [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The random variable Bm k is an indicator whether machine m sends index k to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Then E(Bm) = O � d � k=1 E(Bm k ) ln d � = O � � � � � � i∈S pm i + � j̸∈S pm j � � � ln d � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (33) Since pm j ≤ 3/d for all j ̸∈ S, then � j̸∈S pm j ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Additionally, � i∈S pm i ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Therefore, E(Bm) = O (K ln d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The proof is similar to that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We first show that the conditions of Lemmas 2 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Then, we derive a lower bound for the probability of Algorithm 2 achieving exact support recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Recall that here the threshold is τ = √ 2r ln d, where r is the SNR introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let us start by proving that the condition of Lemma 2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We need to show that pm i ≥ 8 ln d M for all i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (31) in the proof of Theorem 2, pm i ≥ Φc(τ − ϑm i ) − ϵ(τ) where ϑm i is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 29 By condition C2, ϑm i ≥ √nθmin/√cΩ = √ 2r ln d, where θmin and cΩ are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (14) and (15), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since τ = √ 2r ln d, then τ − ϑm i ≤ 0 and pm i ≥ Φc(0) − ϵ(τ) ≥ 1 2 − ϵ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The inequality pm i ≥ 8 ln d M follows directly from the requirement on M in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (19), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', M ≥ 8 ln d 1/2−ϵ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Next, we verify the conditions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Its first condition that M > 4 ln d holds given Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For the second condition, we need to show that pm j ≤ 1/M for all j ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As in the proof of Theorem 2, pm j ≤ 2Φc(τ) + 2ϵ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Plugging the value τ = √ 2r ln d into the Gaussian tail bound (23) gives pm j ≤ 1 √ πr ln d 1 dr + 2ϵ(τ) (i) ≤ 1 dr , (34) where inequality (i) follows from the assumptions that ϵ(τ) < 1/(4dr) and r > ln(16 ln d)/(ln d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' This latter assumption implies that 1 √ πr ln(d) ≤ 1 2 for sufficienly large d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since we assume that M ≤ dr in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (19), then pm j ≤ 1/dr ≤ 1/M for all j ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, both conditions of Lemma 3 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Applying a union bound and the result of Lemmas 2 and 3, it follows that Pr( ˆS = S) ≥ Pr �� max j̸∈S Vj ≤ 4 ln d � ∩ � min i∈S Vi ≥ 4 ln d �� ≥ 1 − (K + 1) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Finally, let us analyze the average communication per machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Let B de- note the number of bits sent by a single machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Following the same steps used to compute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (33), the expectation of B may be bounded as E(B) ≤ O �� K + d−K dr � ln d � , where the factor 1/dr is due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hence, the ex- pected communication of a single machine is O � d1−r ln d � bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Proof of Corollary 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' We proceed similar to Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018) in the proof of their Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' By the law of total probability, for any constant C′ > 0, Pr � ∥ˆθ − ˆθ LS∥2 > C′ √ M max{K, ln N} N � ≤ Pr �� ∥ˆθ − ˆθ LS∥2 > C′ √ M max{K, ln N} N � ∩ { ˆS = S} � + Pr( ˆS ̸= S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 30 By Theorem 2, Pr( ˆS ̸= S) ≤ (K + 1)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In addition, when ˆS = S, then ˆθj = ˆθLS j = 0 for all j ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Consequently, ∥ˆθ − ˆθLS∥2 = ∥ˆθS − ˆθLS S ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Furthermore, the first term on the right hand side above may be bounded by Pr � ∥ˆθS − ˆθ LS S ∥2 > C′ √ M max{K, ln N} N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The next step is to apply a result from the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='1 of Battey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018), which appeared in the last line of their proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' For clarity we state it here as a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Consider the linear model in dimension K, y = X⊤β∗ + σw, where w ∼ N(0, 1), X ∼ N(0, Σ), and Σ ∈ RK×K satisfies 0 < Cmin ≤ σmin(Σ) ≤ σmax(Σ) ≤ Cmax < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Suppose N i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' samples from this model are uniformly distributed to M machines, with n > K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Denote by ˆβm the least squares solution at the m-th machine and ˆβLS the centralized least squares solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' If the number of machines satisfies M = O � NK (max{K,ln N})2 � , then Pr ���� 1 M � m ˆβm − ˆβ LS��� 2 > C′ √ M max{K,ln N} N � ≤ cMe− max{K ln N} + Me−c N M , where c, C′ > 0 are constants that do not depend on K or N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Applying this lemma to our case gives Pr � ∥ˆθS − ˆθ LS S ∥2 > C′ √ M max{K, ln N} N � ≤ cMe− max{K ln N} + Me−c N M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Since M = O � NK (max{K,ln N})2 � , it follows that ∥ˆθ − ˆθLS∥2 = OP �� K N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' As the oracle estimator has rate ∥ˆθLS − θ∗∥2 = OP �� K N � , by the triangle inequality ∥ˆθ − θ∗∥2 = OP �� K N � as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' References Acharya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', De Sa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Foster, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Sridharan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed Learning with Sublinear Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' arXiv preprint arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='11259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 31 Amiraz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Krauthgamer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Nadler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed sparse normal means estimation with sublinear communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Information and In- ference: A Journal of the IMA iaab030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Barghi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Najafi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Motahari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed sparse feature selection in communication-restricted networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='02802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Battey, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed testing and estimation under sparse high dimensional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Annals of Statistics 46 1352–1382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Boucheron, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Lugosi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Massart, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Concentration Inequal- ities: A Nonasymptotic Theory of Independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Oxford University Press, Ox- ford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Boyd, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Parikh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Chu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Peleato, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Eckstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed optimization and statistical learning via the alternating direction method of multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Foundations and Trends in Machine learning 3 1–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Braverman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Garg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Nguyen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Woodruff, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Communication lower bounds for statistical estimation problems via a distributed data processing inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing 1011–1020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Bunea, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Tsybakov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Wegkamp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Sparsity oracle inequal- ities for the Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Electronic Journal of Statistics 1 169–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Candes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Tao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Decoding by linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' IEEE Transactions on Information Theory 51 4203–4215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Xie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A split-and-conquer approach for analysis of extraordinarily large data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Statistica Sinica 24 1655–1684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Mao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed high- dimensional regression under a quantile loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of Machine Learning Research 21 1–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Dobriban, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Sheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' WONDER: Weighted one-shot dis- tributed ridge regression in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of Machine Learning Research 21 1–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Dobriban, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Sheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed linear regression by averag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Annals of Statistics 49 918–943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Zou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Statistical Foundations of Data Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' CRC Press, Boca Raton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A review of distributed statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Statistical Theory and Related R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 32 Fields 6 89–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Guestrin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Bodik, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Thibaux, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Paskin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Madden, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed regression: an efficient framework for modeling sensor network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' In Third International Symposium on Information Processing in Sensor Networks 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hastie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Wainwright, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Statistical Learning with Sparsity: The Lasso and Generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' CRC Press, Boca Raton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Heinze, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', McWilliams, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Meinshausen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Krummenacher, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' LOCO: Distributing ridge regression with random projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' arXiv preprint arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='3469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Huo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Cao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Aggregated inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Wiley Interdisciplinary Reviews: Computational Statistics 11 e1451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Javanmard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Montanari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Hypothesis testing in high- dimensional regression under the Gaussian random design model: Asymptotic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' IEEE Transactions on Information Theory 60 6522–6554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Javanmard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Montanari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Confidence intervals and hy- pothesis testing for high-dimensional regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The Journal of Machine Learning Research 15 2869–2909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Javanmard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Montanari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Debiasing the lasso: Optimal sample size for Gaussian designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The Annals of Statistics 46 2593–2622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Communication-efficient dis- tributed statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of the American Statistical Association 114 668–681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Communication- efficient sparse regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The Journal of Machine Learning Research 18 1– 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Cho, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Cai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Integrative high dimensional multiple testing with heterogeneity under data sharing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of Machine Learning Research 22 1–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lopes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A sharp lower-tail bound for Gaussian maxima with application to bootstrap methods in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Electronic Journal of Statistics 16 58–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Lv, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Lian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Debiased distributed learning for sparse partial linear models in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of Machine Learning Research 23 1–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Mateos, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Bazerque, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Giannakis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed sparse linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' IEEE Transactions on Signal Processing 58 5262– R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Fonseca and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Nadler/Distributed Sparse Linear Regression 33 5276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Predd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Kulkarni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Poor, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Distributed learning in wireless sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' IEEE Signal Processing Magazine 23 56–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Rosenblatt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Nadler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' On the optimality of averaging in distributed statistical learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Information and Inference: A Journal of the IMA 5 379–404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Scaled sparse linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Biometrika 99 879–898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Regression shrinkage and selection via the Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Jour- nal of the Royal Statistical Society: Series B 58 267–288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' van de Geer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and B¨uhlmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' On the conditions used to prove oracle results for the Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Electronic Journal of Statistics 3 1360– 1392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' van de Geer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', B¨uhlmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Ritov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Dezeure, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' On asymptotically optimal confidence regions and tests for high-dimensional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' The Annals of Statistics 42 1166–1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Wainwright, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Sharp thresholds for high-dimensional and noisy sparsity recovery using ℓ1 - constrained quadratic programming (Lasso).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' IEEE Transactions on Information Theory 55 2183–2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Duchi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Wainwright, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Communication- efficient algorithms for statistical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of Machine Learning Research 14 3321–3363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Confidence intervals for low dimen- sional parameters in high dimensional linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B 76 217–242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=', Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' and Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Least-square approximation for a dis- tributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} +page_content=' Journal of Computational and Graphical Statistics 30 1004– 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfowif/content/2301.04022v1.pdf'} diff --git a/JdE0T4oBgHgl3EQfSACX/content/tmp_files/2301.02216v1.pdf.txt b/JdE0T4oBgHgl3EQfSACX/content/tmp_files/2301.02216v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5dcf0793871d2473601820b96ed11063954185c6 --- /dev/null +++ b/JdE0T4oBgHgl3EQfSACX/content/tmp_files/2301.02216v1.pdf.txt @@ -0,0 +1,1950 @@ +arXiv:2301.02216v1 [gr-qc] 5 Jan 2023 +Logarithm Corrections and Thermodynamics for Horndeski gravity like Black Holes +Riasat Ali,1, ∗ Zunaira Akhtar,2, † Rimsha Babar,3, ‡ G. Mustafa,4, § and Xia Tiecheng1, ¶ +1Department of Mathematics, Shanghai University, +Shanghai-200444, Shanghai, People’s Republic of China +2Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan +3Division of Science and Technology, University of Education, Township, Lahore-54590, Pakistan +4Department of Physics Zhejiang Normal University Jinhua 321004, People’s Republic of China +In this paper, we compute the Hawking temperature by applying quantum tunneling approach for +the Horndeski like black holes. We utilize the semi-classical phenomenon and WKB approximation +to the Lagrangian field equation involving generalized uncertainty principle (GUP) and compute +the tunneling rate as well as Hawking temperature. +For the zero gravity parameter, we obtain +results consistent without correction parameter or original tunneling. +Moreover, we study the +thermal fluctuations of the considered geometry and examine the stable state of the system by +heat capacity technique. +We also investigate the behaviour of thermodynamic quantities under +the influence of thermal fluctuations. We observe from the graphical analysis, the corresponding +system is thermodynamically stable with these correction terms. +keywords: +Horndeski like black holes; +Quantum gravity; +Tunneling radiation, +Thermal +fluctuations, Corrected entropy, Phase transition. +I. +INTRODUCTION +Tunneling is the semi-classical mechanism in which particles have radiated from black hole (BH) outer horizon. +Some analysis shows the keen interest in the Hawking temperature (TH) via tunneling method from different BHs. +The main aspect to examine the TH is the imaginary part of classical action which leads to the tunneling radiation +of boson particles appearing from the Horndeski like BHs. +The quantum tunneling and TH of charged fermions in BHs has been observed [1]. In this paper, they examined that +the tunneling and TH depend on charges of electric and magnetic, acceleration, rotation, mass and NUT parameter of +the charged pair BHs. The tunneling strategy from Reissner Nordstom-de Sitter BH like solution in global monopole +has been analyzed [2]. +In this article, the authors observed that the modified TH depends on the parameter of +global monopole. The BH thermodynamics have been examined [3] with some parameters like acceleration, NUT +and rotation. The researchers studied thermodynamical quantities like the area, entropy, surface gravity and TH. +The tunneling spectrum of bosonic particles has been computed from the modified BHs horizon by utilizing the +Proca field equation. Hawking evaluated tunneling probability from BH [4] by utilizing theoretical technique and +later, it has been explained by Parikh and Wilczek [5, 6]. The important of this radiation represents that vacuum +thermal fluctuation produce pairs of particle (particle and anti-particle) from the horizon. Hawking considered that +the particle’s have ability to emit from the BH and the anti-particles have no ability to radiate from the horizon. +Parikh and Wilczek explained a mathematical approach by utilizing WKB approximation. This phenomenon use +geometrical optic approximation which is another view of eikonal approximation in wave clarification [7]. The set of +all particles remain at the front boundary and with the emission of these particles, the BH mass reduces in the form +of particles energy. +In the Parikh-Wilczek method, a precisely tunneling was established and there were as still unanswered problems +like information release, temperature unitary and divergence. +Many authors have made efforts on the tunneling +strategy and semi-classical phenomenon from the different BHs horizon; one of the important explanations can be +checked in [8]-[31]. The radiate particles for many BHs have been analyzed and also computed the radiate particles +with the influences of the geometry of BH with different parameters. It is possible to study modified thermodynamic +properties of BH by considering generalized uncertainty principle (GUP) influences [32]. The GUP implies high energy +∗Electronic address: riasatyasin@gmail.com +†Electronic address: zunaira.math.pu@gmail.com +‡Electronic address: rimsha.babar10@gmail.com +§Electronic address: gmustafa3828@gmail.com +¶Electronic address: xiatc@shu.edu.cn + +2 +result to thermodynamic of BH, by considering the quantum gravity theory with a minimal length. By considering +the GUP influences, it is viable to examine the modified thermodynamic of BHs. +It is a well known fact that thermal fluctuations are a result of statistical perturbations in the dense matter. With +the emission of Hawking radiations from the BH, the size of BH reduces and consequently its temperature increases. +Faizal and his colleague [33] have studied the thermodynamics and thermal fluctuations of generalized Schwarschild +BH, (i.e., Reissner-Nordstrom, Kerr and charged AdS BHs) with the help first-order corrections and discussed the +stability of BHs. +The thermodynamics of rotating Kerr-AdS BH and its phase transition have been studied by +Pourhassan and Faizal [34]. They concluded that the entropy corrections are very helpful to examine the geometry +of small BHs. By applying the stability test of heat capacity and Hessian matrix, the phase transition as well as +thermodynamics of non-minimal regular BHs in the presence of cosmological constant has been investigated [35] and +the authors concluded that the local and global stability of the corresponding BHs increases for higher values of +correction parameters. Zhang and Pradhan [36, 37] have investigated the corrected entropy and second order phase +transition via thermal fluctuations on the charged accelerating BHs. +Moreover, the thermodynamics and geometrical analysis of new Schwarzschild BHs have been studied [38, 39]. By +using the tunneling approach under the influence of quantum gravity the Hawking temperture for different types +of BH have been discussed [40]-[44]. Sharif and Zunaira [45, 46] have computed the thermodynamics, quasi-normal +modeand thermal fluctuations of charged BHs with the help of Weyl corrections. The authors found that the system +is unstable for the small radii of BHs under the influence of first order corrections and by using the heat capacity and +Hessian matrix technique, they have also studied the stable conditions of the system. The authors in [47, 48] have +investigated the thermodynamics, phase transition and local/global stability of NUT BH via charged, accelerating as +well as rotating pairs. Ilyas et al.[49–52] discussed the energy conditions and calculated the new solutions for stellar +structures by taking black hole geometry as exterior spacetime in the background of different modified theories of +gravity. Recently, Ditta et al. [53] discussed the thermal stability and Joule–Thomson expansion of regular BTZ-like +black hole. +The main intention of this paper is to investigate the tunneling radiation without self-gravity and back-reaction and +also explain the modified tunneling rate. The tunneling radiation is evaluated under the conditions of charge-energy +conservation, Horndeski parameter and GUP parameter influences. +The modified TH depends on the Horndeski +parameter as well as GUP parameter and also investigated the behaviour of thermodynamic quantities via thermal +fluctuations. +This paper is based on the analysis of quantum tunneling, TH, stability and instability conditions for the Horndeski +like BH. The paper is outlined as follows: in Sec. II, we study the tunneling radiation of bosonic particles for 4D +Horndeski like BH and also calculate the effects of GUP parameters on tunneling and TH. In section III, we study +the graphical presentation of tunneling radiation for this type of BH and analyze the stable and unstable conditions +for Horndeski like BH. In section IV, we investigate the behaviour of thermodynamic quantities under the effects of +thermal fluctuations. In section V, we express the discussion and conclusion of the whole analysis. +II. +HORNDESKI LIKE BLACK HOLES +Hui and his coauthor Nicolis [58] argued that the no-hair theorems cannot be applied on a Galileon field, as it is +coupled to gravity under the effect of peculiar derivative interactions. Further, they demonstrated that static and +spherically symmetric spacetime defining the geometry of the black hole could not sustain nontrivial Galileon profiles. +Babichev and Charmousis [59] examined the no-hair theorem in Ref. [58] by considering Horndeski theories and +beyond. Furthermore, they provided the Lagrangian of Horndeski theory which can be expressed as a generalized +Galileon Lagrangian, which is defined as +S = +� √−g +� +Q2(χ) + Q3(χ)□φ + Q4(χ)R + Q4,χ +� +(□φ)2�� +. +− (∇ǫ∇εφ) (∇ǫ∇εφ)] + Q5(χ)Gǫε∇ǫ∇εφ +− 1 +6Q5,χ +� +(□φ)3 − 3(□φ) (∇ǫ∇vφ) (∇ǫ∇εφ) ++2 (∇ǫ∇εφ) (∇v∇γφ) (∇γ∇ǫφ)]} d4x. +(1) +where Q2, Q3, Q4, and Q5 are the arbitrary functions of the scalar field φ and χ = −∂ǫφ∂ǫφ/2 represents the canonical +kinetic term. Additionally, in the current analysis, fχ stands for ∂f(χ)/∂χ, Gǫε is the Einstein tensorR is the Ricci +scalar, and other relations are defined as: +(∇ǫ∇εφ)2 ≡ ∇ǫ∇εφ∇ε∇ǫφ +(∇ǫ∇εφ)3 ≡ ∇ǫ∇εφ∇ε∇ρφ∇ρ∇ǫφ +(2) + +3 +The scalar field admits the Galilean shift symmetry ∂ǫφ → ∂ǫφ + bǫ in flat spacetime for Q2 ∼ Q3 ∼ χ and Q4 ∼ +Q5 ∼ χ2, which resembles the Galilean symmetry [60]. In the current study, we investigate the tunneling radiation of +spin-1 massive boson particles from Horndeski-like BH. For this purpose, we adopted the procedure, which is already +reported in [57] for Horndeski spacetime. Finally, we have the following spacetime: +ds2 = − +� +1 − 2rM(r) +Σ +� +dt2 + +1 +∇(r)Σdr2 + Σ2dθ2 − A +Σ sin2 dφ2 − 4ar +Σ M(r) sin2 θdtdφ, +(3) +with Σ2 = a2 cos2 θ + r2 and ∇(r) = a2 − 2rM(r)+ r2, M(r) = M − 1 +2QIn r +r0 and A = (a2 + r2)2 − ∇a2 sin2 θ, while a, +Q and M represent the rotation parameter, Horndeski parameter and mass of BH, respectively. If Q → 0, the metric +(3) goes over to the Kerr BH [61] and if Q = a = 0 the metric (3) also goes over to the Schwarzschild metric. The +line-element (3) can be re-written as +ds2 = −f(r)dt2 + g−1(r)dr2 + I(r)dφ2 + h(r)dθ2 + 2R(r)dtdφ +(4) +where +f(r) = +� +1 − 2rM(r) +Σ +� +, +g−1(r) = 1 +∇Σ, +h(r) = Σ2, +I(r) = −A +Σ sin2, +R(r) = −2ar +Σ M(r) sin2 θ. +We study the tunneling radiation of spin-1 particles from four-dimensional Horndeski like BHs. +By utilizing the +Hamilton-Jacobi ansatz and the WKB approximation to the modified field equation for the Horndeski space-time, the +tunneling phenomenon is successfully applied. We study the modified filed equation on a four dimensional space-time +with the background of rotation parameter, Horndeski parameter and evaluated for the radial function. As a result, +we get the tunneling probability of the radiated particles and derive the modified TH of Horndeski like BHs. The +modified filed equation is expressed by [27, 30] +∂µ +�√−gΨνµ� ++ √−g m2 +ℏ2 Ψν + √−g i +ℏAµΨνµ + √−g i +ℏeF νµΨµ + ℏ2β∂0∂0∂0 +�√−gg00Ψ0ν� +−ℏ2β∂i∂i∂i +�√−ggiiΨiν� += 0, +(5) +here Ψνµ, m and g present the anti-symmetric tensor, bosonic particle mass and determinant of coefficient matrix, so +Ψνµ = +� +1 − ℏ2β∂2 +ν +� +∂νΨµ − +� +1 − ℏ2β∂2 +µ +� +∂µΨν + +� +1 − ℏ2β∂2 +ν +� i +ℏeAνΨµ +− +� +1 − ℏ2β∂2 +ν +� i +ℏeAµΨν, +and Fνµ = ∇νAµ − ∇µAν, +with β, e , ∇µ and Aµ are the GUP parameter(quantum gravity), bosonic particle charge, covariant derivative and +BH potential, respectively. The Ψνµ can be computed as +Ψ0 = −IΨ0 + RΨ3 +fI + R2 +, +Ψ1 = +1 +g−1 Ψ1, +Ψ2 = 1 +hΨ2, +Ψ3 = RΨ0 + fΨ3 +fI + R2 +, +Ψ01 = +˜ +−DΨ01 + RΨ13 +(R2 + fI)g−1 , +Ψ02 = +˜ +−DΨ02 +(R2 + fI)h, +Ψ03 = (f 2 − fI)Ψ03 +(fI + R2)2 , +Ψ12 = +1 +g−1hΨ12, +Ψ13 = +1 +g−1(fI + R2)Ψ13, +Ψ23 = fΨ23 + RΨ02 +(fI + R2)h . +In order to observe the bosonic tunneling, we have assumed Lagrangian gravity equation. Further, we utilized the +WKB approximation to the Lagrangian gravity equation and computed set of equations. +Furthermore, we have +utilized the variable separation action to get required solutions. The approximation of WKB is defined [? ] as +Ψν = ην exp +� i +ℏK0(t, r, φ, θ) + ΣℏnKn(t, r, φ, θ) +� +. +(6) +we get set of equations in Appendix A. Utilizing variable separation technique, we can take +K0 = −(E − Lω)t + W(r) + Lφ + ν(θ), +(7) + +4 +where E and L present the particle energy and particle angular, respectively, corresponding to angle φ. +After considering Eq. (7) into Eqs. (22)-(25), we reach a matrix in the form +U(η0, η1, η2, η3)T = 0, +which express a 4 × 4 matrix presented as ”U”, whose elements are given as follows: +U00 = +−I +g−1(fI + R2) +� +W 2 +1 + βW 4 +1 +� +− +I +(fI + R2)h +� +L2 + βL4� +− +fI +(fI + R2)2 +� +ν2 +1 + βν4 +1 +� +− +m2I +(fI + R2), +U01 = +−I +g−1(fI + R2) +� +((E − Lω) + (E − Lω)3β + A0e + (E − Lω)2βeA0 +� +W1 + +R +g−1(fI + R2) + +� +ν1 + βν3 +1 +� +, +U02 = +−I +h(fI + R2) +� +(E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0 +� +L, +U03 = +−R +g−1(fI + R2) +� +W 2 +1 + βW 4 +1 +� +− +fI +h(fI + R2)2 +� +(E − Lω)3β − (E − Lω)2βeA0 + (E − Lω) − eA0 +� +ν1 ++ +m2R +(fI + R2)2 , +U12 = +1 +g−1h +� +W1 + βW 3 +1 +� +L, +U11 = +−I +g−1(fI + R2) +� +β(E − Lω)4 − βeA0EW 2 +1 + (E − Lω)2 − eA0(E − Lω) +� ++ +R +(fI + R2)g−1 ++ +� +ν1 + βν3 +1 +� +(E − Lω) − +1 +g−1h +� +L2 + βL4� +− +1 +(fI + R2)g−1 +� +ν1 + βν3 +1 +� +− m2 +g−1 − +eA0I +(fI + R2)g−1 +× +� +(E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0 +� ++ +eA0R +g−1(fI + R2) +� +ν1 + βν3 +1 +� +, +U13 = +−R +g−1(fI + R2) +� +W1 + βW 3 +1 +� +(E − Lω) + +1 +g−1(fI + R2)2 +� +W1 + βW 3 +1 +� +ν1 + +ReA0 +g−1(fI + R2) +� +W1 + βW 3 +1 +� +, +U20 = +I +h(fI + R2) +� +(E − Lω)L + β(E − Lω)L3� ++ +R +h(fI + R2) +� +(E − Lω) + β(E − Lω)3ν2 +1 +� +− +IeA0 +h(fI + R2) +� +L + βL3� +, +U22 = +I +h(R2 + fI) +� +βE4 − βeA0E + E2 − eA0(E − Lω) +� +− +1 +g−1h + +R +h(R2 + fI) +� +(E − Lω)3β ++ −(E − Lω)2βeA0 − A0e + (E − Lω) +� +ν1 − +f +h(R2 + fI) +� +ν2 +1 + βν4 +1 +� +− m2 +h − +eA0I +h(fI + R2) +� +(E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0 +� +, +U23 = +f +h(fI + R2) +� +L + βL3� +ν1, +U30 = +(fI − f 2) +(fI + R2)2 +� +ν1 + βν3 +1 +� +E + +R +h(fI + R2) +� +L2 + βL4� +− +m2R +(fI + R2) − eA0(fI − f 2) +(fI + R2)2 +� +ν1 + βν3 +1 +� +, +U31 = +1 +g−1(fI + R2) +� +ν1 + βν3 +1 +� +W1, +U32 = +R +h(R2 + fI) +� +L + βL3� +E + +f +h(R2 + fI) +� +ν1 + βν3 +1 +� +L, +U33 = +(fI − f 2) +(fI + R2) +� +(E − Lω)2 − eA0(E − Lω) + β(E − Lω)4 − βeA0(E − Lω)3� +− +1 +g−1(R2 + fI) +� +W 2 +1 + βW 4 +1 +� +− +f +(R2 + fI)h +� +L2 + βL4� +− +m2f +(fI + R2) − eA0(fI − f 2) +(fI + R2) +× +� +(E − Lω) + β(E − Lω)3 − eA0(E − Lω)2� +, + +5 +with ∂tK0 = (E − Lω), +∂φK0 = L, W1 = ∂rK0 and ν1 = ∂θK0. For non-trivial solution, we get +ImW ± = ± +� +� +� +� +� +� +E − LΩ − eA0 +�2 ++ Z1 +� +1 + β Z2 +Z1 +� +(fI + R2)gI−1 +dr, += ±iπ +� +R − LΩ − A0e +� ++ +� +1 + βA +� +2k(r+) +, +(8) +where +Z1 = (E − Lω)ν1 +g−1R +fI + R2 + +fg−1 +fI + R2 ν2 +1 − g−1m2, +Z2 = +g−1I +fI + R2 +� +(R − LΩ)4 + (eA0)2(R − LΩ)2 − 2A0e(R − LΩ)3� ++ +g−1R +h(fI + R2) +� +(R − LΩ)3 − eA0(R − LΩ)2� +ν1 − +fg−1 +fI + R2 ν4 +1 − W 4 +1 . +and A is a arbitrary parameter. In particular case, we take the radial component of the action of particle, for this aim +we choose a components of matrix equals to zero. Since, we have found the tunneling radiation (related to Horndeski +gravity and quantum gravity) for BH. This tunneling and TH quantities relate on the Horndeski gravity and quantum +gravity of this particular physical object. Thus, we have found the corresponding TH which important as a component +of leading metric with Horndeski gravity and quantum gravity. Such that, in this method we are not concerned in +order of higher for Planck’s constant only obtained appropriate result. The generalized tunneling depends on the BHs +metric and Horndeski gravity and GUP parameter. The generalized tunneling for Horndeski like BH can be written +as +T = Temission +Tabsorption += exp +� +−2π(R − LΩ − A0e) +k(r+) +� +[1 + βA] , +(9) +with +k(r+) = 4πr+ +� +a2 + r2 ++ +� +Qr+ + −a2 + r2 ++ +(10) +In the presence of GUP terms, we calculate the TH of the Horndeski gravity BHs. by taking the Boltzmann factor +TB = exp [(E − Lω − eA0)/TH] as +TH = −a2 + Qr+ + r2 ++ +4πr+ +� +a2 + r2 ++ +� [1 − βA] . +(11) +The above result shows that the TH depends on the Horndeski gravity, GUP parameter, rotation parameter, Horndeski +gravity, arbitrary parameter A and radius (r+) of BH. When β = 0, we obtain the general TH in [57]. In the absence +of charge i.e., Q = 0, the above temperature reduces into Kerr BH temperature [58, 59]. For β = 0 and a = 0, +the temperature reduces into Reissner Nordstr¨om BH. Moreover, when Q = 0 = a, we recover the temperature of +Schwarzschild BH [60]. The quantum corrections slow down the increase in TH throughout the radiation phenomenon. +A. +TH versus r+ +We observe the geometrical presentation of TH w.r.t r+ for the 4D Horndeski like metric. Moreover, we observe +the physical significance of these graphs under Horndeski gravity and GUP parameter and study the stability and +instability analysis of corresponding TH. For β equals to zero, the tunneling radiation will be independent of GUP +parameter. In the left plot of Fig. 1, the TH increases with increasing β in small region of horizon 0 ≤ r+ ≤ 5, that +indicates the stable state of BH till r+ → ∞. In the right plot of Fig. 1, the rotating parameter and β are fixed, +then we take changing values of hairy parameter of Horndeski gravity and get the completely unstable form of BH +with negative temperature. + +6 +β=10 +β=20 +β=30 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +6 +r+ +TH +Q=0.5 +Q=1 +Q=1.5 +0 +1 +2 +3 +4 +5 +- 6 +- 4 +- 2 +0 +2 +r+ +TH +Figure 1: TH w.r.t horizon r+ for a = 5, Q = 0.5 and Ξ = 1 and left β = 10 (black), β = 20 (blue), β = 30 (red). +Right a = 0.5, Ξ = 1, β = 5, Q = 0.5 (black), a = 0.5, Ξ = 1, β = 5, Q = 1 (blue), a = 0.5, Ξ = 1, β = 5, Q = 1.5 +(red). +III. +THERMODYNAMICS AND EFFECTS OF FIRST ORDER CORRECTIONS +Thermal fluctuations plays important role on the study of BH thermodynamics. With the concept of Euclidean +quantum gravity, the temporal coordinates shifts towards complex plan. To check the effects of these correction in +entropy, we find Hawking temperature and usual entropy of the given system with the help of first law of thermody- +namics +S = π +� +a2 + r2 ++ +� +, +T = −a2 + Qr+ + r2 ++ +4πr+ +� +a2 + r2 ++ +� +(12) +To check the corrected entropy along these thermal fluctuations, the partition function is Z(µ) in terms of density of +states η(E) is given as [37] +Z(µ) = +� ∞ +0 +exp(−µE)η(E)dE, +(13) +where T+ = 1 +µ and E is the mean energy of thermal radiations. By using the Laplace inverse transform, the expression +of density takes the form +ρ(E) = +1 +2πi +� µ0+i∞ +µ0−i∞ +Z(µ) exp(µE)dµ = +1 +2πi +� µ0+i∞ +µ0−i∞ +exp( ˜S(µ))dµ, +(14) +where ˜S(µ) = µE + ln Z(µ) represents the modified entropy of the considered system that is dependent on Hawking +temperature. Moreover, the expression of entropy gets modified with the help of steepest decent method, +˜S(µ) = S + 1 +2(µ − µ0)2 ∂2 ˜S(µ) +∂µ2 +��� +µ=µ0 + higher-order terms. +(15) +Using the conditions ∂ ˜S +∂µ = 0 and ∂2 ˜S +∂µ2 > 0, the corrected entropy relation under the first-order corrections modified. +By neglecting higher order terms, the exact expression of entropy is expressed as +˜S = S − δ ln(ST 2), +(16) +where δ is called correction parameter, the usual entropy of considered system is attained by fixing δ = 0 that is +without influence of these corrections. Furthermore, inserting the Eq. (12) into (16), we have +˜S = (a2 + r2 ++) − δ log +�� +a2 − r+ (Q + r+) +�2 +16πr2 ++ +� +a2 + r2 ++ +� +� +. +(17) + +7 +δ=0 +δ=0.4 +δ=0.6 +δ=0.6 +0.0 +0.1 +0.2 +0.3 +0.4 +10 +20 +30 +40 +r+ +S +˜ +Figure 2: Corrected entropy versus r+ for a=0.2, Q=0.4. . +In the Fig. 2, the graph of corrected entropy is monotonically increasing throughout the considered domain. It is +noted the graph (black) of usual entropy is increasing just for small value of horizon radius but corrected expression +of energy is increasing smoothly. Thus, these corrections terms are more effective for small BHs. Now, using the +expression of corrected entropy and check the other other thermodynamic quantities via thermal fluctuations. In this +way, the the Helmholtz energy (F = − � ˜SdT ) leads to the form +F = +� +a4 − r2 ++ +� +−4a2 + 2Qr+ + r2 ++ +�� � +δ log +� +(a2−r+(Q+r+)) +2 +{ +r2 ++ +� +a2 + r2 ++ +� +} +� +− a2 − δ log(16π) − r2 ++ +� +4πr2 ++ +� +a2 + r2 ++ +�2 +. +(18) +δ=0 +δ=0.4 +δ=0.6 +δ=0.6 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +35 +40 +45 +50 +55 +60 +65 +r+ +F +Figure 3: Helmholtz free energy versus r+ for a=0.2, Q=0.4.. +The Fig. 3 shows the graph of Helmholtz free energy versus horizon radius. It is observed that the behaviour of +energy is gradually decreases for the different correction parameter δ values. While the graph of usual entropy shows +opposite behaviour as the graph is increasing. This behaviour means, the considered system shifts its state towards +equilibrium, thus, no more work can be extract from it. The expression of internal energy (E = F + T ˜S) for the +corresponding geometry is given by [37] +E = +� +r+ +� +a2 + r2 ++ +� � +r+ (Q + r+) − a2� +� +δ +� +log(16π) − log +�� +a2 − r+ (Q + r+) +� 2 +r2 ++ +� +a2 + r2 ++ +� +�� ++ π +� +a2 + r2 ++ +� +� +− +� +a4 − r2 ++ +� +−4a2 + 2Qr+ + r2 ++ +�� +� +−δ log +�� +a2 − r+ (Q + r+) +�2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +� � +� +4πr2 ++ +� +a2 + r2 ++ +�2 �−1 +. +(19) + +8 +δ=0.2 +δ=0.4 +δ=0.6 +δ=0.8 +0.0 +0.1 +0.2 +0.3 +0.4 +-2 +0 +2 +4 +6 +r+ +E +Figure 4: Internal Energy w.r.t r+ for a=0.2, Q=0.4. +The graphical behaviour of internal energy for the different choices of horizon radius is shown in Fig. +4. +It is +observable that for the small values of radii, the graph is gradually decreases even shifts towards negative side, While +the corrected internal energy depicts positive behaviour. This mean that the considered BH absorbing more and +more heat from the surrounding to maintain its state. Since, BHs considered as a thermodynamic system, so there +is another important thermodynamic quantity that is pressure. In this regard, there is deep connection between +voulme (V = +2π(r2 +++a2)(2r2 +++a2) +3r+ +) and pressure. The Expression of BH pressure (P = − dF +dV ) under the effect of thermal +fluctuations takes the form +P = +� +2r2 ++ +� +a2 + r2 ++ +�� +− 4a2 + 3Qr+ + 2r2 ++ +�� +a2 − r+ +� +Q + r+ +��2� +− δ log +��� +a2 − r+(Q + r+) +�2� +(r2 ++(a2 + r2 ++))−1� ++ a2 + δ log(16π) + r2 ++ +� +− 2 +� +r+ +� +Q + r+ +� +− a2�� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +− a4δ + Qr3 ++ +� +a2 + δ +� +− r2 ++ +� +a4 + 3a2δ +� ++ Qr5 ++ + r6 ++ +� ++ 4r2 ++ +� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +a2 − r+ +� +Q + r+ +��2� +− δ log +× +��� +a2 − r+ +� +Q + r+ +��2�� +r2 ++ +� +a2 + r2 ++ +��−1� ++ a2 + δ log(16π) + r2 ++ +� ++ 2 +� +a2 + r2 ++ +�� +a4 − r2 ++ +� +− 4a2 ++ 2Qr+ + r2 ++ +��� +a2 − r+ +� +Q + r+ +��2� +− δ log +��� +a2 − r+ +� +Q + r+ +��2� +(r2 ++ +� +a2 + r2 ++ +��−1� ++ a2 + δ log(16π) ++ r2 ++ +��� +4πr3 ++ +� +a2 + r2 ++ +�3� +a2 − r+ +� +Q + r+ +��2�−1 +. +(20) +δ=0 +δ=0.2 +δ=0.4 +δ=0.6 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +×108 +2 +�108 +3 +�108 +4 +�108 +5 +�108 +r+ +P +Figure 5: Pressure versus r+ for a=0.2, Q=0.4. +In the Fig. 5, the graph of pressure is just coincides the state of equilibrium. For the different values of correction +parameter, the pressure is significantly increases for the considered system. +Further, there is another important +thermodynamic quantity enthalpy (H = E + PV ) is given in Appendix B. + +9 +δ=0 +δ=0.4 +δ=0.6 +δ=0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-10 +-5 +0 +5 +r+ +H +Figure 6: Enthalpy versus r+ for a=0.2, Q=0.4. +From Fig. 6, it can observed that the graph of usual enthalpy is coincide with the plots of corrected one and abruptly +decreases even shifts towards negative side. This means that there exists a exothermic reactions means there will be +huge amount of energy release into its surroundings. By taking into account the thermal fluctuations, the Gibbs free +energy (G = H − T ˜S) is expressed in Appendix B. +δ=0 +δ=0.4 +δ=0.6 +δ=0.8 +0 +5 +10 +15 +20 +5 +10 +15 +20 +r+ +G +Figure 7: Gibbs free energy versus r+ for a=0.2, Q=0.4. +The graphical analysis of Gibbs free energy with respect to horizon radius is shows in Fig. 7. The positivity of this +energy is sign of occurrence of non-spontaneous reactions means this system requires more energy to gain equilibrium +state. After the detail discussion of thermodynamics quantities, there is another important concept is the stability of +the system that is checked by specific heat. The specific heat (C ˜S = dE +dT ) is given as +C ˜S = +� +2 +� +a2 + 3r2 ++ +�� +r+ +� +r+ +� +a2� +− δ(Q − 4) log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2(πQ − 5) + δ(Q − 4) log(16π) +� ++ r+ ++ +� +− πa4 − 2δQ log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ r+ +� +− δ(Q + 1) log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ r+ +� +− δ log +× +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ πa2 + δ log(16π) + r+ +� +πQ + πr+ + 1 +� ++ 2Q +� ++ a2(2πQ − 3) + δ(Q + 1) log(16π) +� ++ 2a2Q + 2δQ log(16π) +�� +− a4� +− δ log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ πa2 + δ log(16π) +�� +− a4� +− δ log +× +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) +���� +r+ +� +a2 + r2 ++ +�� +− a4 − 4a2r2 ++ + 2Qr3 ++ + r4 ++ +��−1 +. +(21) + +10 +δ=0 +δ=0.4 +δ=0.6 +δ=0.8 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +-0.10 +-0.05 +0.00 +0.05 +r+ +CS +~ +Figure 8: Specific heat versus r+ for a=0.2, Q=0.4. +From Fig. +8, the behaviour of specific heat is observed with respect to horizon radius and different choices of +correction parameter δ. It can be observed that the uncorrected quantity (black) depicts negative behaviour means +the system is unstable while the corrected specific heat shows positive behaviour throughout the considered domain. +The positivity of this plot is indication of stable region. It can be concluded that these correction terms makes the +system stable under thermal fluctuations. +IV. +DISCUSSION AND RESULT +In this paper, we have utilized Lagrangian gravity equation to observe the tunneling of bosonic particles through +the horizon of Horndeski like BH. We have used the metric from Ref. [57]. We have considered a new version of +black hole with Horndeski parameter Q and a rotation parameter a, due to the presence of these parameters, we call +the metric a new type of spacetime. Our results are also in terms of these parameters, therefore they are different +from the previous literature related about the thermodynamics of this black hole. Assuming to relativistic quantum +mechanics and the region of vacuum, where particles are produced continuously in the phenomenon of annihilated. +The tunneling radiation as a quantum mechanical processes can be observed as a tunneling phenomenon, where +positive boson particle radiate the horizon and the negative energy boson particle move inward and absorbed by the +BH. The incoming and outgoing boson particles movement carried out by the action of particle’s is real and complex, +respectively. The emission rate of these tunneling radiation corresponding to the Horndeski like BH configuration is +associated to the imaginary part of the action of particles, which is associated to the factor of Boltzmann, this factor +gives TH for Horndeski like BH. From our investigation, we have observed that, in rotating case of BH, the TH at +which boson particles tunnel through the BH horizon is not dependent at any types of particles. In special case when +particles have different (zero or upward or down) spins, the tunneling rate will be alike by assuming the semi-classical +approach. Thus, their corresponding TH must be similar for all types of particle. Therefore, one can say tunneling +radiation is independent of all kinds of the particles and this result also holds for different frame of coordinates by +utilizing the transformations of particular coordinate. For this procedure, the tunneling particles is associated to the +energy of particles, momentum, quantum gravity, hairy parameter of Horndeski gravity and BH surface gravity, while +the temperature depends on hairy parameter of Horndeski gravity, rotation and quantum gravity parameters. It is +very important to mention here that, when β = 0, we obtain the standard temperature for Horndeski like BH. In +the absence of charge i.e., Q = 0, the above temperature reduces into Kerr BH temperature. For β = 0 and a = 0, +the temperature reduces into Reissner Nordstr¨om BH. Moreover, when Q = 0 = a, we recover the temperature of +Schwarzschild BH. For the changing values of β from 10 to 30 in the region 0 ≤ r+ ≤ 5, we have observed that the +Horndeski like BH is stable and for changing Q from 0.5 to 1.5, the Horndeski like BH is un-stable with negative +temperature. Moreover, the temperature increases with the increasing values of quantum gravity β. +Moreover, we have computed TH as well as heat capacity for Horndeski gravity like BHs. +Firstly, the TH is +calculated through entropy and the density of state is also calculated with help of inverse Laplace transformation. +We have observed that the exact entropy of the system depends on Hawking temperature by applying the method of +steepest descent under different conditions. In graphically, we have studied the monotonically increasing entropy of +the metric throughout the assumed domain. It is observed that a decrease in entropy for a certain value of the radius, +the corrected expression of energy is increasing smoothly and also studied that the small BHs are more effective for +thermal fluctuations. +It is observed that the behaviour of energy gradually decreases for the different correction parameter δ values and +the graph of usual entropy shows opposite behaviour as the graph increases. Therefore, the considered system shifts +its state towards equilibrium. It is observable that for the small values of radii, the graph gradually decreases even + +11 +shifts towards negative side, while the corrected internal energy depicts positive behaviour. So, the considered BH +absorbed more and more heat from the surrounding to maintain its state. The graph of pressure coincides the state +of equilibrium. For the different values of correction parameter, the pressure significantly increases for the considered +system. +It is observed that the graph of usual enthalpy coincides with the plots of corrected one and abruptly decreases even +shifts towards negative side. It can be concluded that there exists a exothermic reactions means a huge amount of +energy released into its surroundings. The positivity of this energy is sign of occurrence of non-spontaneous reactions +means the system required more energy to gain equilibrium state. The behaviour of specific heat with respect to +horizon radius and different choices of correction parameter δ has been observed. The uncorrected quantity depicts +negative behaviour means the system is unstable while the corrected specific heat shows positive behaviour throughout +the considered domain. The positivity of the plots for specific heat is indication of stable region. It can be concluded +that these correction terms makes the system stable under thermal fluctuations. +Appendix A +We have utilized the Lagrangian equation in the approximation of WKB to get following solutions, +I +(R2 + fI)g−1 +� +η1(∂0K0)(∂1K0) + βη1(∂0K0)3(∂1K0) − η0(∂1K0)2 − βη0(∂1K0)4 + η1eA0(∂1K0) + +η1βeA0(∂0K0)2(∂1K0) +� +− +R +g−1(fI + R2) +� +η3(∂1K0)2 + βη3(∂1K0)4 − η1(∂1K0)(∂3K0) − βη1(∂1K0)(∂3K0)2� ++ +I +h(fI + R2) +� +η2(∂0K0)(∂2K0) + βη2(∂0K0)3(∂2K0) − η0(∂2K0)2 − βη0(∂2K0)4 + η2eA0(∂2K0) ++ η2eA0β(∂0K0)2(∂1K0) +� ++ +fI +(fI + R2)2 +� +η3(∂0K0)(∂3K0) + βη3(∂0K0)3(∂3K0) − η0(∂3K0)2 − βη0(∂3K0)4 ++ η3eA0(∂3K0) + η3eA0(∂0K0)2(∂3K0) +� +− m2 Iη0 − Rη3 +(fI + R2) = 0, +(22) +−I +g−1(fI + R2) +� +η1(∂0K0)2 + βη1(∂0K0)4 − η0(∂0K0)(∂1K0) − βη0(∂0K0)(∂1K0)3 + η1eA0(∂0K0) + βη1eA0(∂0K0)3� ++ +R +g−1(fI + R2) +� +η3(∂0K0)(∂1K0) + βη3(∂0K0)(∂1K0)3 − η1(∂0K0)(∂3K0) − βη1(∂0K0)(∂3K0)3� ++ +1 +g−1h [η2(∂1K0)(∂2K0) + βη2(∂1K0)(∂2K0)3 − η1(∂2K0)2 − βη1(∂2K0)4� ++ +1 +g−1(fI + R2) +� +η3(∂1K0)(∂3K0) + βη3(∂1K0)(∂3K0)3 − η1(∂3K0)2 − βη1(∂3K0)4� +− m2η1 +g−1 + +eA0I +g−1(fI + R2) +� +η1(∂0K0) + βη1(∂0K0)3 − η0(∂1K0) − βη0(∂1K0)3 + eA0η1 + βη1eA0(∂0K0)2) +� ++ +eA0R +g−1(fI + R2) +� +η3(∂1K0) + βη3(∂1K0)3 − η1(∂3K0) − βη1(∂1K0)3� += 0, +(23) + +12 +I +h(fI + R2) +� +η2(∂0K0)2 + βη2(∂0K0)4 − η0(∂0K0)(∂2K0) − βη0(∂0K0)(∂2K0)3 + η2eA0(∂0K0) + βη2eA0(∂0K0)3� ++ +1 +g−1h +� +η2(∂1K0)2 + βη2(∂1K0)4 − η1(∂1K0)(∂2K0) − βη1(∂1K0)(∂2K0)3� +− +R +h(fI + R2) +� +η2(∂0K0)(∂3K0) ++ βη2(∂0K0)3(∂3K0) − η0(∂0K0)(∂3K0) − βη0(∂0K0)3(∂3K0) + η2eA0(∂3K0) + βη2eA0(∂3K0)3� ++ +f +h(fI + R2) +� +η3(∂2K0)(∂3K0) + βη3(∂2K0)3(∂3K0) − η2(∂3K0)2 − βη2(∂3K0)4� ++ +eA0I +h(fI + R2) +� +η2(∂0K0) ++ βη2(∂0K0)3 − η0(∂2K0) − βη0(∂2K0)3 + η2eA0 + η2βeA0(∂0K0)2� +− m2η2 +h += 0, +(24) +(fI − f 2) +(fI + R2)2 +� +η3(∂0K0)2 + βη3(∂0K0)4 − η0(∂0K0)(∂3K0) − βη0(∂0K0)(∂3K0)3 + eA0η3(∂0K0) + βη3eA0(∂0K0)3� +− +I +h(fI + R2) +� +η3(∂1K0)2 + βη3(∂1K0)4 − η1(∂1K0)(∂3K0) − βη1(∂1K0)(∂3K0)3� +− +R +h(fI + R2) +� +η2(∂0K0)(∂2K0) + βη2(∂0K0)3(∂2K0) − η0(∂2K0)2 + βη0(∂2K0)4 + eA0η2(∂2K0) + βη2eA0∂0K0)2(∂2K0) +� +− +eA0f +h(fI + R2) +� +η3(∂2K0)2 + βη3(∂2K0)4 − η2(∂2K0)(∂3K0) − βη2(∂0K0)(∂3K0)3� +− m2(Rη0 − fη3 +(fI + R2) ++eA0(fI − f 2) +(fI + R2)2 +� +η3(∂0K0) + βη3(∂0K0)3 − η0(∂3K0) − βη0(∂3K0)3 + η3eA0 + η3βeA0(∂0K0)2� += 0, +(25) +Appendix B +The thermodynamic quantity enthalpy is given as +H = +� +r+ +� +a2 + r2 ++ +�� +r+ +� +a2 + r2 ++ +�� +r+ +� +Q + r+ +� +− a2�� +δ +� +log(16π) − log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +�� ++ π +� +a2 + r2 ++ +�� +− +� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +− δ log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +�� ++ +� +2r2 ++ +� +a2 + r2 ++ +�� +− 4a2 + 3Qr+ + 2r2 ++ +�� +a2 − r+ +� +Q + r+ +��2� +− δ log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 ++ δ log(16π) + r2 ++ +� +− 2 +� +r+ +� +Q + r+ +� +− a2�� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +− a4δ + Qr3 ++ +� +a2 + δ +� +− r2 ++ +� +a4 + 3a2δ +� ++ Qr5 ++ + r6 ++ +� ++ 4r2 ++ +� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +a2 − r+ +� +Q + r+ +��2� +− δ log +× +� +� +a2 − r+ +� +Q + r+ +�� +2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +� ++ 2 +� +a2 + r2 ++ +�� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +�� +× +� +a2 − r+ +� +Q + r+ +��2� +− δ log +� +� +a2 − r+ +� +Q + r+ +��2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +���� +a2 − r+ +� +Q + r+ +��2�−1 +) +× +� +4πr3 ++ +� +a2 + r2 ++ +�3�−1 +. +(26) + +13 +The Gibbs free energy is expressed as +G = +� +− r2 ++ +� +a2 + r2 ++ +� +2� +r+ +� +Q + r+ +� +− a2�� +− δ log +� +� +a2 − r+ +� +Q + r+ +�� +2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +� ++ r+ +� +a2 + r2 ++ +� +× +� +r+ +� +a2 + r2 ++ +�� +r+ +� +Q + r+ +� +− a2�� +δ +� +log(16π) − log +� +� +a2 − r+ +� +Q + r+ +�� +2 +r2 ++ +� +a2 + r2 ++ +� +�� ++ π +� +a2 + r2 ++ +�� +− +� +a4 − r2 ++ +× +� +− 4a2 + 2Qr+ + r2 ++ +��� +− δ log +� +� +a2 − r+ +� +Q + r+ +�� +2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +�� ++ +� +2r2 ++ +� +a2 + r2 ++ +� +× +� +− 4a2 + 3Qr+ + 2r2 ++ +�� +a2 − r+ +� +Q + r+ +�� +2� +− δ log +� +� +a2 − r+ +� +Q + r+ +�� +2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +� +− 2 +� +r+ +� +Q + r+ +� +− a2�� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +− a4δ + Qr3 ++ +� +a2 + δ +� +− r2 ++ +� +a4 + 3a2δ +� ++ Qr5 ++ + r6 ++ +� ++ 4r2 ++ +� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +a2 − r+ +� +Q + r+ +��2� +− δ log +� +� +a2 − r+ +� +Q + r+ +�� +2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) ++ r2 ++ +� ++ 2 +� +a2 + r2 ++ +�� +a4 − r2 ++ +� +− 4a2 + 2Qr+ + r2 ++ +��� +a2 − r+ +� +Q + r+ +��2� +− δ log +� +� +a2 − r+ +� +Q + r+ +�� +2 +r2 ++ +� +a2 + r2 ++ +� +� ++ a2 + δ log(16π) + r2 ++ +���� +a2 − r+ +� +Q + r+ +�� +2��� +4πr3 ++ +� +a2 + r2 ++ +� +3�−1 +. +(27) +[1] M. Sharif, W. Javed, Eur. Phys. J. C 72, 1997(2012). +[2] M. Sharif, W. Javed, Int. J. Mod. Phys. Conf. Ser. 23(2013)271. +[3] M. Sharif, W. Javed, Can. J. Phys. 91(2013)236. +[4] S. W. Hawking, Commun. Math. Phys. 43(1975)199. +[5] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85(2000)5042. +[6] M. K. Parikh, Int. J. Mod. Phys. D 13(2004)2351. +[7] J. T. Firouzjaee and G. F. R. Ellis, Gen. Rel. Grav. 47(2015)6. +[8] W. Javed, R. Babar, Adv. High Energy Phys. 2019(2019)2759641. +[9] W. Javed, R. Babar, Chinese Journal of Phys. 61(2019)138. +[10] W. Javed, R. Babar, The Fifteenth Marcel Grossmann Meeting, World Scientific 3(2022)905. +[11] W. Javed, R. Babar, Punjab University Journal of Mathematics 52(2020)6. +[12] W. Javed, R. Babar, A. ¨Ovg¨un, Mod. Phys. Lett. A 34(2019)1950057. +[13] R. Babar, W. Javed, A. ¨Ovg¨un, Mod. Phys. Lett. A 35(2020)2050104. +[14] M. Sharif, W. Javed, J. Korean Phys. Soc. 57(2010)217. +[15] A. ¨Ovg¨un, K. Jusufi, Eur. Phys. J. Plus. 132(2017)298. +[16] A. ¨Ovg¨un, Int. J. Theor. Phys. 55(2016)2919. +[17] K. Jusufi, A. Ovgun, G. Apostolovska, Adv. High Energy Phys. 2017(2017)8798657. +[18] I. Sakalli, A. ¨Ovg¨un, K. Jusufi, Astrophys Space Sci. 361(2016)330. +[19] I. Sakalli, A. ¨Ovg¨un, General Relativity and Gravitation 48(2016)1. +[20] A. ¨Ovg¨un, I. Sakalli, Int. J. Theor. Phys. 57(2018)322. +[21] A. ¨Ovg¨un, I. Sakalli, J. Saavedra, C. Leiva, Mod. Phys. Lett. A 35(2020)2050163. +[22] I. Sakalli, A. Ovgun, J. Exp. Theor. Phys. 121(2015)404. +[23] R. Ali, K. Bamba, M. Asgher, M. F. Malik, S. A. A. Shah, Symmetry. 12(2020)1165. +[24] R. Ali, K. Bamba, M. Asgher, S. A. A. Shah, Int. J. Mod. Phys. D 30(2021)2150002. +[25] R. Ali, M. Asgher, M. F. Malik, Mod. Phys. Lett. A 35(2020)2050225. +[26] W. Javed, R. Ali, R. Babar, A. ¨Ovg¨un, Eur. Phys. J. Plus 134(2019)511. +[27] W. Javed, R. Ali, R. Babar, A. ¨Ovg¨un, Chinese Phys. C 44(2020)015104. +[28] K. Jusufi, A. ¨Ovg¨un, Astrophys Space Sci. 361(2016)207. + +14 +[29] R. Ali, M. Asgher, New Astronomy 93(2022)101759 +[30] R. Ali, R. Babar and P. K. Sahoo, Physics of the Dark Universe 35(2022)100948. +[31] R. Ali, R. Babar, M. Asgher and S. A. A. Shah, Int. J. Geom. Methods Mod. Phys. 19(2022)2250017. +[32] J. M. Bardeen, in Conference Proceedings of GR5 (Tbilisi, URSS, 1968), p. 174. +[33] M. Faizal and M. M. Khalil, Int. J. Mod. Phys. A 30(2015)1550144. +[34] B. Pourhassan and M. Faizal, Nucl. Phys. B 913(2016)834. +[35] A. Jawad and M. U. Shahzad, Eur. Phys. J. C 77(2017)349. +[36] M. Zhang, Nucl. Phys. B 935(2018)170. +[37] P. Pradhan, Universe 5(2019)57. +[38] K. Ghaderi, B. Malakolkalami, Nucl. Phys. B 903(2016)10. +[39] W. X. Chen, Y. G. Zheng, arXiv preprint arXiv:2204.05470. +[40] R. Ali, R. Babar, M. Asgher, and S. A. A. Shah, Annals of Physics, 432(2021)168572. +[41] W. Javed, G. Abbas, R. Ali, Eur. Phys. J. C 77(2017)296. +[42] A. ¨Ovg¨un, W. Javed, R. Ali, Adv. High Energy Phys. 2018(2018)11. +[43] W. Javed, R. Ali, G. Abbas, Can. J. Phys. 97(2018)176. +[44] R. Ali, K. Bamba, S. A. A. Shah, Symmetry. 631(2019)11. +[45] M. Sharif and Z. Akhtar, Phys. Dark Universe 29(2020)100589. +[46] M. Sharif and Z. Akhtar, Chin. J. Phys 71(2021)669. +[47] W. Javed, Z. Yousaf and Z. Akhtar, Mod. Phys. Lett. A 33(2018) 1850089. +[48] Z. Yousaf, K. Bamba, Z. Akhtar and W. Javed, Int. J. Geom. Methods Mod. 19(2022)2250102. +[49] K. Bamba, M. Ilyas, M.Z.Bhatti, and Z. Yousaf, General Relativity and Gravitation, 49(8) (2017), pp.1-17. +[50] M. Ilyas, Eur. Phys. J. C 78, 757 (2018). +[51] M. Ilyas, International Journal of Modern Physics A 36.24 (2021): 2150165. +[52] M. Ilyas, International Journal of Geometric Methods in Modern Physics 16.10 (2019): 1950149. +[53] A. Ditta et al., Eur. Phys. J. C (2022) 82:756. +[54] L. Hui, A. Nicolis, Phys. Rev. Lett. 110, 241104 (2013) +[55] E. Babichev, C. Charmousis, A. Leh´ebel, JCAP 04, 027 (2017) +[56] A. Nicolis, R. Rattazzi, E. Trincherini, Phys. Rev. D 79, 064036 (2009) +[57] R. K. Walia, S. D. Maharaj and S. G. Ghosh, Eur. Phys. J. C 82(2022)547. +[58] R. Kumar, S. G. Ghosh, A. Wang, Phys. Rev. D 101(2020)104001. +[59] R. Kumar, S. G. Ghosh, Eur. Phys. J. C 78(2018)750. +[60] D. Y. Chen, Q. Q. Jiang, X. T. Zua, Phys. Lett. B 665(2008)106. +[61] R. P. Kerr, Phys. Rev. Lett. 11(1963)237. + diff --git a/JdE0T4oBgHgl3EQfSACX/content/tmp_files/load_file.txt b/JdE0T4oBgHgl3EQfSACX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e03e40b6c99fd88973b75b0add6038189cb90a0 --- /dev/null +++ b/JdE0T4oBgHgl3EQfSACX/content/tmp_files/load_file.txt @@ -0,0 +1,1549 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf,len=1548 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='02216v1 [gr-qc] 5 Jan 2023 Logarithm Corrections and Thermodynamics for Horndeski gravity like Black Holes Riasat Ali,1, ∗ Zunaira Akhtar,2, † Rimsha Babar,3, ‡ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Mustafa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' § and Xia Tiecheng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¶ 1Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shanghai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shanghai-200444,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' People’s Republic of China 2Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' University of the Punjab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Quaid-e-Azam Campus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lahore 54590,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Pakistan 3Division of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' University of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Township,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lahore-54590,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Pakistan 4Department of Physics Zhejiang Normal University Jinhua 321004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' People’s Republic of China In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' we compute the Hawking temperature by applying quantum tunneling approach for the Horndeski like black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We utilize the semi-classical phenomenon and WKB approximation to the Lagrangian field equation involving generalized uncertainty principle (GUP) and compute the tunneling rate as well as Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For the zero gravity parameter, we obtain results consistent without correction parameter or original tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, we study the thermal fluctuations of the considered geometry and examine the stable state of the system by heat capacity technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We also investigate the behaviour of thermodynamic quantities under the influence of thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We observe from the graphical analysis, the corresponding system is thermodynamically stable with these correction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' keywords: Horndeski like black holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Quantum gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Tunneling radiation, Thermal fluctuations, Corrected entropy, Phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' INTRODUCTION Tunneling is the semi-classical mechanism in which particles have radiated from black hole (BH) outer horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Some analysis shows the keen interest in the Hawking temperature (TH) via tunneling method from different BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The main aspect to examine the TH is the imaginary part of classical action which leads to the tunneling radiation of boson particles appearing from the Horndeski like BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The quantum tunneling and TH of charged fermions in BHs has been observed [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In this paper, they examined that the tunneling and TH depend on charges of electric and magnetic, acceleration, rotation, mass and NUT parameter of the charged pair BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The tunneling strategy from Reissner Nordstom-de Sitter BH like solution in global monopole has been analyzed [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In this article, the authors observed that the modified TH depends on the parameter of global monopole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The BH thermodynamics have been examined [3] with some parameters like acceleration, NUT and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The researchers studied thermodynamical quantities like the area, entropy, surface gravity and TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The tunneling spectrum of bosonic particles has been computed from the modified BHs horizon by utilizing the Proca field equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Hawking evaluated tunneling probability from BH [4] by utilizing theoretical technique and later, it has been explained by Parikh and Wilczek [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The important of this radiation represents that vacuum thermal fluctuation produce pairs of particle (particle and anti-particle) from the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Hawking considered that the particle’s have ability to emit from the BH and the anti-particles have no ability to radiate from the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Parikh and Wilczek explained a mathematical approach by utilizing WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' This phenomenon use geometrical optic approximation which is another view of eikonal approximation in wave clarification [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The set of all particles remain at the front boundary and with the emission of these particles, the BH mass reduces in the form of particles energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the Parikh-Wilczek method, a precisely tunneling was established and there were as still unanswered problems like information release, temperature unitary and divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Many authors have made efforts on the tunneling strategy and semi-classical phenomenon from the different BHs horizon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' one of the important explanations can be checked in [8]-[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The radiate particles for many BHs have been analyzed and also computed the radiate particles with the influences of the geometry of BH with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is possible to study modified thermodynamic properties of BH by considering generalized uncertainty principle (GUP) influences [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The GUP implies high energy ∗Electronic address: riasatyasin@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='com †Electronic address: zunaira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='pu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='com ‡Electronic address: rimsha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='babar10@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='com §Electronic address: gmustafa3828@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='com ¶Electronic address: xiatc@shu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='cn 2 result to thermodynamic of BH, by considering the quantum gravity theory with a minimal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' By considering the GUP influences, it is viable to examine the modified thermodynamic of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is a well known fact that thermal fluctuations are a result of statistical perturbations in the dense matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' With the emission of Hawking radiations from the BH, the size of BH reduces and consequently its temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Faizal and his colleague [33] have studied the thermodynamics and thermal fluctuations of generalized Schwarschild BH, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=', Reissner-Nordstrom, Kerr and charged AdS BHs) with the help first-order corrections and discussed the stability of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The thermodynamics of rotating Kerr-AdS BH and its phase transition have been studied by Pourhassan and Faizal [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' They concluded that the entropy corrections are very helpful to examine the geometry of small BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' By applying the stability test of heat capacity and Hessian matrix, the phase transition as well as thermodynamics of non-minimal regular BHs in the presence of cosmological constant has been investigated [35] and the authors concluded that the local and global stability of the corresponding BHs increases for higher values of correction parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Zhang and Pradhan [36, 37] have investigated the corrected entropy and second order phase transition via thermal fluctuations on the charged accelerating BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, the thermodynamics and geometrical analysis of new Schwarzschild BHs have been studied [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' By using the tunneling approach under the influence of quantum gravity the Hawking temperture for different types of BH have been discussed [40]-[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sharif and Zunaira [45, 46] have computed the thermodynamics, quasi-normal modeand thermal fluctuations of charged BHs with the help of Weyl corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The authors found that the system is unstable for the small radii of BHs under the influence of first order corrections and by using the heat capacity and Hessian matrix technique, they have also studied the stable conditions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The authors in [47, 48] have investigated the thermodynamics, phase transition and local/global stability of NUT BH via charged, accelerating as well as rotating pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ilyas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [49–52] discussed the energy conditions and calculated the new solutions for stellar structures by taking black hole geometry as exterior spacetime in the background of different modified theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Recently, Ditta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [53] discussed the thermal stability and Joule–Thomson expansion of regular BTZ-like black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The main intention of this paper is to investigate the tunneling radiation without self-gravity and back-reaction and also explain the modified tunneling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The tunneling radiation is evaluated under the conditions of charge-energy conservation, Horndeski parameter and GUP parameter influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The modified TH depends on the Horndeski parameter as well as GUP parameter and also investigated the behaviour of thermodynamic quantities via thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' This paper is based on the analysis of quantum tunneling, TH, stability and instability conditions for the Horndeski like BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The paper is outlined as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' II, we study the tunneling radiation of bosonic particles for 4D Horndeski like BH and also calculate the effects of GUP parameters on tunneling and TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In section III, we study the graphical presentation of tunneling radiation for this type of BH and analyze the stable and unstable conditions for Horndeski like BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In section IV, we investigate the behaviour of thermodynamic quantities under the effects of thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In section V, we express the discussion and conclusion of the whole analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' HORNDESKI LIKE BLACK HOLES Hui and his coauthor Nicolis [58] argued that the no-hair theorems cannot be applied on a Galileon field, as it is coupled to gravity under the effect of peculiar derivative interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Further, they demonstrated that static and spherically symmetric spacetime defining the geometry of the black hole could not sustain nontrivial Galileon profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babichev and Charmousis [59] examined the no-hair theorem in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [58] by considering Horndeski theories and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Furthermore, they provided the Lagrangian of Horndeski theory which can be expressed as a generalized Galileon Lagrangian, which is defined as S = � √−g � Q2(χ) + Q3(χ)□φ + Q4(χ)R + Q4,χ � (□φ)2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' − (∇ǫ∇εφ) (∇ǫ∇εφ)] + Q5(χ)Gǫε∇ǫ∇εφ − 1 6Q5,χ � (□φ)3 − 3(□φ) (∇ǫ∇vφ) (∇ǫ∇εφ) +2 (∇ǫ∇εφ) (∇v∇γφ) (∇γ∇ǫφ)]} d4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (1) where Q2, Q3, Q4, and Q5 are the arbitrary functions of the scalar field φ and χ = −∂ǫφ∂ǫφ/2 represents the canonical kinetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Additionally, in the current analysis, fχ stands for ∂f(χ)/∂χ, Gǫε is the Einstein tensorR is the Ricci scalar, and other relations are defined as: (∇ǫ∇εφ)2 ≡ ∇ǫ∇εφ∇ε∇ǫφ (∇ǫ∇εφ)3 ≡ ∇ǫ∇εφ∇ε∇ρφ∇ρ∇ǫφ (2) 3 The scalar field admits the Galilean shift symmetry ∂ǫφ → ∂ǫφ + bǫ in flat spacetime for Q2 ∼ Q3 ∼ χ and Q4 ∼ Q5 ∼ χ2, which resembles the Galilean symmetry [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the current study, we investigate the tunneling radiation of spin-1 massive boson particles from Horndeski-like BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For this purpose, we adopted the procedure, which is already reported in [57] for Horndeski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Finally, we have the following spacetime: ds2 = − � 1 − 2rM(r) Σ � dt2 + 1 ∇(r)Σdr2 + Σ2dθ2 − A Σ sin2 dφ2 − 4ar Σ M(r) sin2 θdtdφ, (3) with Σ2 = a2 cos2 θ + r2 and ∇(r) = a2 − 2rM(r)+ r2, M(r) = M − 1 2QIn r r0 and A = (a2 + r2)2 − ∇a2 sin2 θ, while a, Q and M represent the rotation parameter, Horndeski parameter and mass of BH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' If Q → 0, the metric (3) goes over to the Kerr BH [61] and if Q = a = 0 the metric (3) also goes over to the Schwarzschild metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The line-element (3) can be re-written as ds2 = −f(r)dt2 + g−1(r)dr2 + I(r)dφ2 + h(r)dθ2 + 2R(r)dtdφ (4) where f(r) = � 1 − 2rM(r) Σ � , g−1(r) = 1 ∇Σ, h(r) = Σ2, I(r) = −A Σ sin2, R(r) = −2ar Σ M(r) sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We study the tunneling radiation of spin-1 particles from four-dimensional Horndeski like BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' By utilizing the Hamilton-Jacobi ansatz and the WKB approximation to the modified field equation for the Horndeski space-time, the tunneling phenomenon is successfully applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We study the modified filed equation on a four dimensional space-time with the background of rotation parameter, Horndeski parameter and evaluated for the radial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' As a result, we get the tunneling probability of the radiated particles and derive the modified TH of Horndeski like BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The modified filed equation is expressed by [27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 30] ∂µ �√−gΨνµ� + √−g m2 ℏ2 Ψν + √−g i ℏAµΨνµ + √−g i ℏeF νµΨµ + ℏ2β∂0∂0∂0 �√−gg00Ψ0ν� −ℏ2β∂i∂i∂i �√−ggiiΨiν� = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (5) here Ψνµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' m and g present the anti-symmetric tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' bosonic particle mass and determinant of coefficient matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' so Ψνµ = � 1 − ℏ2β∂2 ν � ∂νΨµ − � 1 − ℏ2β∂2 µ � ∂µΨν + � 1 − ℏ2β∂2 ν � i ℏeAνΨµ − � 1 − ℏ2β∂2 ν � i ℏeAµΨν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' and Fνµ = ∇νAµ − ∇µAν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' with β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' e ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ∇µ and Aµ are the GUP parameter(quantum gravity),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' bosonic particle charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' covariant derivative and BH potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The Ψνµ can be computed as Ψ0 = −IΨ0 + RΨ3 fI + R2 , Ψ1 = 1 g−1 Ψ1, Ψ2 = 1 hΨ2, Ψ3 = RΨ0 + fΨ3 fI + R2 , Ψ01 = ˜ −DΨ01 + RΨ13 (R2 + fI)g−1 , Ψ02 = ˜ −DΨ02 (R2 + fI)h, Ψ03 = (f 2 − fI)Ψ03 (fI + R2)2 , Ψ12 = 1 g−1hΨ12, Ψ13 = 1 g−1(fI + R2)Ψ13, Ψ23 = fΨ23 + RΨ02 (fI + R2)h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In order to observe the bosonic tunneling, we have assumed Lagrangian gravity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Further, we utilized the WKB approximation to the Lagrangian gravity equation and computed set of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Furthermore, we have utilized the variable separation action to get required solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The approximation of WKB is defined [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ] as Ψν = ην exp � i ℏK0(t, r, φ, θ) + ΣℏnKn(t, r, φ, θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (6) we get set of equations in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Utilizing variable separation technique, we can take K0 = −(E − Lω)t + W(r) + Lφ + ν(θ), (7) 4 where E and L present the particle energy and particle angular, respectively, corresponding to angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' After considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (7) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (22)-(25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' we reach a matrix in the form U(η0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' η3)T = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' which express a 4 × 4 matrix presented as ”U”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' whose elements are given as follows: U00 = −I g−1(fI + R2) � W 2 1 + βW 4 1 � − I (fI + R2)h � L2 + βL4� − fI (fI + R2)2 � ν2 1 + βν4 1 � − m2I (fI + R2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U01 = −I g−1(fI + R2) � ((E − Lω) + (E − Lω)3β + A0e + (E − Lω)2βeA0 � W1 + R g−1(fI + R2) + � ν1 + βν3 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U02 = −I h(fI + R2) � (E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0 � L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U03 = −R g−1(fI + R2) � W 2 1 + βW 4 1 � − fI h(fI + R2)2 � (E − Lω)3β − (E − Lω)2βeA0 + (E − Lω) − eA0 � ν1 + m2R (fI + R2)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U12 = 1 g−1h � W1 + βW 3 1 � L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U11 = −I g−1(fI + R2) � β(E − Lω)4 − βeA0EW 2 1 + (E − Lω)2 − eA0(E − Lω) � + R (fI + R2)g−1 + � ν1 + βν3 1 � (E − Lω) − 1 g−1h � L2 + βL4� − 1 (fI + R2)g−1 � ν1 + βν3 1 � − m2 g−1 − eA0I (fI + R2)g−1 × � (E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0 � + eA0R g−1(fI + R2) � ν1 + βν3 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U13 = −R g−1(fI + R2) � W1 + βW 3 1 � (E − Lω) + 1 g−1(fI + R2)2 � W1 + βW 3 1 � ν1 + ReA0 g−1(fI + R2) � W1 + βW 3 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U20 = I h(fI + R2) � (E − Lω)L + β(E − Lω)L3� + R h(fI + R2) � (E − Lω) + β(E − Lω)3ν2 1 � − IeA0 h(fI + R2) � L + βL3� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U22 = I h(R2 + fI) � βE4 − βeA0E + E2 − eA0(E − Lω) � − 1 g−1h + R h(R2 + fI) � (E − Lω)3β + −(E − Lω)2βeA0 − A0e + (E − Lω) � ν1 − f h(R2 + fI) � ν2 1 + βν4 1 � − m2 h − eA0I h(fI + R2) � (E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U23 = f h(fI + R2) � L + βL3� ν1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U30 = (fI − f 2) (fI + R2)2 � ν1 + βν3 1 � E + R h(fI + R2) � L2 + βL4� − m2R (fI + R2) − eA0(fI − f 2) (fI + R2)2 � ν1 + βν3 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U31 = 1 g−1(fI + R2) � ν1 + βν3 1 � W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U32 = R h(R2 + fI) � L + βL3� E + f h(R2 + fI) � ν1 + βν3 1 � L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U33 = (fI − f 2) (fI + R2) � (E − Lω)2 − eA0(E − Lω) + β(E − Lω)4 − βeA0(E − Lω)3� − 1 g−1(R2 + fI) � W 2 1 + βW 4 1 � − f (R2 + fI)h � L2 + βL4� − m2f (fI + R2) − eA0(fI − f 2) (fI + R2) × � (E − Lω) + β(E − Lω)3 − eA0(E − Lω)2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 5 with ∂tK0 = (E − Lω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ∂φK0 = L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' W1 = ∂rK0 and ν1 = ∂θK0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For non-trivial solution, we get ImW ± = ± � � � � � � E − LΩ − eA0 �2 + Z1 � 1 + β Z2 Z1 � (fI + R2)gI−1 dr, = ±iπ � R − LΩ − A0e � + � 1 + βA � 2k(r+) , (8) where Z1 = (E − Lω)ν1 g−1R fI + R2 + fg−1 fI + R2 ν2 1 − g−1m2, Z2 = g−1I fI + R2 � (R − LΩ)4 + (eA0)2(R − LΩ)2 − 2A0e(R − LΩ)3� + g−1R h(fI + R2) � (R − LΩ)3 − eA0(R − LΩ)2� ν1 − fg−1 fI + R2 ν4 1 − W 4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' and A is a arbitrary parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In particular case, we take the radial component of the action of particle, for this aim we choose a components of matrix equals to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Since, we have found the tunneling radiation (related to Horndeski gravity and quantum gravity) for BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' This tunneling and TH quantities relate on the Horndeski gravity and quantum gravity of this particular physical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Thus, we have found the corresponding TH which important as a component of leading metric with Horndeski gravity and quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Such that, in this method we are not concerned in order of higher for Planck’s constant only obtained appropriate result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The generalized tunneling depends on the BHs metric and Horndeski gravity and GUP parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The generalized tunneling for Horndeski like BH can be written as T = Temission Tabsorption = exp � −2π(R − LΩ − A0e) k(r+) � [1 + βA] , (9) with k(r+) = 4πr+ � a2 + r2 + � Qr+ + −a2 + r2 + (10) In the presence of GUP terms, we calculate the TH of the Horndeski gravity BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' by taking the Boltzmann factor TB = exp [(E − Lω − eA0)/TH] as TH = −a2 + Qr+ + r2 + 4πr+ � a2 + r2 + � [1 − βA] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (11) The above result shows that the TH depends on the Horndeski gravity, GUP parameter, rotation parameter, Horndeski gravity, arbitrary parameter A and radius (r+) of BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' When β = 0, we obtain the general TH in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the absence of charge i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=', Q = 0, the above temperature reduces into Kerr BH temperature [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For β = 0 and a = 0, the temperature reduces into Reissner Nordstr¨om BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, when Q = 0 = a, we recover the temperature of Schwarzschild BH [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The quantum corrections slow down the increase in TH throughout the radiation phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' TH versus r+ We observe the geometrical presentation of TH w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='t r+ for the 4D Horndeski like metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, we observe the physical significance of these graphs under Horndeski gravity and GUP parameter and study the stability and instability analysis of corresponding TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For β equals to zero, the tunneling radiation will be independent of GUP parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 1, the TH increases with increasing β in small region of horizon 0 ≤ r+ ≤ 5, that indicates the stable state of BH till r+ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 1, the rotating parameter and β are fixed, then we take changing values of hairy parameter of Horndeski gravity and get the completely unstable form of BH with negative temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 6 β=10 β=20 β=30 0 1 2 3 4 5 0 1 2 3 4 5 6 r+ TH Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 Q=1 Q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 0 1 2 3 4 5 6 4 2 0 2 r+ TH Figure 1: TH w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='t horizon r+ for a = 5, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 and Ξ = 1 and left β = 10 (black), β = 20 (blue), β = 30 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Right a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5, Ξ = 1, β = 5, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 (black), a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5, Ξ = 1, β = 5, Q = 1 (blue), a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5, Ξ = 1, β = 5, Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' THERMODYNAMICS AND EFFECTS OF FIRST ORDER CORRECTIONS Thermal fluctuations plays important role on the study of BH thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' With the concept of Euclidean quantum gravity, the temporal coordinates shifts towards complex plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' To check the effects of these correction in entropy, we find Hawking temperature and usual entropy of the given system with the help of first law of thermody- namics S = π � a2 + r2 + � , T = −a2 + Qr+ + r2 + 4πr+ � a2 + r2 + � (12) To check the corrected entropy along these thermal fluctuations, the partition function is Z(µ) in terms of density of states η(E) is given as [37] Z(µ) = � ∞ 0 exp(−µE)η(E)dE, (13) where T+ = 1 µ and E is the mean energy of thermal radiations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' By using the Laplace inverse transform, the expression of density takes the form ρ(E) = 1 2πi � µ0+i∞ µ0−i∞ Z(µ) exp(µE)dµ = 1 2πi � µ0+i∞ µ0−i∞ exp( ˜S(µ))dµ, (14) where ˜S(µ) = µE + ln Z(µ) represents the modified entropy of the considered system that is dependent on Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, the expression of entropy gets modified with the help of steepest decent method, ˜S(µ) = S + 1 2(µ − µ0)2 ∂2 ˜S(µ) ∂µ2 ��� µ=µ0 + higher-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (15) Using the conditions ∂ ˜S ∂µ = 0 and ∂2 ˜S ∂µ2 > 0, the corrected entropy relation under the first-order corrections modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' By neglecting higher order terms, the exact expression of entropy is expressed as ˜S = S − δ ln(ST 2), (16) where δ is called correction parameter, the usual entropy of considered system is attained by fixing δ = 0 that is without influence of these corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Furthermore, inserting the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (12) into (16), we have ˜S = (a2 + r2 +) − δ log �� a2 − r+ (Q + r+) �2 16πr2 + � a2 + r2 + � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (17) 7 δ=0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 10 20 30 40 r+ S ˜ Figure 2: Corrected entropy versus r+ for a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2, Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 2, the graph of corrected entropy is monotonically increasing throughout the considered domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is noted the graph (black) of usual entropy is increasing just for small value of horizon radius but corrected expression of energy is increasing smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Thus, these corrections terms are more effective for small BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Now, using the expression of corrected entropy and check the other other thermodynamic quantities via thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In this way, the the Helmholtz energy (F = − � ˜SdT ) leads to the form F = � a4 − r2 + � −4a2 + 2Qr+ + r2 + �� � δ log � (a2−r+(Q+r+)) 2 { r2 + � a2 + r2 + � } � − a2 − δ log(16π) − r2 + � 4πr2 + � a2 + r2 + �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (18) δ=0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='30 35 40 45 50 55 60 65 r+ F Figure 3: Helmholtz free energy versus r+ for a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2, Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='. The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 3 shows the graph of Helmholtz free energy versus horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is observed that the behaviour of energy is gradually decreases for the different correction parameter δ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' While the graph of usual entropy shows opposite behaviour as the graph is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' This behaviour means, the considered system shifts its state towards equilibrium, thus, no more work can be extract from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The expression of internal energy (E = F + T ˜S) for the corresponding geometry is given by [37] E = � r+ � a2 + r2 + � � r+ (Q + r+) − a2� � δ � log(16π) − log �� a2 − r+ (Q + r+) � 2 r2 + � a2 + r2 + � �� + π � a2 + r2 + � � − � a4 − r2 + � −4a2 + 2Qr+ + r2 + �� � −δ log �� a2 − r+ (Q + r+) �2 r2 + � a2 + r2 + � � + a2 + δ log(16π) + r2 + � � � 4πr2 + � a2 + r2 + �2 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (19) 8 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 2 0 2 4 6 r+ E Figure 4: Internal Energy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='t r+ for a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2, Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The graphical behaviour of internal energy for the different choices of horizon radius is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is observable that for the small values of radii, the graph is gradually decreases even shifts towards negative side, While the corrected internal energy depicts positive behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' This mean that the considered BH absorbing more and more heat from the surrounding to maintain its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Since, BHs considered as a thermodynamic system, so there is another important thermodynamic quantity that is pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In this regard, there is deep connection between voulme (V = 2π(r2 ++a2)(2r2 ++a2) 3r+ ) and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The Expression of BH pressure (P = − dF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='dV ) under the effect of thermal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='fluctuations takes the form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='P = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 3Qr+ + 2r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+(Q + r+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+(a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+))−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a4δ + Qr3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 + 3a2δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ Qr5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ + r6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 4r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4πr3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (20) δ=0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='0 0 1 ×108 2 �108 3 �108 4 �108 5 �108 r+ P Figure 5: Pressure versus r+ for a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2, Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 5, the graph of pressure is just coincides the state of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For the different values of correction parameter, the pressure is significantly increases for the considered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Further, there is another important thermodynamic quantity enthalpy (H = E + PV ) is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 9 δ=0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='0 10 5 0 5 r+ H Figure 6: Enthalpy versus r+ for a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2, Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 6, it can observed that the graph of usual enthalpy is coincide with the plots of corrected one and abruptly decreases even shifts towards negative side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' This means that there exists a exothermic reactions means there will be huge amount of energy release into its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' By taking into account the thermal fluctuations, the Gibbs free energy (G = H − T ˜S) is expressed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' δ=0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='8 0 5 10 15 20 5 10 15 20 r+ G Figure 7: Gibbs free energy versus r+ for a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2, Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The graphical analysis of Gibbs free energy with respect to horizon radius is shows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The positivity of this energy is sign of occurrence of non-spontaneous reactions means this system requires more energy to gain equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' After the detail discussion of thermodynamics quantities, there is another important concept is the stability of the system that is checked by specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The specific heat (C ˜S = dE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='dT ) is given as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='C ˜S = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + 3r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ(Q − 4) log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2(πQ − 5) + δ(Q − 4) log(16π) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− πa4 − 2δQ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ(Q + 1) log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ πa2 + δ log(16π) + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='πQ + πr+ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 2Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2(2πQ − 3) + δ(Q + 1) log(16π) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 2a2Q + 2δQ log(16π) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ πa2 + δ log(16π) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a4 − 4a2r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ + 2Qr3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ + r4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (21) 10 δ=0 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='6 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='05 r+ CS ~ Figure 8: Specific heat versus r+ for a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2, Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 8, the behaviour of specific heat is observed with respect to horizon radius and different choices of correction parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It can be observed that the uncorrected quantity (black) depicts negative behaviour means the system is unstable while the corrected specific heat shows positive behaviour throughout the considered domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The positivity of this plot is indication of stable region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It can be concluded that these correction terms makes the system stable under thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' DISCUSSION AND RESULT In this paper, we have utilized Lagrangian gravity equation to observe the tunneling of bosonic particles through the horizon of Horndeski like BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We have used the metric from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We have considered a new version of black hole with Horndeski parameter Q and a rotation parameter a, due to the presence of these parameters, we call the metric a new type of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Our results are also in terms of these parameters, therefore they are different from the previous literature related about the thermodynamics of this black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Assuming to relativistic quantum mechanics and the region of vacuum, where particles are produced continuously in the phenomenon of annihilated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The tunneling radiation as a quantum mechanical processes can be observed as a tunneling phenomenon, where positive boson particle radiate the horizon and the negative energy boson particle move inward and absorbed by the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The incoming and outgoing boson particles movement carried out by the action of particle’s is real and complex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The emission rate of these tunneling radiation corresponding to the Horndeski like BH configuration is associated to the imaginary part of the action of particles, which is associated to the factor of Boltzmann, this factor gives TH for Horndeski like BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' From our investigation, we have observed that, in rotating case of BH, the TH at which boson particles tunnel through the BH horizon is not dependent at any types of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In special case when particles have different (zero or upward or down) spins, the tunneling rate will be alike by assuming the semi-classical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Thus, their corresponding TH must be similar for all types of particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Therefore, one can say tunneling radiation is independent of all kinds of the particles and this result also holds for different frame of coordinates by utilizing the transformations of particular coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For this procedure, the tunneling particles is associated to the energy of particles, momentum, quantum gravity, hairy parameter of Horndeski gravity and BH surface gravity, while the temperature depends on hairy parameter of Horndeski gravity, rotation and quantum gravity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is very important to mention here that, when β = 0, we obtain the standard temperature for Horndeski like BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In the absence of charge i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=', Q = 0, the above temperature reduces into Kerr BH temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For β = 0 and a = 0, the temperature reduces into Reissner Nordstr¨om BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, when Q = 0 = a, we recover the temperature of Schwarzschild BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For the changing values of β from 10 to 30 in the region 0 ≤ r+ ≤ 5, we have observed that the Horndeski like BH is stable and for changing Q from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='5, the Horndeski like BH is un-stable with negative temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, the temperature increases with the increasing values of quantum gravity β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Moreover, we have computed TH as well as heat capacity for Horndeski gravity like BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Firstly, the TH is calculated through entropy and the density of state is also calculated with help of inverse Laplace transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' We have observed that the exact entropy of the system depends on Hawking temperature by applying the method of steepest descent under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' In graphically, we have studied the monotonically increasing entropy of the metric throughout the assumed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is observed that a decrease in entropy for a certain value of the radius, the corrected expression of energy is increasing smoothly and also studied that the small BHs are more effective for thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is observed that the behaviour of energy gradually decreases for the different correction parameter δ values and the graph of usual entropy shows opposite behaviour as the graph increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Therefore, the considered system shifts its state towards equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is observable that for the small values of radii, the graph gradually decreases even 11 shifts towards negative side, while the corrected internal energy depicts positive behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' So, the considered BH absorbed more and more heat from the surrounding to maintain its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The graph of pressure coincides the state of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' For the different values of correction parameter, the pressure significantly increases for the considered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It is observed that the graph of usual enthalpy coincides with the plots of corrected one and abruptly decreases even shifts towards negative side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It can be concluded that there exists a exothermic reactions means a huge amount of energy released into its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The positivity of this energy is sign of occurrence of non-spontaneous reactions means the system required more energy to gain equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The behaviour of specific heat with respect to horizon radius and different choices of correction parameter δ has been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The uncorrected quantity depicts negative behaviour means the system is unstable while the corrected specific heat shows positive behaviour throughout the considered domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' The positivity of the plots for specific heat is indication of stable region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' It can be concluded that these correction terms makes the system stable under thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Appendix A We have utilized the Lagrangian equation in the approximation of WKB to get following solutions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' I (R2 + fI)g−1 � η1(∂0K0)(∂1K0) + βη1(∂0K0)3(∂1K0) − η0(∂1K0)2 − βη0(∂1K0)4 + η1eA0(∂1K0) + η1βeA0(∂0K0)2(∂1K0) � − R g−1(fI + R2) � η3(∂1K0)2 + βη3(∂1K0)4 − η1(∂1K0)(∂3K0) − βη1(∂1K0)(∂3K0)2� + I h(fI + R2) � η2(∂0K0)(∂2K0) + βη2(∂0K0)3(∂2K0) − η0(∂2K0)2 − βη0(∂2K0)4 + η2eA0(∂2K0) + η2eA0β(∂0K0)2(∂1K0) � + fI (fI + R2)2 � η3(∂0K0)(∂3K0) + βη3(∂0K0)3(∂3K0) − η0(∂3K0)2 − βη0(∂3K0)4 + η3eA0(∂3K0) + η3eA0(∂0K0)2(∂3K0) � − m2 Iη0 − Rη3 (fI + R2) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='−I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η1(∂0K0)2 + βη1(∂0K0)4 − η0(∂0K0)(∂1K0) − βη0(∂0K0)(∂1K0)3 + η1eA0(∂0K0) + βη1eA0(∂0K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂0K0)(∂1K0) + βη3(∂0K0)(∂1K0)3 − η1(∂0K0)(∂3K0) − βη1(∂0K0)(∂3K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1h [η2(∂1K0)(∂2K0) + βη2(∂1K0)(∂2K0)3 − η1(∂2K0)2 − βη1(∂2K0)4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂1K0)(∂3K0) + βη3(∂1K0)(∂3K0)3 − η1(∂3K0)2 − βη1(∂3K0)4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− m2η1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='eA0I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η1(∂0K0) + βη1(∂0K0)3 − η0(∂1K0) − βη0(∂1K0)3 + eA0η1 + βη1eA0(∂0K0)2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='eA0R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂1K0) + βη3(∂1K0)3 − η1(∂3K0) − βη1(∂1K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η2(∂0K0)2 + βη2(∂0K0)4 − η0(∂0K0)(∂2K0) − βη0(∂0K0)(∂2K0)3 + η2eA0(∂0K0) + βη2eA0(∂0K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='g−1h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η2(∂1K0)2 + βη2(∂1K0)4 − η1(∂1K0)(∂2K0) − βη1(∂1K0)(∂2K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η2(∂0K0)(∂3K0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ βη2(∂0K0)3(∂3K0) − η0(∂0K0)(∂3K0) − βη0(∂0K0)3(∂3K0) + η2eA0(∂3K0) + βη2eA0(∂3K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂2K0)(∂3K0) + βη3(∂2K0)3(∂3K0) − η2(∂3K0)2 − βη2(∂3K0)4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='eA0I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η2(∂0K0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ βη2(∂0K0)3 − η0(∂2K0) − βη0(∂2K0)3 + η2eA0 + η2βeA0(∂0K0)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− m2η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(24) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(fI − f 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(fI + R2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂0K0)2 + βη3(∂0K0)4 − η0(∂0K0)(∂3K0) − βη0(∂0K0)(∂3K0)3 + eA0η3(∂0K0) + βη3eA0(∂0K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂1K0)2 + βη3(∂1K0)4 − η1(∂1K0)(∂3K0) − βη1(∂1K0)(∂3K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η2(∂0K0)(∂2K0) + βη2(∂0K0)3(∂2K0) − η0(∂2K0)2 + βη0(∂2K0)4 + eA0η2(∂2K0) + βη2eA0∂0K0)2(∂2K0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='eA0f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='h(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂2K0)2 + βη3(∂2K0)4 − η2(∂2K0)(∂3K0) − βη2(∂0K0)(∂3K0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− m2(Rη0 − fη3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(fI + R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+eA0(fI − f 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(fI + R2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='η3(∂0K0) + βη3(∂0K0)3 − η0(∂3K0) − βη0(∂3K0)3 + η3eA0 + η3βeA0(∂0K0)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Appendix B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='The thermodynamic quantity enthalpy is given as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='H = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='log(16π) − log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 3Qr+ + 2r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a4δ + Qr3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 + 3a2δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ Qr5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ + r6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 4r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4πr3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�3�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='(26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='The Gibbs free energy is expressed as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='G = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='log(16π) − log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 3Qr+ + 2r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− a4δ + Qr3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 + 3a2δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ Qr5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ + r6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 4r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a4 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− 4a2 + 2Qr+ + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='− δ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ a2 + δ log(16π) + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 − r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Q + r+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='2��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='4πr3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='a2 + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='3�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' (27) [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sharif, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C 72, 1997(2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sharif, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 23(2013)271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sharif, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 91(2013)236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Hawking, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 43(1975)199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Parikh and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Wilczek, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 85(2000)5042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Parikh, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' D 13(2004)2351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Firouzjaee and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ellis, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 47(2015)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 2019(2019)2759641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, Chinese Journal of Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 61(2019)138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, The Fifteenth Marcel Grossmann Meeting, World Scientific 3(2022)905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, Punjab University Journal of Mathematics 52(2020)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A 34(2019)1950057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A 35(2020)2050104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sharif, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Korean Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 57(2010)217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Jusufi, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Plus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 132(2017)298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 55(2016)2919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Jusufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ovgun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Apostolovska, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 2017(2017)8798657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sakalli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Jusufi, Astrophys Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 361(2016)330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [19] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sakalli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, General Relativity and Gravitation 48(2016)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sakalli, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 57(2018)322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sakalli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Saavedra, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Leiva, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A 35(2020)2050163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [22] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sakalli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ovgun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 121(2015)404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Bamba, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Asgher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Malik, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shah, Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 12(2020)1165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Bamba, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Asgher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shah, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' D 30(2021)2150002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Asgher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Malik, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A 35(2020)2050225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Plus 134(2019)511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [27] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, Chinese Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C 44(2020)015104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [28] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Jusufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, Astrophys Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 361(2016)207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 14 [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Asgher, New Astronomy 93(2022)101759 [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sahoo, Physics of the Dark Universe 35(2022)100948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Asgher and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shah, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 19(2022)2250017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Bardeen, in Conference Proceedings of GR5 (Tbilisi, URSS, 1968), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Faizal and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Khalil, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A 30(2015)1550144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [34] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Pourhassan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Faizal, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' B 913(2016)834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Jawad and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shahzad, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C 77(2017)349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Zhang, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' B 935(2018)170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Pradhan, Universe 5(2019)57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [38] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ghaderi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Malakolkalami, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' B 903(2016)10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [39] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Zheng, arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='05470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Asgher, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shah, Annals of Physics, 432(2021)168572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [41] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Abbas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C 77(2017)296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' ¨Ovg¨un, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 2018(2018)11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [43] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Abbas, Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 97(2018)176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ali, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Bamba, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Shah, Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 631(2019)11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sharif and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Akhtar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Dark Universe 29(2020)100589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Sharif and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Akhtar, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys 71(2021)669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [47] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Yousaf and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Akhtar, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' A 33(2018) 1850089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [48] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Yousaf, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Bamba, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Akhtar and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Javed, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 19(2022)2250102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [49] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Bamba, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ilyas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='Bhatti, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Yousaf, General Relativity and Gravitation, 49(8) (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='1-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ilyas, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C 78, 757 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [51] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ilyas, International Journal of Modern Physics A 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='24 (2021): 2150165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ilyas, International Journal of Geometric Methods in Modern Physics 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content='10 (2019): 1950149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ditta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=', Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C (2022) 82:756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [54] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Hui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Nicolis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 110, 241104 (2013) [55] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Babichev, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Charmousis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Leh´ebel, JCAP 04, 027 (2017) [56] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Nicolis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Rattazzi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Trincherini, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' D 79, 064036 (2009) [57] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Walia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Maharaj and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ghosh, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C 82(2022)547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [58] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ghosh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' D 101(2020)104001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [59] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Ghosh, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' C 78(2018)750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [60] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Zua, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' B 665(2008)106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' [61] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Kerr, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE0T4oBgHgl3EQfSACX/content/2301.02216v1.pdf'} +page_content=' 11(1963)237.' 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a/K9E0T4oBgHgl3EQfigH5/content/tmp_files/2301.02448v1.pdf.txt b/K9E0T4oBgHgl3EQfigH5/content/tmp_files/2301.02448v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..011f543b5ae15ba6edc1a9b08a26f3c6cc0a82c4 --- /dev/null +++ b/K9E0T4oBgHgl3EQfigH5/content/tmp_files/2301.02448v1.pdf.txt @@ -0,0 +1,2484 @@ +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for composite +quantile regression with distributed data +Xiaohui Yuan1, Shiting Zhou1† and Yue Wang1*† +1*School of Mathematics and Statistics, Changchun University of +Technology, Changchun, 130012, Jilin, China. +*Corresponding author(s). E-mail(s): wangyueccut@gmail.com; +Contributing authors: yuanxh@ccut.edu.cn; +zhoushiting1999@outlook.com; +†These authors contributed equally to this work. +Abstract +For massive data stored at multiple machines, we propose a dis- +tributed subsampling procedure for the composite quantile regres- +sion. By establishing the consistency and asymptotic normality of +the +composite +quantile +regression +estimator +from +a +general +sub- +sampling algorithm, we derive the optimal subsampling probabili- +ties and the optimal allocation sizes under the L-optimality cri- +teria. A two-step algorithm to approximate the optimal subsam- +pling +procedure +is +developed. +The +proposed +methods +are +illus- +trated through numerical experiments on simulated and real datasets. +Keywords: Composite quantile regression, Distributed data, Massive data, +Optimal subsampling +1 Introduction +With the rapid development of science and technology, extremely large datasets +are ubiquitous and lays heavy burden on storage and computation facilities. +Many efforts have been made to deal with these challenge. There are three +main directions from the view of statistical applications: divide-and-conquer, +online updating, and subsampling. Among them, subsampling has been found +1 +arXiv:2301.02448v1 [stat.CO] 6 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Optimal subsampling algorithm for CQR with distributed data +to be useful for reducing computational burden and extracting information +from massive data. +The idea of subsampling was first proposed by Jones (1956)[5]. A key +tactic of subsampling methods is to specify nonuniform sampling probabil- +ities to include more informative data points with higher probabilities. For +example, the leverage score-based subsampling in Ma et al. (2015)[6], the +information based optimal subdata selection in Wang et al. (2019)[12], and +the optimal subsampling method under the A-optimality criterion in Wang et +al. (2018)[11]. Recently, Fang et al. (2021)[2] applied subsampling to a weak- +signal-assisted procedure for variable selection and statistical inference. Ai et +al. (2021)[1] studied the optimal subsampling method for generalized linear +models under the A-optimality criterion. Shao et al. (2022)[8] employed the +optimal subsampling method to ordinary quantile regression. +Due to the large scale and fast arrival speed of data stream, massive data +are often partitioned across multiple servers. For example, Walmart stores +produce a large number of data sets from different locations around the world, +which need to be processed. However, it is difficult to transmit these datasets to +a central location. For these datasets, it is common to analyze them on multiple +machines. Qiu et al. (2020)[7] constructed a data stream classification model +based on distributed processing. Sun et al. (2021)[10] proposed a data mining +scheme for edge computing based on distributed integration strategy. Zhang +and Wang (2021)[17] proposed a distributed subdata selection method for +big data linear regression model. Zuo et al. (2021)[19] proposed a distributed +subsampling procedure for the logistic regression. Yu et al. (2022)[16] derived +a optimal distributed Poisson subsampling procedure for the maximum quasi- +likelihood estimators with massive data. +In the paper, we investigate the optimal distributed subsampling for com- +posite quantile regression (CQR; Zou and Yuan (2008)[18]) in massive data. +In a linear model, composite quantile regression can uniformly estimate the +regression coefficients under heavy tail error. Moreover, since the asymptotic +variance of the composite quantile regression estimate does not depend on +the moment of the error distribution, the CQR estimator is robust. The CQR +method is widely used in many fields. For massive data, Jiang et al. (2018)[3] +proposed a divide-and-conquer CQR method. Jin and Zhao (2021)[4] proposed +a divide-and-conquer CQR neural network method. Wang et al. (2021)[13] +proposed a distributed CQR method for the massive data. Shao and Wang +(2022)[9] and Yuan et al. (2022)[15] developed the subsampling for composite +quantile regression. To the best of our knowledge, there is almost no work on +random subsampling for composite quantile regression with distributed data. +Based on the above motivation, we investigate the optimal subsampling +for the composite quantile regression in massive data when the datasets are +stored at different sites. We propose a distributed subsampling method in the +context of CQR, and then study the optimal subsampling technology for data +in each machine. The main advantages of our method are as follows: First, +we establish the convergence rate of the subsample-based estimator, which + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +3 +ensures the consistency of our proposed method. Second, it avoids the impact +of different intercept items in data sets stored at different sites. Third, the +computational speed of our subsampling method is much faster than the full +data approach. +The rest of this article is organized as follows. In Section 2, we propose the +distributed subsampling algorithm based on composite quantile regression. The +asymptotic properties of estimators based on subsamples are also established. +We present a subsampling strategy with optimal subsampling probability and +optimal allocation size. The simulation studies are given in Section 3. In Section +4, we study the real data sets. The content of the article is summarized in +Section 5. All proofs are given in the Appendix. +2 Methods +2.1 Model and notation +Consider the following linear model +yik = xT +ikβ0 + εik, i = 1, · · · , nk, k = 1, · · · , K, +(1) +where xik denotes a p-dimensional covariate vector, β0 = (β1, · · · , βp)T ∈ Θ +is a p-dimensional vector of regression coefficients, nk is the sample size of +the kth dataset, n = �K +k=1 nk is the total sample size, and K is the number +of distributed datasets. Assume that the random error εik has cumulative +distribution function F(·) and probability density function f(·). +Let M be the composite level of composite quantile regression, which does +not depend on the sample size n. Given M, let τm, m = 1, · · · , M be the speci- +fied quantile levels such that τ1 < · · · < τM. Write θ0 = (θ01, · · · , θ0(p+M))T = +(βT +0 , bT +0 )T and b0 = (b01, · · · , b0M)T, where b0m = inf{u : F(u) ≥ τm} for +m = 1, · · · , M. In this paper, we assume that xik’s are nonrandom and are +interested in inferences about the unknown θ0 from the observed dataset +Dn = {Dkn = {(xT +ik, yik), i = 1 · · · , n}, k = 1, · · · , K}. +For τ ∈ (0, 1), u ∈ Rp, let ρτ(u) = u{τ − I(u < 0)} be the check loss function +for the τ-th quantile level. The CQR estimator of θ based on the full dataset +Dn is given by +ˆθF = (ˆβ +T +F , ˆb +T +F )T = arg min +β,b +K +� +k=1 +nk +� +i=1 +M +� +m=1 +ρτm(yik − bm − xT +ikβ), +(2) +Our aim is to construct a subsample-based estimator, which can be used to +effectively approximate the full data estimator ˆθF . + +Springer Nature 2021 LATEX template +4 +Optimal subsampling algorithm for CQR with distributed data +2.2 +Subsampling algorithm and asymptotic properties +In this subsection, we propose a distributed subsampling algorithm to approx- +imate the ˆθF . First we propose a subsampling method in Algorithm 1, which +can reasonably select a subsample from distributed data. +Algorithm 1 Distributed Subsampling Algorithm£º +• Sampling: Assign subsampling probabilities {πik}nk +i=1 for the kth dataset +Dk = {(yik, xik), i = 1, · · · , nk} with �nk +i=1 πik = 1, where k = 1, · · · , K. +Given total sampling size r, draw a random subsample of size rk with replace- +ment from Dk according to {πik}nk +i=1, where {rk}K +k=1 are allocation sizes +with �K +k=1 rk = r. For i = 1, · · · , nk and k = 1, · · · , K, we denote the cor- +responding responses, covariates, and subsampling probabilities as y∗ +ik, x∗ +ik +and π∗ +ik, respectively. +• Estimation: Based on the subsamples {(y∗ +ik, x∗ +ik, π∗ +ik), i = 1, · · · , rk}K +k=1, and +calculate the estimate ˜θs = (˜βs, ˜bs) = arg minθ Q∗(θ), where +Q∗(θ) = 1 +n +K +� +k=1 +r +rk +rk +� +i=1 +M +� +m=1 +ρτm(y∗ +ik − βTx∗ +ik − bm) +π∗ +ik +. +To establish asymptotic properties of the subsample-based estimator ˜θs, +we need the following assumptions: +(A.1) Assume that f(t) is continuous with respect to t and 0 < f(b0m) < ++∞ for 1 ≤ m ≤ M. Let ˜xik,m = (xT +ik, eT +m)T, where em denotes a M ×1 vector, +which has a one only in its mth coordinate and is zero elsewhere. Define +En = 1 +n +K +� +k=1 +nk +� +i=1 +M +� +m=1 +f(b0m)˜xik,m(˜xik,m)T. +(3) +Assume that there exist positive definite matrices E, such that +En −→ E, +and +max +1≤k≤K,1≤i≤nk ∥xik∥ = o(n1/2). +(A.2) Assume that, for k = 1, · · · , K. +max +1≤k≤K,1≤i≤nk +∥xik∥ + 1 +rkπik += op +� n +r1/2 +� +. +(4) +Define +V π = 1 +n2 +K +� +k=1 +r +rk +nk +� +i=1 +1 +πik +� M +� +m=1 +{I(εik < b0m) − τm}˜xik,m +�⊗2 +, +(5) + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +5 +where for a vector a, a⊗2 = aaT. Assume that there exist positive definite +matrices V such that +V π +p +−→ V , +where +p +−→ means convergence in probability. +Theorem 1. If Assumptions (A.1) and (A.2) hold, conditional on Dn, as +n → ∞ and r → ∞, if r/n = o(1), then we have +Σ−1/2√r(˜θs − θ0) +d +−→ N(0, I), +(6) +where +d +−→ denotes convergence in distribution, Σ = E−1 +n V πE−1 +n . +2.3 Optimal subsampling strategy +Given r, we specify the subsampling probablities {πik}nk +i=1, and the alloca- +tion sizes {rk}K +k=1 in Algorithm 1. A naive choice is the uniform subsampling +strategy with {πik = 1/nk}nk +i=1 and {rk = [rnk/n]}K +k=1, where [·] denotes the +rounding operation. However, this uniform subsampling method is not opti- +mal. As suggested by Wang et al. (2018)[11], we adopted the nonuniform +subsampling strategy to determine the optimal allocation sizes and optimal +subsampling probabilities by minimizing the trace of Σ in Theorem 1. +Since Σ = E−1 +n V πE−1 +n , the optimal allocation sizes and subsampling prob- +abilities require the calculation of En, which depend on the unknown density +function f(·). Following Wang and Ma (2021)[14], we derive optimal subsam- +pling probabilities under the L-optimality criterion. Note that En and V π are +nonnegative definite. Simple matrix algebra yields that tr(Σ) = tr(V πE−2 +n ) = +tr(E−2 +n )tr(V π). Σ depends on rk and πik only through V π, and En is free of +rk and πik. Hence, we suggest to determine the optimal allocation sizes and +optimal subsampling probabilites by directly minimizing tr(V π) rather than +tr(Σ), which can effectively speed up our subsampling algorithm. +Theorem 2. If rk and πik, i = 1, · · · , nk, k = 1, · · · , K, are chosen as +πLopt +ik += πLopt +ik +(θ0) = +∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +�nk +i=1 ∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +, +(7) +and +rLopt +k += r +�nk +i=1 ∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +�K +k=1 +�nk +i=1 ∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +, +(8) +then tr(V π)/n attains its minimum. + +Springer Nature 2021 LATEX template +6 +Optimal subsampling algorithm for CQR with distributed data +2.4 Two-step algorithm +Note that the optimal subsampling probabilities and allocation sizes depend +depends on εik = yik − xT +ikβ0 and b0m, m = 1, · · · , M. The L-optimal weight +result is not directly implementable. To deal with this problem, we use a pilot +estimator ˜θ to replace θ0. In the following, we propose a two-step subsampling +procedure in Algorithm 2. +Algorithm 2 Two-Step Algorithm£º +• Step 1: Given r0, we run Algorithm 1 with subsampling size rk = [r0 +nk +n ] to +obtain a pilot estimator ˜θ, using πik = 1/nk, where [·] denotes the rounding +operation. Replace θ0 with ˜θ0 in (7) and (8) to get the allocation sizes rk(˜θ) +and subsampling probabilities πik(˜θ), for i = 1, · · · , nk and k = 1, · · · , K, +respectively. +• Step 2: Based on {rk(˜θ)}K +k=1 and {πik(˜θ)}nk +i=1 in Step 1, we can select a sub- +sample {(y∗ +ik, x∗ +ik, π∗ +ik) : i = 1, · · · , rk}K +k=1 from the full data Dn. Minimizes +the following weighted function +Q∗(θ) = +K +� +k=1 +r +rk(˜θ) +rk(˜θ) +� +i=1 +M +� +m=1 +ρτm(y∗ +ik − βTx∗ +ik − bm) +π∗ +ik +, +to get a two-step subsample estimate ˆθLopt, where ˆθLopt = (ˆβLopt, ˆbLopt) = +arg min Q∗(θ). +For the subsample-based estimator ˆθLopt in Algorithm 2, we give its +asymptotic distribution in the following theorem. +Theorem 3. If Assumptions (A.1) and (A.2) hold, then as r0 → ∞, r → +∞, and n → ∞, then we have +Σ−1/2√r(ˆθLopt − θ0) +d +−→ N(0, I), +(9) +where +d +−→ denotes convergence in distribution, Σ = E−1 +n V πE−1 +n . Here +V π = 1 +n2 +K +� +k=1 +r +rLopt +k +nk +� +i=1 +1 +πLopt +ik +� M +� +m=1 +{I(εik < b0m) − τm}˜xik,m +�⊗2 +, (10) +where +πLopt +ik += +∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +�nk +i=1 ∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +, +and +rLopt +k += r +�nk +i=1 ∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +�K +k=1 +�nk +i=1 ∥ �M +m=1{τm − I(εik < b0m)}˜xik,m ∥ +. +For the statistical inference about θ0, to avoid estimating f(b0m), we +propose the following iterative sampling procedure. + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +7 +Firstly, using {πLopt +ik +(˜θ)}nk +i=1 proposed in Algorithm 2, we sample with +replacement to obtain B subsamples, {(y∗,j +ik , x∗,j +ik , π∗,j +ik ), i = 1, · · · , rLopt +k +(˜θ), k = +1, · · · , K} for j = 1, · · · , B. Next, we calculate the jth estimate of θ0 through +ˆθLopt,j = (ˆβLopt,j, ˆbLopt,j) += arg min +θ +K +� +k=1 +r +rLopt +k +(˜θ) +rLopt +k +(˜θ) +� +i=1 +M +� +m=1 +ρτm(y∗,j +ik − βTx∗,j +ik − bm) +π∗,j +ik +. +The combined estimate can be obtained by +ˆθL = (ˆβ +T +L, ˆb +T +L)T = 1 +B +B +� +j=1 +ˆθLopt,j +(11) +and its variance-covariance matrix Ω = cov(ˆθL) can be estimated by +ˆΩ = +1 +refB(B − 1) +B +� +j=1 +(ˆθLopt,j − ˆθL)⊗2, +(12) +where ref is the effective subsample size ratio (Wang & Ma, 2021[14]) given by +ref = 1 +K +K +� +k=1 +� +1 − rkB − 1 +2 +nk +� +i=1 +{πLopt +ik +(˜θ)}2 +� +. +From Theorem 3, for any fixed B, the conditional distribution of +√ +rB(ˆθL− +θ0) satisfies +{E−1 +n V πE−1 +n }−1/2√ +rB(ˆθL − θ0) +d +−→ N(0, I). +The distribution of ˆθLopt can be approximated by the empirical distribution +of {˜θLopt,j}B +j=1. For s = 1, · · · , p + K, the 100 × (1 − α)% confidence interval +of θ0s can be approximated by [ˆθL,s − ˆω1/2 +ss z1−α/2, ˆθL,s + ˆω1/2 +ss z1−α/2], where +ˆθL,s is the sth element of ˆθL, ˆωss is the (s, s)th element of ˆΩ and z1−α/2 is +the 1 − α/2 quantile of the standard normal distribution. +3 Numerical studies +In this section, we conduct a simulation study to evaluate the performances of +the proposed optimal subsampling algorithm. Simulations were performed on +a laptop running Window 10 with an Intel i7 processor and 16 GB memory. +Full data are generated from the model +yik = xT +ikβ0 + εik, i = 1, · · · , nk, k = 1, · · · , K, + +Springer Nature 2021 LATEX template +8 +Optimal subsampling algorithm for CQR with distributed data +with the true parameter β0 = (1, 1, 1, 1, 1)T . We consider the following four +cases for the error term ε: (1) the standard normal distribution, N(0, 1); (2) +the mixture normal distribution, 0.5N(0, 1) + 0.5N(0, 9); (3) the Student¡¯s +t distribution with three degrees of freedom, t(3); (4) the standard Cauchy +distribution, Cauchy(0,1). +We consider the following four cases for the covariate x: +Case I: xik ∼ N(0, Σ), where Σ = (0.5|s−t|)s,t. +Case II: xik ∼ N(0, Σ), where Σ = (0.5I(s̸=t))s,t. +Case III: xik ∼ t3(0, Σ) with three degrees of freedom and Σ = (0.5|s−t|)s,t. +Case IV: Set K = 5, xi1 ∼ N5(0, I), xi2 ∼ N5(0, Σ1), xi3 ∼ N5(0, Σ2), +xi4 ∼ t3(0, Σ1) and xi5 ∼ t5(0, Σ1), where Σ1 = (0.5|s−t|)s,t, Σ2 = +(0.5I(s̸=t))s,t. +Note that in Cases I-III, the covariate distributions are identical for all +distributed datasets. In Case IV, the covariates have different distributions for +distributed datasets. +All the simulation are based on 1000 replications. We set the sample size +of each datasets as {nk = [nuk/ �K +k=1 uk]}K +k=1, where [·] denotes the rounding +operation, uk are generated from the uniform distribution over (1, 2) with K = +5 and 10, respectively. We use the quantile levels τm = m/16, m = 1, · · · , 15 +for the composite quantile regression. +In Tables 1, we report the simulation results on subsample-based estimator +for β1 (other βi’s are similar and omitted) with K = 5 and K = 10 respec- +tively, including the estimated bias (Bias) and the standard deviation (SD) of +the estimates where r0 = 200, n = 106 in Case I. The bias and SDs of the pro- +posed subsample estimate for Case IV with n = 106 and n = 107 are presented +in Tabel 2. The subsample sizes r = 200, 400, 600, 800 and 1000, respectively. +It can be seen from the results that the subsample-based estimator is unbi- +ased. The performance of our estimator becomes better as r increases, which +confirms the theoretical result on consistency of the subsampling methods. +For comparison, we consider the uniform subsampling method (Uniform) +with πik = +1 +nk , and rk = [rnk/n] for i = 1, · · · , nk and k = 1, · · · , K. We cal- +culate empirical mean square error (MSE) of uniform subsampling estimator +(Unif) and our optimal subsampling estimator (Lopt) based on 1000 repeti- +tions of the simulation. Figures 1 and 2 present the MSEs of each method for +Case I with K = 5 and K = 10, where n = 106. Figures 3 presents the MSEs of +the subsampling estimator for Case IV with n = 106, n = 107 and ε ∼ N(0, 1). +From the above results, we can see that the MSEs of our method (Lopt) are +much smaller than those of Uniform subsampling method (Unif). The results +indicate that our method also works well with heterogeneous covariates, i.e., +the covariates can have different distributions in different data blocks. +In the following, we evaluate the computational efficiency of our two-step +subsampling algorithm. The mechanism of data generation is the same as +the above mentioned situation. For fair comparison, we count the CPU time +with one core based on the mean calculation time of 1000 repetitions of each +subsample-based method. In Table 3, we report the results for Case I and + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +9 +the normal error with n = 106, K = 5, r0 = 200 and different r, respectively. +The computing time for the full data method is also given in the last row. +Note that the uniform subsampling requires the least computing time, because +its subampling probabilities πik = +1 +nk , and allocation sizes rk = [rnk/n], do +not take time to compute. Our subsampling algorithm has great computation +advantage over the full data method. To further investigate the computational +gain of the subsampling approach, we increase the dimension p to 30 with the +true parameter β0 = (0.5, · · · , 0.5)T. Table 4 presents the computing time for +Case I and normal error with r0 = 200, r = 1000, K = 5, n = 104, 105, 106 and +107, respectively. It is clear that both subsampling methods take significantly +less computing times than the full data approach. +To investigate the performance of ˆΩ in (12), we compare the empirical mean +square error (EMSE, s−1 �1000 +s=1 ∥ ˆβ +s +L−β0 ∥2) and the average estimated mean +square error(AMSE) of ˆβL in (11) with different B. In Tables 5, we report the +average length of the confidence intervals and 95% coverage probabilities (CP) +of our subsample-based estimator for β1 (other βi’s are similar and omitted) +with n = 106, r = 1000 and K = 5. Figures 4-7 present the EMSEs and AMSEs +of ˆβL. For all cases, the AMSEs are very close to the EMSEs, and the EMSEs +and AMSEs become smaller as B increases. +4 A real data example +In this section, we apply our method to the USA airline data, which are pub- +licly available at http://stat-computing.org/datastore/2009/the-data.html. +The data include detailed information on the arrivals and departures of all +commercial flights in the USA from 1987 to 2008, and they are stored in 22 +separate files (K = 22). The raw dataset is as large as 10 GB on a hard +drive. We use the composite regression to model the relationship between the +arrival delay time, y, and three covariate variables: x1, weekend/weekday sta- +tus (binary; 1 if departure occurred during the weekend, 0 otherwise), x2, the +departure delay time and x3, the distance. Since the y, x2 and x3 in the data +set are on different scales, we normalize them first. In addition, we drop the +NA values in the dataset and we have n = 115, 257, 291 observations with +completed information on y and x. Table 6 shows the cleaned data. +We use the quantile levels τm = m/16, m = 1, · · · , 15 for the composite +quantile regression. For comparison, the full-data estimate of the regression +parameters is given by ˆβF = (−0.0451, 0.9179, −0.0248)T. The proposed point +estimate ˆβL and corresponding confident intervals with different r and B are +presented in Table 7. It can be seen from Table 7 that the subsample estimator +ˆβL is close to ˆβF . In Figure 8, we present the MSEs of both subsampling +methods based on 1000 subsamples with r = 200, 400, 600, 800 and 1000, +respectively. The MSEs of the the optimal subsampling estimator are smaller +than those of the uniform subsampling estimator. + +Springer Nature 2021 LATEX template +10 +Optimal subsampling algorithm for CQR with distributed data +5 Conclusion +We have studied the statistical properties of a subsampling algorithm for the +composite quantile regression model with distributed massive data. We derived +the optimal subsampling probabilities and optimal allocation sizes. The asymp- +totic properties of the subsample estimator were established. Some simulations +and a real data example were provided to check the performance of our method. +Appendix +Proof of Theorem 1 +Define +A∗ +r(u) = 1 +n +K +� +k=1 +r +rk +rk +� +i=1 +M +� +m=1 +1 +π∗ +ik +A∗ +ik,m(u), +where A∗ +ik,m(u) = ρτm(ε∗ +ik − b0m − uT˜x∗ +ik,m/√r) − ρτm(ε∗ +ik − b0m), ˜x∗ +ik,m = +(x∗T +ik , eT +m)T, and ε∗ +ik = y∗ +ik − βT +0 x∗ +ik, i = 1, · · · , rk. Since A∗ +r(u) is a convex +function of u, its minimizer is √r(˜θs − θ0), we can focus on A∗ +r(u) when +evaluating the properties of √r(˜θs − θ0). +Let ψτ(u) = τ − I(u < 0). By Knight’s identity (Knight, 1998), +ρτ(u − v) − ρτ(u) = −vψτ(u) + +� v +0 +{I(u ≤ s) − I(u ≤ 0)}ds, +we can rewrite A∗ +ik,m(u) as +A∗ +ik,m(u) = ρτm(ε∗ +ik − b0m − uT˜x∗ +ik,m/√r) − ρτm(ε∗ +ik − b0m) += − 1 +√ruT˜x∗ +ik,m{τm − I(ε∗ +ik − b0m < 0)} ++ +� uT ˜x∗ +ik,m/√r +0 +{I(ε∗ +ik − b0m ≤ s) − I(ε∗ +ik − b0m ≤ 0)}ds. +Thus, we have +A∗ +r(u) += −uT 1 +√r +1 +n +K +� +k=1 +r +rk +M +� +m=1 +rk +� +i=1 +1 +π∗ +ik +{τm − I(ε∗ +ik − b0m < 0)}˜x∗ +ik,m ++ 1 +n +K +� +k=1 +r +rk +M +� +m=1 +rk +� +i=1 +1 +π∗ +ik +� uT ˜xik,m/√r +0 +{I(ε∗ +ik − b0m ≤ s) − I(ε∗ +ik − b0m ≤ 0)}ds += uTZ∗ +r + A∗ +2r(u), +(1) + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +11 +where +Z∗ +r = − 1 +√r +1 +n +K +� +k=1 +r +rk +M +� +m=1 +rk +� +i=1 +1 +π∗ +ik +{τm − I(ε∗ +ik − b0m < 0)}˜x∗ +ik,m, +A∗ +2r(u) = 1 +n +K +� +k=1 +r +rk +rk +� +i=1 +1 +π∗ +ik +A∗ +k,i(u), +A∗ +k,i(u) = +M +� +m=1 +� uT ˜x∗ +ik,m/√r +0 +{I(ε∗ +ik − b0m ≤ s) − I(ε∗ +ik − b0m ≤ 0)}ds. +Firstly, we prove the asymptotic normality of Z∗ +r. Denote +η∗ +ik = − +r +rknπ∗ +ik +M +� +m=1 +{τm − I(ε∗ +ik − b0m < 0)}˜x∗ +ik,m, +then Z∗ +r can be written as Z∗ +r = +1 +√r +�K +k=1 +�rk +i=1 η∗ +ik. Direct calculation yields +E(η∗ +ik | Dn) = − r +rkn +nk +� +i=1 +M +� +m=1 +{τm − I(εik − b0m < 0)}˜xik,m = Op +� +rn−1/2 +k +rkn +� +, +cov(η∗ +ik | Dn) = E{(η∗ +ik)⊗2 | Dn} − {E(η∗ +ik | Dn)}⊗2 += +nk +� +i=1 +r2 +r2 +kn2πik +� +M +� +m=1 +[τm − I(εik − b0m < 0)]˜xik,m +�⊗2 +− {E(η∗ +ik | Dn)}⊗2 += +nk +� +i=1 +r2 +r2 +kn2πik +� +M +� +m=1 +[τm − I(εik − b0m < 0)]˜xik,m +�⊗2 +− op(1). +It is easy to verify that +E{E(η∗ +ik | Dn)} = 0, +cov{E(η∗ +ik | Dn)} = +r2 +r2 +kn2 +nk +� +i=1 +cov +� M +� +m=1 +[τm − I(εik < b0m)] ˜xik,m +� +. +Denote the (s, t) th element of cov{E(η∗ +ik | Dn)} as σst. Using the Cauchy +inequality, it is easy to obtain +| σst |≤ √σss +√σtt ≤ +r2 +r2 +kn2 +nk +� +i=1 +M(∥xi∥2 + 1) = Op +�r2nk +r2 +kn2 +� +. + +Springer Nature 2021 LATEX template +12 +Optimal subsampling algorithm for CQR with distributed data +By Assumption 1 and Chebyshev’s inequality, +E(η∗ +ik | Dn) = Op +� +rn1/2 +k +rkn +� +. +Under the conditional distribution given Dn, we check Lindeberg’s condi- +tions (Theorem 2.27 of van der Vaart, 1998). Specifically, for ϵ > 0, we want +to prove that +K +� +k=1 +rk +� +i=1 +E{∥r−1/2η∗ +ik∥2I(∥η∗ +ik∥ > √rϵ) | Dn} = op(1). +(2) +Note that +K +� +k=1 +rk +� +i=1 +E{∥r−1/2η∗ +ik∥2I(∥η∗ +ik∥ > √rϵ) | Dn} += +K +� +k=1 +rk +� +i=1 +E +����� +r1/2 +rknπ∗ +ik +M +� +m=1 +˜x∗ +ik,m{τm − I(εik − b0m < 0)} +���� +2 +×I +����� +r−1/2 +rknπ∗ +ikϵ +M +� +m=1 +˜x∗ +ik,m{τm − I(εik − b0m < 0)} +���� > 1 +�����Dn +� += +K +� +k=1 +nk +� +i=1 +r +rkn2πik +���� +M +� +m=1 +{τm − I(εik − b0m < 0)}˜xik,m +���� +2 +×I +� +r1/2 +rknπikϵ +���� +M +� +m=1 +{τm − I(εik − b0m < 0)}˜xik,m +���� > 1 +� +. +(3) +By Assumption (A.2), +max +1≤k≤K max +1≤i≤nk +∥xik∥ + 1 +rkπik += op +� n +r1/2 +� +, +M 2 +K +� +k=1 +nk +� +i=1 +(1 + ∥xik∥)2 +n2πik += Op(1), +the right hand side of (3) satisfies +K +� +k=1 +nk +� +i=1 +r +rkn2πik +���� +M +� +m=1 +{τm − I(εik < b0m)}˜xik,m +���� +2 +×I +� +r1/2 +rknπikϵ +���� +M +� +m=1 +{τm − I(εik < b0m)}˜xik,m +���� > 1 +� + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +13 +≤ M 2 +K +� +k=1 +n +� +i=1 +r +rkn2πik +(1 + ∥xik∥)2I +�M(1 + ∥xik∥)r1/2 +rknπikϵ +> 1 +� +≤ I +� +max +1≤k≤K max +1≤i≤nk +∥xik∥ + 1 +rkπik +> +nϵ +r1/2M +� +×M 2 +K +� +k=1 +nk +� +i=1 +r(1 + ∥xik∥)2 +rkn2πik += op(1). +(4) +Thus, the Lindeberg’s conditions hold with probability approaching one. +Note that η∗ +ik, i = 1, · · · , rk, are independent and identically distributed +with mean E(η∗ +ik | Dn) and the covariance cov(η∗ +ik | Dn) when given Dn. +Based on this result, as r, n → ∞, we get +V −1/2 +π +{Z∗ +r − √r +K +� +k=1 +E(η∗ +ik | Dn)} +d +−→ N(0, I). +Since √r �K +k=1 E(η∗ +ik | Dn) = Op +� +r1/2 +n1/2 +�K +k=1 +rn1/2 +k +rkn1/2 +� += op(1), it is easy +to verify that +V −1/2 +π +Z∗ +r +d +−→ N(0, I). +(5) +Next, we prove that +A∗ +2r(u) = 1 +2uTEu + op(1). +Write the conditional expectation of A∗ +2r(u) as +E{A∗ +2r(u) | Dn} += r +n +K +� +k=1 +nk +� +i=1 +E{Ak,i(u)} + r +n +K +� +k=1 +nk +� +i=1 +[Ak,i(u) − E{A2r,i(u)}]. +(6) +By Assumption (A.1), +max +1≤k≤K max +1≤i≤nk ∥xik∥ = o(max(n1/2 +1 +, · · · , n1/2 +K )) = o(n1/2), +we can get +r +n +K +� +k=1 +nk +� +i=1 +E(Ak,i(u)) + +Springer Nature 2021 LATEX template +14 +Optimal subsampling algorithm for CQR with distributed data += r +n +K +� +k=1 +nk +� +i=1 +M +� +m=1 +� uT ˜xik,m/√r +0 +{F(b0m + s) − F(b0m)}ds += +√r +n +K +� +k=1 +nk +� +i=1 +M +� +m=1 +� uT ˜xik,m +0 +{F(b0m + t/√r) − F(b0m)}dt += 1 +2uT +� +1 +n +K +� +k=1 +nk +� +i=1 +M +� +m=1 +f(b0m)˜xik,m˜xT +ik,m +� +u + o(1) += 1 +2uTEu + o(1). +(7) +Furthermore, we have +E +� +r +n +K +� +k=1 +nk +� +i=1 +� +Ak,i(u) − E{Ak,i(u)} +�� += 0, +and +var +� r +n +K +� +k=1 +nk +� +i=1 +[Ak,i(u) − E{Ak,i(u)}] +� +≤ r2 +n2 +K +� +k=1 +nk +� +i=1 +E{A2 +k,i(u)}. +(8) +Since Ak,i(u) is nonnegative, it is easy to obtain +Ak,i(u) ≤ +���� +M +� +m=1 +� uT ˜xik,m/√r +0 +{I(εik ≤ b0m + s) − I(εik ≤ b0m)}ds +���� +≤ +M +� +m=1 +� uT ˜xik,m/√r +0 +����{I(εik ≤ b0m + s) − I(εik ≤ b0m)} +����ds +≤ +1 +√r +M +� +m=1 +| uT˜xik,m | . +(9) +By Assumption (A.1), +max +1≤k≤K max +1≤i≤nk ∥xik∥ = o(max(n1/2 +1 +, · · · , n1/2 +K )) = o(n1/2), +together with (8) and (9), we get +var +� r +n +K +� +k=1 +nk +� +i=1 +[Ak,i(u) − E{Ak,i(u)}] +� +≤ +� +M ∥u∥ +√n (1 + max +1≤k≤K max +1≤i≤nk ∥xik∥) +� K +� +k=1 +r3/2 +n3/2 +nk +� +i=1 +E{Ak,i(u)} + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +15 += o(1). +(10) +Combining the Chebyshev’s inequality, it follows from (6), (7) and (10) that +E {A∗ +2r(u) | Dn} = 1 +2uTEu + op(1). +(11) +Next, we derive the conditional variance of A∗ +2r(u), i.e., var {A∗ +2r(u) | Dn}. +Observing that A∗ +k,i(u), i = 1, · · · , rk are independent and identically dis- +tributed when given Dn, +var {A∗ +2r(u) | Dn} = +K +� +k=1 +r2 +(rkn)2 +rk +� +i=1 +var +�A∗ +k,i(u) +π∗ +ik +����Dn +� +≤ +K +� +k=1 +r2rk +r2 +kn2 E +��A∗ +k,i(u) +π∗ +ik +�2����Dn +� +. +(12) +By (9), the right hand of (12) satisfies +K +� +k=1 +r2rk +r2 +kn2 +nk +� +i=1 +A2 +k,i(u) +πik +≤ r2 +n2 +K +� +k=1 +nk +� +i=1 +Ak,i(u) +� 1 +√r +M +� +m=1 +| uT˜xik,m | +rkπik +� +≤ +�r1/2 +n M∥u∥ max +1≤k≤K max +1≤i≤nk +∥xik∥ + 1 +rkπik +� r +n +K +� +k=1 +nk +� +i=1 +Ak,i(u). +(13) +Together with (7), (13) and Assumption (A.2), we have +var +� +A∗ +2r(u) | Dn +� += op(1). +(14) +Together with (9), (14) and Chebyshev’s inequality, we can obtain +A∗ +2r(u) = 1 +2uTEu + op|Dn(1), +(15) +Here op|Dn(1) means if a = op|Dn(1), then a converges to 0 in conditional +probability given Dn in probability, in other words, for any δ > 0, P(| a |> +δ | Dn) +p +−→ 0 as n → +∞. Since 0 ≤ P(| a |> δ | Dn) ≤ 1, then it converges +to 0 in probability if and only P(| a |> δ) = E{P(| a |> δ | Dn)} → 0. Thus, +a = op|Dn(1) is equivalent to a = op(1). + +Springer Nature 2021 LATEX template +16 +Optimal subsampling algorithm for CQR with distributed data +It follows from (1) and (15) that +A∗ +2r(u) = uTZ∗ +r + 1 +2uTEu + op(1). +Since A∗ +2r(u) is a convex function, we have +√r(˜θs − θ0) = −E−1 +n Z∗ +r + op(1). +Based on the above results, we can prove that +{E−1 +n V πE−1 +n }−1/2√r(˜θs − θ0) = −{E−1 +n V πE−1 +n }−1/2E−1 +n Z∗ +r + op(1). +By Slutsky’s Theorem, for any a ∈ Rp+M, from (5) we have that +P[{E−1 +n V πE−1 +n }−1/2√r(˜θs − θ0) ≤ a | Dn] +p +−→ Φp+M(a), +(16) +where Φp+M(a) denotes the standard p + M dimensional multivariate normal +distribution function. And the conditional probability in (16) is a bounded +random variable, then convergence in probability to a constant implies +convergence in the mean. Therefore, for any a ∈ Rp+M, +P[{E−1 +n V πE−1 +n }−1/2√r(˜θs − θ0) ≤ a] += E(P[{E−1 +n V πE−1 +n }−1/2√r(˜θs − θ0) ≤ a | Dn]) +→ Φp+M(a). +We complete the proof of Theorem 1. +Proof the Theorem 2 +We can prove that +tr(V π) = 1 +n2 +K +� +k=1 +r +rk +nk +� +i=1 +1 +πik +tr +� +� +� M +� +m=1 +{I(εik < b0m) − τm}˜xik,m +�⊗2� +� += 1 +n2 +K +� +k=1 +r +rk +� nk +� +i=1 +πik +� � nk +� +i=1 +1 +πik +���� +M +� +m=1 +[I(εik < b0m) − τm]˜xik,m +���� +2� +≥ 1 +n2 +K +� +k=1 +r +rk +� nk +� +i=1 +���� +M +� +m=1 +{I(εik < b0m) − τm}˜xik,m +���� +2� += 1 +n2 +� K +� +k=1 +rk +� K +� +k=1 +1 +rk +� nk +� +i=1 +���� +M +� +m=1 +[I(εik < b0m) − τm]˜xik,m +���� +2� +≥ 1 +n2 +K +� +k=1 +nk +� +i=1 +���� +M +� +m=1 +{I(εik < b0m) − τm}˜xik,m +���� +2 +, + +Springer Nature 2021 LATEX template +Optimal subsampling algorithm for CQR with distributed data +17 +with Cauchy-Schwarz inequality and the equality in it holds if and only if +when πik ∝ ∥ �M +m=1[I(εik < b0m)−τm]˜xik,m∥ and rk ∝ �nk +i=1 ∥ �M +m=1[I(εik < +b0m) − τm]˜xik,m∥, respectively. We complete the proof of Theorem 2. + +Springer Nature 2021 LATEX template +18 +REFERENCES +References +Ai M, Yu J, Zhang H, Wang H (2019) Optimal subsampling algorithms for +big data regressions. Statistica Sinica 31: 749-772 +Fang F, Zhao J, Ahmed S E, Qu A (2021) A weak-signal-assisted proce- +dure for variable selection and statistical inference with an informative +subsample. Biometrics 77(3): 996-1010 +Jiang R, Hu X, Yu K, Qian W (2018) Composite quantile regression for +massive datasets. Statistics 52(5): 980-1004 +Jin J, Zhao Z (2021) Composite Quantile Regression Neural Network for +Massive Datasets. Mathematical Problems in Engineering 2021 +Jones H L (1956) Investigating the properties of a sample mean by employ- +ing random subsample means. Journal of the American Statistical +Association 51(273): 54-83 +Ma P, Mahoney M W, Yu B (2015) A statistical perspective on algorithmic +leveraging. Journal of Machine Learning Research 16: 861-919 +Qiu Y, Du G, Chai S (2020) A novel algorithm for distributed data stream +using big data classification model. International Journal of Information +Technology and Web Engineering 15(4): 1-17 +Shao L, Song S, Zhou Y (2022) Optimal subsampling for large-sample +quantile regression with massive data. Canadian Journal of Statistics +https://doi.org/10.1002/cjs.11697 +Shao Y, Wang L (2022) Optimal subsampling for composite quantile regres- +sion model in massive data. Statistical Papers 63(4): 1139¨C1161 +Sun X, Xu R, Wu L, Guan Z (2021) A differentially private distributed data +mining scheme with high efficiency for edge computing. Journal of Cloud +Computing 10(1): 1-12 +Wang H Y, Zhu R, Ma P (2018) Optimal subsampling for large sample logistic +regression. Journal of the American Statistical Association 113(522): 829- +844 +Wang H Y, Yang M, Stufken J (2019) Information-based optimal sub- +data selection for big data linear regression. Journal of the American +Statistical Association 114(525): 393-405 +Wang K, Li S, Zhang B (2021) Robust communication-efficient distributed +composite quantile regression and variable selection for massive data. +Computational Statistics & Data Analysis 161: 107262 +Wang H, Ma Y (2021) Optimal subsampling for quantile regression in big +data. Biometrika 108: 99-112 +Yuan X, Li Y, Dong X, Liu T (2022) Optimal subsampling for composite +quantile regression in big data. Statistical Papers 63(5): 1649-1676 +Yu J, Wang H, Ai M, Zhang H (2022) Optimal Distributed Subsampling for +Maximum Quasi-Likelihood Estimators With Massive Data. Journal of +the American Statistical Association 117(537): 265-276 +Zhang H, Wang H (2021) Distributed subdata selection for big data via +sampling-based approach. Computational Statistics and Data Analysis +153: 107072 + +Springer Nature 2021 LATEX template +REFERENCES +19 +Zou H, Yuan M (2008) Composite quantile regression and the oracle model +selection theory. Annals of Statistics 36(3): 1108-1126 +Zuo L, Zhang H, Wang H Y, Sun L (2021) Optimal subsample selection +for massive logistic regression with distributed data. Computational +Statistics 36(4): 2535-2562 + +Springer Nature 2021 LATEX template +20 +REFERENCES +Table 1: The proposed subsample estimate of β1 with n = 106 in Case I. +K = 5 +K = 10 +Error +r +Bias +SD +Bias +SD +200 +0.0006 +0.0769 +0.0010 +0.0737 +400 +-0.0009 +0.0554 +-0.0008 +0.0531 +N(0, 1) +600 +0.0025 +0.0425 +0.0008 +0.0423 +800 +0.0009 +0.0379 +0.0004 +0.0388 +1000 +0.0004 +0.0348 +-0.0014 +0.0338 +200 +0.0023 +0.1405 +0.0049 +0.1336 +400 +-0.0023 +0.0970 +0.0006 +0.0934 +mixNormal +600 +-0.0033 +0.0797 +-0.0004 +0.0822 +800 +0.0028 +0.0688 +-0.0019 +0.0707 +1000 +-0.0002 +0.0600 +-0.0033 +0.0621 +200 +-0.0021 +0.0961 +0.0009 +0.0914 +400 +0.0006 +0.0665 +-0.0004 +0.0645 +t(3) +600 +-0.0015 +0.0552 +-0.0002 +0.0505 +800 +-0.0003 +0.0477 +0.0005 +0.0462 +1000 +0.0024 +0.0415 +0.0013 +0.0423 +200 +-0.0108 +0.1312 +0.0070 +0.1373 +400 +0.0040 +0.0959 +0.0003 +0.0954 +Cauchy +600 +0.0023 +0.0793 +-0.0008 +0.0778 +800 +0.0011 +0.0700 +-0.0005 +0.0674 +1000 +-0.0014 +0.0612 +-0.0018 +0.0637 +Table 2: The proposed subsample estimate of β1 for Case IV and ε ∼ N(0, 1). +n = 106 +n = 107 +r +Bias +SD +Bias +SD +200 +0.0004 +0.0551 +0.0005 +0.0555 +400 +-0.0003 +0.0394 +0.0003 +0.0392 +600 +0.0002 +0.0313 +-0.0020 +0.0312 +800 +0.0012 +0.0273 +-0.0005 +0.0267 +1000 +0.0012 +0.0242 +-0.0011 +0.0256 +Table 3: The CPU time for Case I and ε ∼ N(0, 1) with K = 5, n = 106 (seconds) +r +Methods +200 +400 +600 +800 +1000 +Uniform +0.077 +0.098 +0.145 +0.170 +0.217 +Proposed +0.446 +0.494 +0.552 +0.615 +0.689 +Full data +421.03 +Table 4: The CPU time for Case I and ε ∼ N(0, 1) with r = 1000, K = 5 and +p = 30 (seconds) +n +Methods +104 +105 +106 +107 +Uniform +0.411 +0.417 +0.447 +0.490 +Proposed +0.586 +0.620 +0.922 +5.393 +Full data +4.43 +61.60 +676.08 +4667.22 + +Springer Nature 2021 LATEX template +REFERENCES +21 +Table 5: The CPs and the average lengths (in parenthesis) of the confident interval +of β1 with n = 106, r = 1000 and K = 5. +Error +B +Case I +Case II +Case III +Case IV +20 +0.930(0.030) +0.948(0.034) +0.932(0.014) +0.920(0.021) +40 +0.928(0.021) +0.924(0.024) +0.936(0.010) +0.954(0.015) +N(0, 1) +60 +0.952(0.018) +0.942(0.020) +0.942(0.009) +0.944(0.013) +80 +0.918(0.015) +0.934(0.017) +0.926(0.008) +0.914(0.011) +100 +0.936(0.014) +0.934(0.016) +0.930(0.007) +0.916(0.010) +20 +0.926(0.054) +0.920(0.060) +0.938(0.026) +0.930(0.038) +40 +0.932(0.038) +0.934(0.044) +0.922(0.019) +0.954(0.027) +mixNormal +60 +0.924(0.031) +0.936(0.036) +0.930(0.015) +0.934(0.023) +80 +0.928(0.027) +0.928(0.031) +0.934(0.014) +0.946(0.020) +100 +0.930(0.025) +0.934(0.028) +0.932(0.012) +0.948(0.018) +20 +0.940(0.037) +0.940(0.041) +0.928(0.018) +0.954(0.026) +40 +0.944(0.026) +0.960(0.030) +0.946(0.013) +0.916(0.019) +t(3) +60 +0.946(0.022) +0.968(0.025) +0.936(0.010) +0.936(0.016) +80 +0.940(0.019) +0.944(0.021) +0.946(0.009) +0.940(0.013) +100 +0.948(0.017) +0.944(0.019) +0.934(0.008) +0.914(0.012) +20 +0.932(0.053) +0.944(0.060) +0.918(0.026) +0.936(0.038) +40 +0.926(0.037) +0.932(0.043) +0.922(0.018) +0.944(0.027) +Cauchy +60 +0.924(0.031) +0.942(0.036) +0.930(0.015) +0.926(0.022) +80 +0.938(0.027) +0.946(0.031) +0.934(0.013) +0.924(0.020) +100 +0.942(0.024) +0.952(0.028) +0.926(0.012) +0.928(0.018) +Table 6: The number of yearly data and allocation sizes (r = 1000) +Years +nk +rk +Years +nk +rk +1987 +1,287,333 +11 +1998 +5,227,051 +45 +1988 +5,126,498 +47 +1999 +5,360,018 +45 +1989 +4,925,482 +45 +2000 +5,481,303 +45 +1990 +5,110,527 +46 +2001 +4,873,031 +42 +1991 +4,995,005 +46 +2002 +5,093,462 +45 +1992 +5,020,651 +47 +2003 +6,375,689 +56 +1993 +4,993,587 +46 +2004 +6,987,729 +59 +1994 +5,078,411 +46 +2005 +6,992,838 +58 +1995 +5,219,140 +46 +2006 +7,003,802 +57 +1996 +5,209,326 +44 +2007 +7,275,288 +58 +1997 +5,301,999 +47 +2008 +2,319,121 +19 + +Springer Nature 2021 LATEX template +22 +REFERENCES +Table 7: The estimator and the length of confident interval for ˆβL with different r +and B for the airline data. +B +r +40 +100 +200 +β1 +-0.0524 (-0.0675,-0.0373) +-0.0458 (-0.0545,-0.0370) +β2 +0.9232 (0.9164, 0.9299) +0.9183 (0.9142,0.9225) +β3 +-0.0242 (-0.0320, -0.0164) +-0.0221 (-0.0261,-0.0181) +600 +β1 +-0.0450 (-0.0539,-0.0361) +-0.0479 (-0.0537,-0.0421) +β2 +0.9172 (0.9127,0.9217) +0.9203 (0.9179,0.9227) +β3 +-0.0268 (-0.0309,-0.0228) +-0.0264 (-0.0288,-0.0240) +1000 +β1 +-0.0446 (-0.0509,-0.0383) +-0.0404 (-0.0445,-0.0363) +β2 +0.9192 (0.9163,0.9220) +0.9205 (0.9184,0.9226) +β3 +-0.0238 (-0.0269,-0.0208) +-0.0277 (-0.0297,-0.0257) +Fig. 1: The MSEs for different subsampling methods with K = 5 and n = 106 +(Case 1). + +E~N(0,1) +E-mixNormal +0.040 +Unif-MSE +0.14 +Unif-MSE +Lopt-MSE +Lopt-MSE +0.10 +MSE +0.025 +MSE +0.06 +0.010 +0.02 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +E~t(3) +E-Cauchy +90'0 +Unif-MSE +0.14 +Unif-MSE +Lopt-MSE +Lopt-MSE +0.04 +0.10 +MSE +MSE +0.06 +0.02 +0 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +-Springer Nature 2021 LATEX template +REFERENCES +23 +Fig. 2: The MSEs for different subsampling methods with K = 10 and n = +106(Case 1). + +E~N(0,1) +E-mixNormal +90'0 +Unif-MSE +0.14 +Unif-MSE +Lopt-MSE +Lopt-MSE +0.10 +MSE +0.03 +MSE +0.06 +0.01 +0.02 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +r +( +E~t(3) +E~Cauchy +20'0 +Unif-MSE +0.14 +Unif-MSE +90'0 +Lopt-MSE +Lopt-MSE +0.10 +MSE +MSE +0.03 +0.06 +.01 +0.02 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +-Springer Nature 2021 LATEX template +24 +REFERENCES +Fig. 3: The MSEs for different subsampling methods with ε ∼ N(0, 1)(Case +IV). + +n=106 +n=107 +0.025 +Unif-MSE +0.025 +Unif-MSE +Lopt-MSE +Lopt-MSE +MSE +0.015 +MSE +0.015 +.005 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +rSpringer Nature 2021 LATEX template +REFERENCES +25 +Fig. 4: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 +(Case 1). + +E~N(0,1) +E~mixNormal +4e-04 +0.0020 +EMSE +EMSE +AMSE +AMSE +MSE +2e-04 +MSE +0.0010 +00+a0 +0000'0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +B +8 +E~t(3) +E~Cauchy +6e-04 +0.0020 +EMSE +EMSE +AMSE +AMSE +MSE +3e-04 +MSE +0.0010 +00+a0 +0000'0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +8 +8Springer Nature 2021 LATEX template +26 +REFERENCES +Fig. 5: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 +(Case II). + +E~N(0,1) +E-mixNormal +4e-04 +0.0020 +EMSE +EMSE +AMSE +AMSE +MSE +2e-04 +MSE +0.0010 +00+a0 +0000'0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +8 +8 +E~t(3) +E~Cauchy +6e-04 +0.0020 +EMSE +EMSE +AMSE +AMSE +MSE +3e-04 +MSE +0.0010 +00+a0 +0000'0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +8 +BSpringer Nature 2021 LATEX template +REFERENCES +27 +Fig. 6: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 +(Case III). + +E~N(0,1) +E-mixNorma +05000'0 +90-a8 +EMSE +EMSE +AMSE +AMSE +MSE +MSE +0.00015 +4e-05 +00+a0 +00000'0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +B +B +E~t(3) +E~Cauchy +0.00020 +05000'0 +EMSE +EMSE +AMSE +AMSE +0.00010 +0.00015 +MSE +MSE +00000'0 +00000'0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +8 +8Springer Nature 2021 LATEX template +28 +REFERENCES +Fig. 7: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 +(Case IV). + +E~N(0,1) +E-mixNormal +0.00020 +6e-04 +EMSE +EMSE +AMSE +AMSE +0.00010 +SE +MSE +3e-04 +0.00000 +00+a0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +8 +8 +E~t(3) +E-Cauchy +05000'0 +6e-04 +EMSE +EMSE +AMSE +AMSE +0.00015 +WSE +MSE +3e-04 +00000'0 +00+a0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +8 +8Springer Nature 2021 LATEX template +REFERENCES +29 +Fig. 8: The results of MSEs for the airline data. + +800'0 +Unif-MSE +Lopt-MSE +MSE +0.004 +000'0 +200 +400 +600 +800 +1000 +r \ No newline at end of file diff --git a/K9E0T4oBgHgl3EQfigH5/content/tmp_files/load_file.txt b/K9E0T4oBgHgl3EQfigH5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c684b319588c762073c803d09cf2f3f4f609d88 --- /dev/null +++ b/K9E0T4oBgHgl3EQfigH5/content/tmp_files/load_file.txt @@ -0,0 +1,766 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf,len=765 +page_content='Springer Nature 2021 LATEX template Optimal subsampling algorithm for composite quantile regression with distributed data Xiaohui Yuan1, Shiting Zhou1† and Yue Wang1*† 1*School of Mathematics and Statistics, Changchun University of Technology, Changchun, 130012, Jilin, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' E-mail(s): wangyueccut@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Contributing authors: yuanxh@ccut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' zhoushiting1999@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Abstract For massive data stored at multiple machines, we propose a dis- tributed subsampling procedure for the composite quantile regres- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' By establishing the consistency and asymptotic normality of the composite quantile regression estimator from a general sub- sampling algorithm, we derive the optimal subsampling probabili- ties and the optimal allocation sizes under the L-optimality cri- teria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' A two-step algorithm to approximate the optimal subsam- pling procedure is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The proposed methods are illus- trated through numerical experiments on simulated and real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Keywords: Composite quantile regression, Distributed data, Massive data, Optimal subsampling 1 Introduction With the rapid development of science and technology, extremely large datasets are ubiquitous and lays heavy burden on storage and computation facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Many efforts have been made to deal with these challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' There are three main directions from the view of statistical applications: divide-and-conquer, online updating, and subsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Among them, subsampling has been found 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='02448v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='CO] 6 Jan 2023 Springer Nature 2021 LATEX template 2 Optimal subsampling algorithm for CQR with distributed data to be useful for reducing computational burden and extracting information from massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The idea of subsampling was first proposed by Jones (1956)[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' A key tactic of subsampling methods is to specify nonuniform sampling probabil- ities to include more informative data points with higher probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For example, the leverage score-based subsampling in Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2015)[6], the information based optimal subdata selection in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2019)[12], and the optimal subsampling method under the A-optimality criterion in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2018)[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Recently, Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2021)[2] applied subsampling to a weak- signal-assisted procedure for variable selection and statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Ai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2021)[1] studied the optimal subsampling method for generalized linear models under the A-optimality criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2022)[8] employed the optimal subsampling method to ordinary quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Due to the large scale and fast arrival speed of data stream, massive data are often partitioned across multiple servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For example, Walmart stores produce a large number of data sets from different locations around the world, which need to be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' However, it is difficult to transmit these datasets to a central location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For these datasets, it is common to analyze them on multiple machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2020)[7] constructed a data stream classification model based on distributed processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2021)[10] proposed a data mining scheme for edge computing based on distributed integration strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Zhang and Wang (2021)[17] proposed a distributed subdata selection method for big data linear regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2021)[19] proposed a distributed subsampling procedure for the logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2022)[16] derived a optimal distributed Poisson subsampling procedure for the maximum quasi- likelihood estimators with massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In the paper, we investigate the optimal distributed subsampling for com- posite quantile regression (CQR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Zou and Yuan (2008)[18]) in massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In a linear model, composite quantile regression can uniformly estimate the regression coefficients under heavy tail error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Moreover, since the asymptotic variance of the composite quantile regression estimate does not depend on the moment of the error distribution, the CQR estimator is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The CQR method is widely used in many fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For massive data, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2018)[3] proposed a divide-and-conquer CQR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Jin and Zhao (2021)[4] proposed a divide-and-conquer CQR neural network method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2021)[13] proposed a distributed CQR method for the massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Shao and Wang (2022)[9] and Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2022)[15] developed the subsampling for composite quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' To the best of our knowledge, there is almost no work on random subsampling for composite quantile regression with distributed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Based on the above motivation, we investigate the optimal subsampling for the composite quantile regression in massive data when the datasets are stored at different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We propose a distributed subsampling method in the context of CQR, and then study the optimal subsampling technology for data in each machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The main advantages of our method are as follows: First, we establish the convergence rate of the subsample-based estimator, which Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 3 ensures the consistency of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Second, it avoids the impact of different intercept items in data sets stored at different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Third, the computational speed of our subsampling method is much faster than the full data approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In Section 2, we propose the distributed subsampling algorithm based on composite quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The asymptotic properties of estimators based on subsamples are also established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We present a subsampling strategy with optimal subsampling probability and optimal allocation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The simulation studies are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In Section 4, we study the real data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The content of the article is summarized in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' All proofs are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 2 Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='1 Model and notation Consider the following linear model yik = xT ikβ0 + εik, i = 1, · · · , nk, k = 1, · · · , K, (1) where xik denotes a p-dimensional covariate vector, β0 = (β1, · · · , βp)T ∈ Θ is a p-dimensional vector of regression coefficients, nk is the sample size of the kth dataset, n = �K k=1 nk is the total sample size, and K is the number of distributed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Assume that the random error εik has cumulative distribution function F(·) and probability density function f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Let M be the composite level of composite quantile regression, which does not depend on the sample size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Given M, let τm, m = 1, · · · , M be the speci- fied quantile levels such that τ1 < · · · < τM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Write θ0 = (θ01, · · · , θ0(p+M))T = (βT 0 , bT 0 )T and b0 = (b01, · · · , b0M)T, where b0m = inf{u : F(u) ≥ τm} for m = 1, · · · , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In this paper, we assume that xik’s are nonrandom and are interested in inferences about the unknown θ0 from the observed dataset Dn = {Dkn = {(xT ik, yik), i = 1 · · · , n}, k = 1, · · · , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For τ ∈ (0, 1), u ∈ Rp, let ρτ(u) = u{τ − I(u < 0)} be the check loss function for the τ-th quantile level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The CQR estimator of θ based on the full dataset Dn is given by ˆθF = (ˆβ T F , ˆb T F )T = arg min β,b K � k=1 nk � i=1 M � m=1 ρτm(yik − bm − xT ikβ), (2) Our aim is to construct a subsample-based estimator, which can be used to effectively approximate the full data estimator ˆθF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 Optimal subsampling algorithm for CQR with distributed data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='2 Subsampling algorithm and asymptotic properties In this subsection, we propose a distributed subsampling algorithm to approx- imate the ˆθF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' First we propose a subsampling method in Algorithm 1, which can reasonably select a subsample from distributed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Algorithm 1 Distributed Subsampling Algorithm£º Sampling: Assign subsampling probabilities {πik}nk i=1 for the kth dataset Dk = {(yik, xik), i = 1, · · · , nk} with �nk i=1 πik = 1, where k = 1, · · · , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Given total sampling size r, draw a random subsample of size rk with replace- ment from Dk according to {πik}nk i=1, where {rk}K k=1 are allocation sizes with �K k=1 rk = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For i = 1, · · · , nk and k = 1, · · · , K, we denote the cor- responding responses, covariates, and subsampling probabilities as y∗ ik, x∗ ik and π∗ ik, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Estimation: Based on the subsamples {(y∗ ik, x∗ ik, π∗ ik), i = 1, · · · , rk}K k=1, and calculate the estimate ˜θs = (˜βs, ˜bs) = arg minθ Q∗(θ), where Q∗(θ) = 1 n K � k=1 r rk rk � i=1 M � m=1 ρτm(y∗ ik − βTx∗ ik − bm) π∗ ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' To establish asymptotic properties of the subsample-based estimator ˜θs, we need the following assumptions: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='1) Assume that f(t) is continuous with respect to t and 0 < f(b0m) < +∞ for 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Let ˜xik,m = (xT ik, eT m)T, where em denotes a M ×1 vector, which has a one only in its mth coordinate and is zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Define En = 1 n K � k=1 nk � i=1 M � m=1 f(b0m)˜xik,m(˜xik,m)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (3) Assume that there exist positive definite matrices E, such that En −→ E, and max 1≤k≤K,1≤i≤nk ∥xik∥ = o(n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='2) Assume that, for k = 1, · · · , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' max 1≤k≤K,1≤i≤nk ∥xik∥ + 1 rkπik = op � n r1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (4) Define V π = 1 n2 K � k=1 r rk nk � i=1 1 πik � M � m=1 {I(εik < b0m) − τm}˜xik,m �⊗2 , (5) Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 5 where for a vector a, a⊗2 = aaT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Assume that there exist positive definite matrices V such that V π p −→ V , where p −→ means convergence in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' If Assumptions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='2) hold, conditional on Dn, as n → ∞ and r → ∞, if r/n = o(1), then we have Σ−1/2√r(˜θs − θ0) d −→ N(0, I), (6) where d −→ denotes convergence in distribution, Σ = E−1 n V πE−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='3 Optimal subsampling strategy Given r, we specify the subsampling probablities {πik}nk i=1, and the alloca- tion sizes {rk}K k=1 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' A naive choice is the uniform subsampling strategy with {πik = 1/nk}nk i=1 and {rk = [rnk/n]}K k=1, where [·] denotes the rounding operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' However, this uniform subsampling method is not opti- mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' As suggested by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2018)[11], we adopted the nonuniform subsampling strategy to determine the optimal allocation sizes and optimal subsampling probabilities by minimizing the trace of Σ in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Since Σ = E−1 n V πE−1 n , the optimal allocation sizes and subsampling prob- abilities require the calculation of En, which depend on the unknown density function f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Following Wang and Ma (2021)[14], we derive optimal subsam- pling probabilities under the L-optimality criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Note that En and V π are nonnegative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Simple matrix algebra yields that tr(Σ) = tr(V πE−2 n ) = tr(E−2 n )tr(V π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Σ depends on rk and πik only through V π, and En is free of rk and πik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Hence, we suggest to determine the optimal allocation sizes and optimal subsampling probabilites by directly minimizing tr(V π) rather than tr(Σ), which can effectively speed up our subsampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' If rk and πik, i = 1, · · · , nk, k = 1, · · · , K, are chosen as πLopt ik = πLopt ik (θ0) = ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ �nk i=1 ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ , (7) and rLopt k = r �nk i=1 ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ �K k=1 �nk i=1 ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ , (8) then tr(V π)/n attains its minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 6 Optimal subsampling algorithm for CQR with distributed data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='4 Two-step algorithm Note that the optimal subsampling probabilities and allocation sizes depend depends on εik = yik − xT ikβ0 and b0m, m = 1, · · · , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The L-optimal weight result is not directly implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' To deal with this problem, we use a pilot estimator ˜θ to replace θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In the following, we propose a two-step subsampling procedure in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Algorithm 2 Two-Step Algorithm£º Step 1: Given r0, we run Algorithm 1 with subsampling size rk = [r0 nk n ] to obtain a pilot estimator ˜θ, using πik = 1/nk, where [·] denotes the rounding operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Replace θ0 with ˜θ0 in (7) and (8) to get the allocation sizes rk(˜θ) and subsampling probabilities πik(˜θ), for i = 1, · · · , nk and k = 1, · · · , K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Step 2: Based on {rk(˜θ)}K k=1 and {πik(˜θ)}nk i=1 in Step 1, we can select a sub- sample {(y∗ ik, x∗ ik, π∗ ik) : i = 1, · · · , rk}K k=1 from the full data Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Minimizes the following weighted function Q∗(θ) = K � k=1 r rk(˜θ) rk(˜θ) � i=1 M � m=1 ρτm(y∗ ik − βTx∗ ik − bm) π∗ ik , to get a two-step subsample estimate ˆθLopt, where ˆθLopt = (ˆβLopt, ˆbLopt) = arg min Q∗(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For the subsample-based estimator ˆθLopt in Algorithm 2, we give its asymptotic distribution in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' If Assumptions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='2) hold, then as r0 → ∞, r → ∞, and n → ∞, then we have Σ−1/2√r(ˆθLopt − θ0) d −→ N(0, I), (9) where d −→ denotes convergence in distribution, Σ = E−1 n V πE−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Here V π = 1 n2 K � k=1 r rLopt k nk � i=1 1 πLopt ik � M � m=1 {I(εik < b0m) − τm}˜xik,m �⊗2 , (10) where πLopt ik = ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ �nk i=1 ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ , and rLopt k = r �nk i=1 ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ �K k=1 �nk i=1 ∥ �M m=1{τm − I(εik < b0m)}˜xik,m ∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For the statistical inference about θ0, to avoid estimating f(b0m), we propose the following iterative sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 7 Firstly, using {πLopt ik (˜θ)}nk i=1 proposed in Algorithm 2, we sample with replacement to obtain B subsamples, {(y∗,j ik , x∗,j ik , π∗,j ik ), i = 1, · · · , rLopt k (˜θ), k = 1, · · · , K} for j = 1, · · · , B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Next, we calculate the jth estimate of θ0 through ˆθLopt,j = (ˆβLopt,j, ˆbLopt,j) = arg min θ K � k=1 r rLopt k (˜θ) rLopt k (˜θ) � i=1 M � m=1 ρτm(y∗,j ik − βTx∗,j ik − bm) π∗,j ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The combined estimate can be obtained by ˆθL = (ˆβ T L, ˆb T L)T = 1 B B � j=1 ˆθLopt,j (11) and its variance-covariance matrix Ω = cov(ˆθL) can be estimated by ˆΩ = 1 refB(B − 1) B � j=1 (ˆθLopt,j − ˆθL)⊗2, (12) where ref is the effective subsample size ratio (Wang & Ma, 2021[14]) given by ref = 1 K K � k=1 � 1 − rkB − 1 2 nk � i=1 {πLopt ik (˜θ)}2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' From Theorem 3, for any fixed B, the conditional distribution of √ rB(ˆθL− θ0) satisfies {E−1 n V πE−1 n }−1/2√ rB(ˆθL − θ0) d −→ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The distribution of ˆθLopt can be approximated by the empirical distribution of {˜θLopt,j}B j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For s = 1, · · · , p + K, the 100 × (1 − α)% confidence interval of θ0s can be approximated by [ˆθL,s − ˆω1/2 ss z1−α/2, ˆθL,s + ˆω1/2 ss z1−α/2], where ˆθL,s is the sth element of ˆθL, ˆωss is the (s, s)th element of ˆΩ and z1−α/2 is the 1 − α/2 quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 3 Numerical studies In this section, we conduct a simulation study to evaluate the performances of the proposed optimal subsampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Simulations were performed on a laptop running Window 10 with an Intel i7 processor and 16 GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Full data are generated from the model yik = xT ikβ0 + εik, i = 1, · · · , nk, k = 1, · · · , K, Springer Nature 2021 LATEX template 8 Optimal subsampling algorithm for CQR with distributed data with the true parameter β0 = (1, 1, 1, 1, 1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We consider the following four cases for the error term ε: (1) the standard normal distribution, N(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2) the mixture normal distribution, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5N(0, 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5N(0, 9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (3) the Student¡¯s t distribution with three degrees of freedom, t(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (4) the standard Cauchy distribution, Cauchy(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We consider the following four cases for the covariate x: Case I: xik ∼ N(0, Σ), where Σ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5|s−t|)s,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Case II: xik ∼ N(0, Σ), where Σ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5I(s̸=t))s,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Case III: xik ∼ t3(0, Σ) with three degrees of freedom and Σ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5|s−t|)s,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Case IV: Set K = 5, xi1 ∼ N5(0, I), xi2 ∼ N5(0, Σ1), xi3 ∼ N5(0, Σ2), xi4 ∼ t3(0, Σ1) and xi5 ∼ t5(0, Σ1), where Σ1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5|s−t|)s,t, Σ2 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5I(s̸=t))s,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Note that in Cases I-III, the covariate distributions are identical for all distributed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In Case IV, the covariates have different distributions for distributed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' All the simulation are based on 1000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We set the sample size of each datasets as {nk = [nuk/ �K k=1 uk]}K k=1, where [·] denotes the rounding operation, uk are generated from the uniform distribution over (1, 2) with K = 5 and 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We use the quantile levels τm = m/16, m = 1, · · · , 15 for the composite quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In Tables 1, we report the simulation results on subsample-based estimator for β1 (other βi’s are similar and omitted) with K = 5 and K = 10 respec- tively, including the estimated bias (Bias) and the standard deviation (SD) of the estimates where r0 = 200, n = 106 in Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The bias and SDs of the pro- posed subsample estimate for Case IV with n = 106 and n = 107 are presented in Tabel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The subsample sizes r = 200, 400, 600, 800 and 1000, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' It can be seen from the results that the subsample-based estimator is unbi- ased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The performance of our estimator becomes better as r increases, which confirms the theoretical result on consistency of the subsampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For comparison, we consider the uniform subsampling method (Uniform) with πik = 1 nk , and rk = [rnk/n] for i = 1, · · · , nk and k = 1, · · · , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We cal- culate empirical mean square error (MSE) of uniform subsampling estimator (Unif) and our optimal subsampling estimator (Lopt) based on 1000 repeti- tions of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Figures 1 and 2 present the MSEs of each method for Case I with K = 5 and K = 10, where n = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Figures 3 presents the MSEs of the subsampling estimator for Case IV with n = 106, n = 107 and ε ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' From the above results, we can see that the MSEs of our method (Lopt) are much smaller than those of Uniform subsampling method (Unif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The results indicate that our method also works well with heterogeneous covariates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=', the covariates can have different distributions in different data blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In the following, we evaluate the computational efficiency of our two-step subsampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The mechanism of data generation is the same as the above mentioned situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For fair comparison, we count the CPU time with one core based on the mean calculation time of 1000 repetitions of each subsample-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In Table 3, we report the results for Case I and Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 9 the normal error with n = 106, K = 5, r0 = 200 and different r, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The computing time for the full data method is also given in the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Note that the uniform subsampling requires the least computing time, because its subampling probabilities πik = 1 nk , and allocation sizes rk = [rnk/n], do not take time to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Our subsampling algorithm has great computation advantage over the full data method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' To further investigate the computational gain of the subsampling approach, we increase the dimension p to 30 with the true parameter β0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5, · · · , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='5)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Table 4 presents the computing time for Case I and normal error with r0 = 200, r = 1000, K = 5, n = 104, 105, 106 and 107, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' It is clear that both subsampling methods take significantly less computing times than the full data approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' To investigate the performance of ˆΩ in (12), we compare the empirical mean square error (EMSE, s−1 �1000 s=1 ∥ ˆβ s L−β0 ∥2) and the average estimated mean square error(AMSE) of ˆβL in (11) with different B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In Tables 5, we report the average length of the confidence intervals and 95% coverage probabilities (CP) of our subsample-based estimator for β1 (other βi’s are similar and omitted) with n = 106, r = 1000 and K = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Figures 4-7 present the EMSEs and AMSEs of ˆβL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For all cases, the AMSEs are very close to the EMSEs, and the EMSEs and AMSEs become smaller as B increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 4 A real data example In this section, we apply our method to the USA airline data, which are pub- licly available at http://stat-computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='org/datastore/2009/the-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The data include detailed information on the arrivals and departures of all commercial flights in the USA from 1987 to 2008, and they are stored in 22 separate files (K = 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The raw dataset is as large as 10 GB on a hard drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We use the composite regression to model the relationship between the arrival delay time, y, and three covariate variables: x1, weekend/weekday sta- tus (binary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 1 if departure occurred during the weekend, 0 otherwise), x2, the departure delay time and x3, the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Since the y, x2 and x3 in the data set are on different scales, we normalize them first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In addition, we drop the NA values in the dataset and we have n = 115, 257, 291 observations with completed information on y and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Table 6 shows the cleaned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We use the quantile levels τm = m/16, m = 1, · · · , 15 for the composite quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' For comparison, the full-data estimate of the regression parameters is given by ˆβF = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0451, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='9179, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0248)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The proposed point estimate ˆβL and corresponding confident intervals with different r and B are presented in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' It can be seen from Table 7 that the subsample estimator ˆβL is close to ˆβF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' In Figure 8, we present the MSEs of both subsampling methods based on 1000 subsamples with r = 200, 400, 600, 800 and 1000, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The MSEs of the the optimal subsampling estimator are smaller than those of the uniform subsampling estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 10 Optimal subsampling algorithm for CQR with distributed data 5 Conclusion We have studied the statistical properties of a subsampling algorithm for the composite quantile regression model with distributed massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We derived the optimal subsampling probabilities and optimal allocation sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' The asymp- totic properties of the subsample estimator were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Some simulations and a real data example were provided to check the performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Appendix Proof of Theorem 1 Define A∗ r(u) = 1 n K � k=1 r rk rk � i=1 M � m=1 1 π∗ ik A∗ ik,m(u), where A∗ ik,m(u) = ρτm(ε∗ ik − b0m − uT˜x∗ ik,m/√r) − ρτm(ε∗ ik − b0m), ˜x∗ ik,m = (x∗T ik , eT m)T, and ε∗ ik = y∗ ik − βT 0 x∗ ik, i = 1, · · · , rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Since A∗ r(u) is a convex function of u, its minimizer is √r(˜θs − θ0), we can focus on A∗ r(u) when evaluating the properties of √r(˜θs − θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Let ψτ(u) = τ − I(u < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' By Knight’s identity (Knight, 1998), ρτ(u − v) − ρτ(u) = −vψτ(u) + � v 0 {I(u ≤ s) − I(u ≤ 0)}ds, we can rewrite A∗ ik,m(u) as A∗ ik,m(u) = ρτm(ε∗ ik − b0m − uT˜x∗ ik,m/√r) − ρτm(ε∗ ik − b0m) = − 1 √ruT˜x∗ ik,m{τm − I(ε∗ ik − b0m < 0)} + � uT ˜x∗ ik,m/√r 0 {I(ε∗ ik − b0m ≤ s) − I(ε∗ ik − b0m ≤ 0)}ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' we have A∗ r(u) = −uT 1 √r 1 n K � k=1 r rk M � m=1 rk � i=1 1 π∗ ik {τm − I(ε∗ ik − b0m < 0)}˜x∗ ik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m + 1 n K � k=1 r rk M � m=1 rk � i=1 1 π∗ ik � uT ˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m/√r 0 {I(ε∗ ik − b0m ≤ s) − I(ε∗ ik − b0m ≤ 0)}ds = uTZ∗ r + A∗ 2r(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (1) Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 11 where Z∗ r = − 1 √r 1 n K � k=1 r rk M � m=1 rk � i=1 1 π∗ ik {τm − I(ε∗ ik − b0m < 0)}˜x∗ ik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' A∗ 2r(u) = 1 n K � k=1 r rk rk � i=1 1 π∗ ik A∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='i(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' A∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='i(u) = M � m=1 � uT ˜x∗ ik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m/√r 0 {I(ε∗ ik − b0m ≤ s) − I(ε∗ ik − b0m ≤ 0)}ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Firstly, we prove the asymptotic normality of Z∗ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Denote η∗ ik = − r rknπ∗ ik M � m=1 {τm − I(ε∗ ik − b0m < 0)}˜x∗ ik,m, then Z∗ r can be written as Z∗ r = 1 √r �K k=1 �rk i=1 η∗ ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Direct calculation yields E(η∗ ik | Dn) = − r rkn nk � i=1 M � m=1 {τm − I(εik − b0m < 0)}˜xik,m = Op � rn−1/2 k rkn � , cov(η∗ ik | Dn) = E{(η∗ ik)⊗2 | Dn} − {E(η∗ ik | Dn)}⊗2 = nk � i=1 r2 r2 kn2πik � M � m=1 [τm − I(εik − b0m < 0)]˜xik,m �⊗2 − {E(η∗ ik | Dn)}⊗2 = nk � i=1 r2 r2 kn2πik � M � m=1 [τm − I(εik − b0m < 0)]˜xik,m �⊗2 − op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' It is easy to verify that E{E(η∗ ik | Dn)} = 0, cov{E(η∗ ik | Dn)} = r2 r2 kn2 nk � i=1 cov � M � m=1 [τm − I(εik < b0m)] ˜xik,m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Denote the (s, t) th element of cov{E(η∗ ik | Dn)} as σst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Using the Cauchy inequality, it is easy to obtain | σst |≤ √σss √σtt ≤ r2 r2 kn2 nk � i=1 M(∥xi∥2 + 1) = Op �r2nk r2 kn2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 12 Optimal subsampling algorithm for CQR with distributed data By Assumption 1 and Chebyshev’s inequality, E(η∗ ik | Dn) = Op � rn1/2 k rkn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Under the conditional distribution given Dn, we check Lindeberg’s condi- tions (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='27 of van der Vaart, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Specifically, for ϵ > 0, we want to prove that K � k=1 rk � i=1 E{∥r−1/2η∗ ik∥2I(∥η∗ ik∥ > √rϵ) | Dn} = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (2) Note that K � k=1 rk � i=1 E{∥r−1/2η∗ ik∥2I(∥η∗ ik∥ > √rϵ) | Dn} = K � k=1 rk � i=1 E ����� r1/2 rknπ∗ ik M � m=1 ˜x∗ ik,m{τm − I(εik − b0m < 0)} ���� 2 ×I ����� r−1/2 rknπ∗ ikϵ M � m=1 ˜x∗ ik,m{τm − I(εik − b0m < 0)} ���� > 1 �����Dn � = K � k=1 nk � i=1 r rkn2πik ���� M � m=1 {τm − I(εik − b0m < 0)}˜xik,m ���� 2 ×I � r1/2 rknπikϵ ���� M � m=1 {τm − I(εik − b0m < 0)}˜xik,m ���� > 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (3) By Assumption (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' max 1≤k≤K max 1≤i≤nk ∥xik∥ + 1 rkπik = op � n r1/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' M 2 K � k=1 nk � i=1 (1 + ∥xik∥)2 n2πik = Op(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' the right hand side of (3) satisfies K � k=1 nk � i=1 r rkn2πik ���� M � m=1 {τm − I(εik < b0m)}˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m ���� 2 ×I � r1/2 rknπikϵ ���� M � m=1 {τm − I(εik < b0m)}˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m ���� > 1 � Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 13 ≤ M 2 K � k=1 n � i=1 r rkn2πik (1 + ∥xik∥)2I �M(1 + ∥xik∥)r1/2 rknπikϵ > 1 � ≤ I � max 1≤k≤K max 1≤i≤nk ∥xik∥ + 1 rkπik > nϵ r1/2M � ×M 2 K � k=1 nk � i=1 r(1 + ∥xik∥)2 rkn2πik = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (4) Thus, the Lindeberg’s conditions hold with probability approaching one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Note that η∗ ik, i = 1, · · · , rk, are independent and identically distributed with mean E(η∗ ik | Dn) and the covariance cov(η∗ ik | Dn) when given Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Based on this result, as r, n → ∞, we get V −1/2 π {Z∗ r − √r K � k=1 E(η∗ ik | Dn)} d −→ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Since √r �K k=1 E(η∗ ik | Dn) = Op � r1/2 n1/2 �K k=1 rn1/2 k rkn1/2 � = op(1), it is easy to verify that V −1/2 π Z∗ r d −→ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (5) Next, we prove that A∗ 2r(u) = 1 2uTEu + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Write the conditional expectation of A∗ 2r(u) as E{A∗ 2r(u) | Dn} = r n K � k=1 nk � i=1 E{Ak,i(u)} + r n K � k=1 nk � i=1 [Ak,i(u) − E{A2r,i(u)}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (6) By Assumption (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' max 1≤k≤K max 1≤i≤nk ∥xik∥ = o(max(n1/2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' n1/2 K )) = o(n1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' we can get r n K � k=1 nk � i=1 E(Ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='i(u)) Springer Nature 2021 LATEX template 14 Optimal subsampling algorithm for CQR with distributed data = r n K � k=1 nk � i=1 M � m=1 � uT ˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m/√r 0 {F(b0m + s) − F(b0m)}ds = √r n K � k=1 nk � i=1 M � m=1 � uT ˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m 0 {F(b0m + t/√r) − F(b0m)}dt = 1 2uT � 1 n K � k=1 nk � i=1 M � m=1 f(b0m)˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m˜xT ik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m � u + o(1) = 1 2uTEu + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (7) Furthermore, we have E � r n K � k=1 nk � i=1 � Ak,i(u) − E{Ak,i(u)} �� = 0, and var � r n K � k=1 nk � i=1 [Ak,i(u) − E{Ak,i(u)}] � ≤ r2 n2 K � k=1 nk � i=1 E{A2 k,i(u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (8) Since Ak,i(u) is nonnegative, it is easy to obtain Ak,i(u) ≤ ���� M � m=1 � uT ˜xik,m/√r 0 {I(εik ≤ b0m + s) − I(εik ≤ b0m)}ds ���� ≤ M � m=1 � uT ˜xik,m/√r 0 ����{I(εik ≤ b0m + s) − I(εik ≤ b0m)} ����ds ≤ 1 √r M � m=1 | uT˜xik,m | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (9) By Assumption (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='1), max 1≤k≤K max 1≤i≤nk ∥xik∥ = o(max(n1/2 1 , · · · , n1/2 K )) = o(n1/2), together with (8) and (9), we get var � r n K � k=1 nk � i=1 [Ak,i(u) − E{Ak,i(u)}] � ≤ � M ∥u∥ √n (1 + max 1≤k≤K max 1≤i≤nk ∥xik∥) � K � k=1 r3/2 n3/2 nk � i=1 E{Ak,i(u)} Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 15 = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (10) Combining the Chebyshev’s inequality, it follows from (6), (7) and (10) that E {A∗ 2r(u) | Dn} = 1 2uTEu + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (11) Next, we derive the conditional variance of A∗ 2r(u), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=', var {A∗ 2r(u) | Dn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Observing that A∗ k,i(u), i = 1, · · · , rk are independent and identically dis- tributed when given Dn, var {A∗ 2r(u) | Dn} = K � k=1 r2 (rkn)2 rk � i=1 var �A∗ k,i(u) π∗ ik ����Dn � ≤ K � k=1 r2rk r2 kn2 E ��A∗ k,i(u) π∗ ik �2����Dn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (12) By (9), the right hand of (12) satisfies K � k=1 r2rk r2 kn2 nk � i=1 A2 k,i(u) πik ≤ r2 n2 K � k=1 nk � i=1 Ak,i(u) � 1 √r M � m=1 | uT˜xik,m | rkπik � ≤ �r1/2 n M∥u∥ max 1≤k≤K max 1≤i≤nk ∥xik∥ + 1 rkπik � r n K � k=1 nk � i=1 Ak,i(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (13) Together with (7), (13) and Assumption (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='2), we have var � A∗ 2r(u) | Dn � = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' (14) Together with (9), (14) and Chebyshev’s inequality, we can obtain A∗ 2r(u) = 1 2uTEu + op|Dn(1), (15) Here op|Dn(1) means if a = op|Dn(1), then a converges to 0 in conditional probability given Dn in probability, in other words, for any δ > 0, P(| a |> δ | Dn) p −→ 0 as n → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Since 0 ≤ P(| a |> δ | Dn) ≤ 1, then it converges to 0 in probability if and only P(| a |> δ) = E{P(| a |> δ | Dn)} → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Thus, a = op|Dn(1) is equivalent to a = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 16 Optimal subsampling algorithm for CQR with distributed data It follows from (1) and (15) that A∗ 2r(u) = uTZ∗ r + 1 2uTEu + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Since A∗ 2r(u) is a convex function, we have √r(˜θs − θ0) = −E−1 n Z∗ r + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Based on the above results, we can prove that {E−1 n V πE−1 n }−1/2√r(˜θs − θ0) = −{E−1 n V πE−1 n }−1/2E−1 n Z∗ r + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' By Slutsky’s Theorem, for any a ∈ Rp+M, from (5) we have that P[{E−1 n V πE−1 n }−1/2√r(˜θs − θ0) ≤ a | Dn] p −→ Φp+M(a), (16) where Φp+M(a) denotes the standard p + M dimensional multivariate normal distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' And the conditional probability in (16) is a bounded random variable, then convergence in probability to a constant implies convergence in the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Therefore, for any a ∈ Rp+M, P[{E−1 n V πE−1 n }−1/2√r(˜θs − θ0) ≤ a] = E(P[{E−1 n V πE−1 n }−1/2√r(˜θs − θ0) ≤ a | Dn]) → Φp+M(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Proof the Theorem 2 We can prove that tr(V π) = 1 n2 K � k=1 r rk nk � i=1 1 πik tr � � � M � m=1 {I(εik < b0m) − τm}˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m �⊗2� � = 1 n2 K � k=1 r rk � nk � i=1 πik � � nk � i=1 1 πik ���� M � m=1 [I(εik < b0m) − τm]˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m ���� 2� ≥ 1 n2 K � k=1 r rk � nk � i=1 ���� M � m=1 {I(εik < b0m) − τm}˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m ���� 2� = 1 n2 � K � k=1 rk � K � k=1 1 rk � nk � i=1 ���� M � m=1 [I(εik < b0m) − τm]˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m ���� 2� ≥ 1 n2 K � k=1 nk � i=1 ���� M � m=1 {I(εik < b0m) − τm}˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m ���� 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Optimal subsampling algorithm for CQR with distributed data 17 with Cauchy-Schwarz inequality and the equality in it holds if and only if when πik ∝ ∥ �M m=1[I(εik < b0m)−τm]˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m∥ and rk ∝ �nk i=1 ∥ �M m=1[I(εik < b0m) − τm]˜xik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='m∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' We complete the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 18 REFERENCES References Ai M, Yu J, Zhang H, Wang H (2019) Optimal subsampling algorithms for big data regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Statistica Sinica 31: 749-772 Fang F, Zhao J, Ahmed S E, Qu A (2021) A weak-signal-assisted proce- dure for variable selection and statistical inference with an informative subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Biometrics 77(3): 996-1010 Jiang R, Hu X, Yu K, Qian W (2018) Composite quantile regression for massive datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Statistics 52(5): 980-1004 Jin J, Zhao Z (2021) Composite Quantile Regression Neural Network for Massive Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Mathematical Problems in Engineering 2021 Jones H L (1956) Investigating the properties of a sample mean by employ- ing random subsample means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Journal of the American Statistical Association 51(273): 54-83 Ma P, Mahoney M W, Yu B (2015) A statistical perspective on algorithmic leveraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Journal of Machine Learning Research 16: 861-919 Qiu Y, Du G, Chai S (2020) A novel algorithm for distributed data stream using big data classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' International Journal of Information Technology and Web Engineering 15(4): 1-17 Shao L, Song S, Zhou Y (2022) Optimal subsampling for large-sample quantile regression with massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Canadian Journal of Statistics https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='1002/cjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='11697 Shao Y, Wang L (2022) Optimal subsampling for composite quantile regres- sion model in massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Statistical Papers 63(4): 1139¨C1161 Sun X, Xu R, Wu L, Guan Z (2021) A differentially private distributed data mining scheme with high efficiency for edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Journal of Cloud Computing 10(1): 1-12 Wang H Y, Zhu R, Ma P (2018) Optimal subsampling for large sample logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Journal of the American Statistical Association 113(522): 829- 844 Wang H Y, Yang M, Stufken J (2019) Information-based optimal sub- data selection for big data linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Journal of the American Statistical Association 114(525): 393-405 Wang K, Li S, Zhang B (2021) Robust communication-efficient distributed composite quantile regression and variable selection for massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Computational Statistics & Data Analysis 161: 107262 Wang H, Ma Y (2021) Optimal subsampling for quantile regression in big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Biometrika 108: 99-112 Yuan X, Li Y, Dong X, Liu T (2022) Optimal subsampling for composite quantile regression in big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Statistical Papers 63(5): 1649-1676 Yu J, Wang H, Ai M, Zhang H (2022) Optimal Distributed Subsampling for Maximum Quasi-Likelihood Estimators With Massive Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Journal of the American Statistical Association 117(537): 265-276 Zhang H, Wang H (2021) Distributed subdata selection for big data via sampling-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Computational Statistics and Data Analysis 153: 107072 Springer Nature 2021 LATEX template REFERENCES 19 Zou H, Yuan M (2008) Composite quantile regression and the oracle model selection theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Annals of Statistics 36(3): 1108-1126 Zuo L, Zhang H, Wang H Y, Sun L (2021) Optimal subsample selection for massive logistic regression with distributed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' Computational Statistics 36(4): 2535-2562 Springer Nature 2021 LATEX template 20 REFERENCES Table 1: The proposed subsample estimate of β1 with n = 106 in Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' K = 5 K = 10 Error r Bias SD Bias SD 200 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0637 Table 2: The proposed subsample estimate of β1 for Case IV and ε ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' n = 106 n = 107 r Bias SD Bias SD 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0551 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 1: The MSEs for different subsampling methods with K = 5 and n = 106 (Case 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' E~N(0,1) E-mixNormal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='040 Unif-MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='14 Unif-MSE Lopt-MSE Lopt-MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='10 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='025 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="02 200 400 600 800 1000 200 400 600 800 1000 E~t(3) E-Cauchy 90'0 Unif-MSE 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='14 Unif-MSE Lopt-MSE Lopt-MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='10 MSE MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='02 0 200 400 600 800 1000 200 400 600 800 1000 Springer Nature 2021 LATEX template REFERENCES 23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 2: The MSEs for different subsampling methods with K = 10 and n = 106(Case 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=" E~N(0,1) E-mixNormal 90'0 Unif-MSE 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='14 Unif-MSE Lopt-MSE Lopt-MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='10 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='03 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="02 200 400 600 800 1000 200 400 600 800 1000 r ( E~t(3) E~Cauchy 20'0 Unif-MSE 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="14 Unif-MSE 90'0 Lopt-MSE Lopt-MSE 0." metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' n=106 n=107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='025 Unif-MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='025 Unif-MSE Lopt-MSE Lopt-MSE MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='015 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='015 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='005 200 400 600 800 1000 200 400 600 800 1000 rSpringer Nature 2021 LATEX template REFERENCES 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 4: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 (Case 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' E~N(0,1) E~mixNormal 4e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0020 EMSE EMSE AMSE AMSE MSE 2e-04 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="0010 00+a0 0000'0 20 40 60 80 100 20 40 60 80 100 B 8 E~t(3) E~Cauchy 6e-04 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0020 EMSE EMSE AMSE AMSE MSE 3e-04 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="0010 00+a0 0000'0 20 40 60 80 100 20 40 60 80 100 8 8Springer Nature 2021 LATEX template 26 REFERENCES Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 5: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 (Case II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' E~N(0,1) E-mixNormal 4e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0020 EMSE EMSE AMSE AMSE MSE 2e-04 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="0010 00+a0 0000'0 20 40 60 80 100 20 40 60 80 100 8 8 E~t(3) E~Cauchy 6e-04 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='0020 EMSE EMSE AMSE AMSE MSE 3e-04 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="0010 00+a0 0000'0 20 40 60 80 100 20 40 60 80 100 8 BSpringer Nature 2021 LATEX template REFERENCES 27 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 6: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 (Case III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=" E~N(0,1) E-mixNorma 05000'0 90-a8 EMSE EMSE AMSE AMSE MSE MSE 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="00015 4e-05 00+a0 00000'0 20 40 60 80 100 20 40 60 80 100 B B E~t(3) E~Cauchy 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="00020 05000'0 EMSE EMSE AMSE AMSE 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="00015 MSE MSE 00000'0 00000'0 20 40 60 80 100 20 40 60 80 100 8 8Springer Nature 2021 LATEX template 28 REFERENCES Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 7: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000 (Case IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' E~N(0,1) E-mixNormal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='00020 6e-04 EMSE EMSE AMSE AMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content='00010 SE MSE 3e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="00000 00+a0 20 40 60 80 100 20 40 60 80 100 8 8 E~t(3) E-Cauchy 05000'0 6e-04 EMSE EMSE AMSE AMSE 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="00015 WSE MSE 3e-04 00000'0 00+a0 20 40 60 80 100 20 40 60 80 100 8 8Springer Nature 2021 LATEX template REFERENCES 29 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=' 8: The results of MSEs for the airline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content=" 800'0 Unif-MSE Lopt-MSE MSE 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} +page_content="004 000'0 200 400 600 800 1000 r" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfigH5/content/2301.02448v1.pdf'} diff --git a/KNFOT4oBgHgl3EQfyzQ_/content/tmp_files/2301.12929v1.pdf.txt b/KNFOT4oBgHgl3EQfyzQ_/content/tmp_files/2301.12929v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e679730e0ee0e5a24c5596cef4588dbf5e47e9c8 --- /dev/null +++ b/KNFOT4oBgHgl3EQfyzQ_/content/tmp_files/2301.12929v1.pdf.txt @@ -0,0 +1,2263 @@ +Can Persistent Homology provide an efficient alternative for +Evaluation of Knowledge Graph Completion Methods? +Anson Bastos +cs20resch11002@iith.ac.in +IIT, Hyderabad +India +Kuldeep Singh +kuldeep.singh1@cerence.com +Zerotha Research and +Cerence GmbH +Germany +Abhishek Nadgeri +abhishek22596@gmail.com +Zerotha Research and +RWTH Aachen +Germany +Johannes Hoffart +johannes@hoffart.ai +SAP +Germany +Toyotaro Suzumura +suzumura@acm.org +The University of Tokyo +Japan +Manish Singh +msingh@cse.iith.ac.in +IIT Hyderabad +India +ABSTRACT +In this paper we present a novel method, Knowledge Persistence +(KP), for faster evaluation of Knowledge Graph (KG) completion +approaches. Current ranking-based evaluation is quadratic in the +size of the KG, leading to long evaluation times and consequently a +high carbon footprint. KP addresses this by representing the topol- +ogy of the KG completion methods through the lens of topological +data analysis, concretely using persistent homology. The character- +istics of persistent homology allow KP to evaluate the quality of +the KG completion looking only at a fraction of the data. Experi- +mental results on standard datasets show that the proposed metric +is highly correlated with ranking metrics (Hits@N, MR, MRR). Per- +formance evaluation shows that KP is computationally efficient: +In some cases, the evaluation time (validation+test) of a KG com- +pletion method has been reduced from 18 hours (using Hits@10) +to 27 seconds (using KP), and on average (across methods & data) +reduces the evaluation time (validation+test) by ≈ 99.96%. +ACM Reference Format: +Anson Bastos, Kuldeep Singh, Abhishek Nadgeri, Johannes Hoffart, Toyotaro +Suzumura, and Manish Singh. 2023. Can Persistent Homology provide +an efficient alternative for Evaluation of Knowledge Graph Completion +Methods?. In Proceedings of the Web Conference 2023 (WWW ’23), APRIL 30 - +MAY 4, 2023, Texas, USA. WWW, Texas, USA, 13 pages. https://doi.org/10. +XXXXX/YYYYY.3449917 +1 +INTRODUCTION +Publicly available Knowledge Graphs (KGs) find broad applicability +in several downstream tasks such as entity linking, relation extrac- +tion, fact-checking, and question answering [22, 41]. These KGs are +large graph databases used to express facts in the form of relations +between real-world entities and store these facts as triples (subject, +Permission to make digital or hard copies of part or all of this work for +personal or classroom use is granted without fee provided that copies are +not made or distributed for profit or commercial advantage and that copies +bear this notice and the full citation on the first page. Copyrights for third- +party components of this work must be honored. For all other uses, contact +the owner/author(s). +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +© 2023 Copyright held by the owner/author(s). +ACM ISBN 978-Y-4500-YYYY-7/21/04. +https://doi.org/10.XXXXX/YYYYY.3449917 +relation, object). KGs must be continuously updated because new en- +tities might emerge or facts about entities are extended or updated. +Knowledge Graph Completion (KGC) task aims to fill the missing +piece of information into an incomplete triple of KG [5, 18, 22]. +Several Knowledge Graph Embedding (KGE) approaches have +been proposed to model entities and relations in vector space for +missing link prediction in a KG [55]. KGE methods infer the connec- +tivity patterns (symmetry, asymmetry, etc.) in the KGs by defining a +scoring function to calculate the plausibility of a knowledge graph +triple. While calculating plausibility of a KG triple τ = (𝑒ℎ,𝑟,𝑒𝑡), +the predicted score by scoring function affirms the confidence of a +model that entities 𝑒𝑡 and 𝑒ℎ are linked by 𝑟. +For evaluating KGE methods, ranking metrics have been widely +used [22] which is based on the following criteria: given a KG triple +with a missing head or tail entity, what is the ability of the KGE +method to rank candidate entities averaged over triples in a held- +out test set [28]? These ranking metrics are useful as they intend to +gauge the behavior of the methods in real world applications of KG +completion. Since 2019, over 100 KGE articles have been published +in various leading conferences and journals that use ranking metrics +as evaluation protocol1. +Limitations of Ranking-based Evaluation: The key challenge +while computing ranking metrics for model evaluation is the time +taken to obtain them. Since the (most of) KGE models aim to rank +all the negative triples that are not present in the KG [8, 9], comput- +ing these metrics takes a quadratic time in the number of entities +in the KG. Moreover, the problem gets alleviated in the case of +hyper-relations [62] where more than two entities participate, lead- +ing to exponential computation time. For instance, Ali et al. [2] +spent 24,804 GPU hours of computation time while performing a +large-scale benchmarking of KGE methods. +There are two issues with high model evaluation time. Firstly, +efficiency at evaluation time is not a widely-adapted criterion for +assessing KGE models alongside accuracy and related measures. +There are efforts to make KGE methods efficient at training time +[52, 54]. However, these methods also use ranking-based protocols +resulting in high evaluation time. Secondly, the need for signifi- +cant computational resources for the KG completion task excludes a +large group of researchers in universities/labs with restricted GPU +1https://github.com/xinguoxia/KGE#papers +arXiv:2301.12929v1 [cs.LG] 30 Jan 2023 + +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Bastos, et al. +availability. Such preliminary exclusion implicitly challenges the ba- +sic notion of various diversity and inclusion initiatives for making +the Web and its related research accessible to a wider community. +In past, researchers have worked extensively towards efficient Web- +related technologies such as Web Crawling [12], Web Indexing [25], +RDF processing [17], etc. Hence, for the KG completion task, similar +to other efficient Web-based research, there is a necessity to develop +alternative evaluation protocols to reduce the computation com- +plexity, a crucial research gap in available KGE scientific literature. +Another critical issue in ranking metrics is that they are biased +towards popular entities and such popularity bias is not captured +by current evaluation metrics [28]. Hence, we need a metric which +is efficient than popular ranking metrics and also omits such biases. +Motivation and Contribution: In this work, we focus on ad- +dressing above-mentioned key research gaps and aim for the first +study to make KGE evaluation more efficient. We introduce Knowl- +edge Persistence(KP), a method for characterizing the topology +of the learnt KG representations. It builds upon Topological Data +Analysis [58] based on the concepts from Persistent Homology(PH) +[15], which has been proven beneficial for analyzing deep networks +[29, 36]. PH is able to effectively capture the geometry of the mani- +fold on which the representations reside whilst requiring fraction of +data [15]. This property allows to reduce the quadratic complexity +of considering all the data points (KG triples in our case) for rank- +ing. Another crucial fact that makes PH useful is its stability with +respect to perturbations making KP robust to noise [19] mitigating +the issues due to the open-world problem. Thus we use PH due to +its effectiveness for limited resources and noise [50]. Concretely, +the following are our key contributions: +(1) We propose (KP), a novel approach along with its theoreti- +cal foundations to estimate the performance of KGE models +through the lens of topological data analysis. This allows us +to drastically reduce the computation factor from order of +O(|E|2) to O(|E|). The code is here. +(2) We run extensive experiments on families of KGE methods +(e.g., Translation, Rotation, Bi-Linear, Factorization, Neural +Network methods) using standard benchmark datasets. The +experiments show that KP correlates well with the stan- +dard ranking metrics. Hence, KP could be used for faster +prototyping of KGE methods and paves the way for efficient +evaluation methods in this domain. +In the remainder of the paper, related work is in section 2. Section +3 briefly explains the concept of persistent homology. Section 4 +describes the proposed method. Later, section 5 shows associated +empirical results and we conclude in section 7. +2 +RELATED WORK +Broadly, KG embeddings are classified into translation and semantic +matching models [55]. Translation methods such as TransE [8], +TransH [57], TransR [26] use distance-based scoring functions. +Whereas semantic matching models (e.g., ComplEx [48], Distmult +[60], RotatE [44]) use similarity-based scoring functions. +Kadlec et al. [23] first pointed limitations of KGE evaluation +and its dependency on hyperparameter tuning. [45] with exhaus- +tive evaluation (using ranking metrics) showed issues of scoring +functions of KGE methods whereas [31] studied the effect of loss +function of KGE performance. Jain et al. [20] studied if KGE meth- +ods capture KG semantic properties. Work in [35] provides a new +dataset that allows the study of calibration results for KGE mod- +els. Speranskaya et al. [43] used precision and recall rather than +rankings to measure the quality of completion models. Authors pro- +posed a new dataset containing triples such that their completion +is both possible and impossible based on queries. However, queries +were build by creating a tight dependency on such queries for the +evaluation as pointed by [47]. Rim et al. [37] proposed a capability- +based evaluation where the focus is to evaluate KGE methods on +various dimensions such as relation symmetry, entity hierarchy, +entity disambiguation, etc. Mohamed et al. [28] fixed the popularity +bias of ranking metrics by introducing modified ranking metrics. +The geometric perspective of KGE methods was introduced by [40] +and its correlation with task performance. Berrendorf et al. [6] sug- +gested the adjusted mean rank to improve reciprocal rank, which +is an ordinal scale. Authors do not consider the effect of negative +triples available for a given triple under evaluation. [47] propose +to balances the number of negatives per triple to improve rank- +ing metrics. Authors suggested the preparation of training/testing +splits by maintaining the topology. Work in [24] proposes efficient +non-sampling techniques for KG embedding training, few other +initiatives improve efficiency of KGE training time [52–54], and +hyperparameter search efficiency of embedding models [49, 56, 63]. +Overall, the literature is rich with evaluations of knowledge +graph completion methods [4, 21, 38, 46]. However, to the best +of our knowledge, extensive attempts have not been made to im- +prove KG evaluation protocols’ efficiency, i.e., to reduce run-time +of widely-used ranking metrics for faster prototyping. We position +our work orthogonal to existing attempts such as [40], [47], [28], +and [37]. In contrast with these attempts, our approach provides a +topological perspective of the learned KG embeddings and focuses +on improving the efficiency of KGE evaluations. +3 +PRELIMINARIES +We now briefly describe concepts used in this paper. +Ranking metrics have been used for evaluating KG embedding +methods since the inception of the KG completion task [8]. These +metrics include the Mean Rank (MR), Mean Reciprocal Rank (MRR) +and the cut-off hit ratio (Hits@N (N=1,3,10)). MR reports the average +predicted rank of all the labeled triples. MRR is the average of the +inverse rank of the labelled triples. Hits@N evaluates the fraction +of the labeled triples that are present in the top N predicted results. +Persistent Homology (PH) [15, 19]: studies the topological +features such as components in 0-dimension (e.g., a node), holes in +1-dimension (e.g., a void area bounded by triangle edges) and so +on, spread over a scale. Thus, one need not choose a scale before- +hand. The number(rank) of these topological features(homology +group) in every dimension at a particular scale can be used for +downstream applications. Consider the simplicial complex ( e.g., +point is a 0-simplex, an edge is a 1-simplex, a triangle is a 2-simplex +) 𝐶 with weights 𝑎0 ≤ 𝑎1 ≤ 𝑎2 . . . 𝑎𝑚−1, which could represent the +edge weights, for example, the triple score from the KG embed- +ding method in our case. One can then define a Filtration process +[15], which refers to generating a nested sequence of complexes +𝜙 ⊆ 𝐶1 ⊆ 𝐶2 ⊆ . . .𝐶𝑚 = 𝐶 in time/scale as the simplices below + +Can Persistent Homology provide an efficient alternative +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Figure 1: Calculating Knowledge Persistence(KP) score from the given KG and KG embedding method. The KG is sampled for +positive(G+) and negative(G−) triples (step one), keeping the order O(|E|). The edge weights represent the score obtained from +the KG embedding method. In step two, the persistence diagram (PD) is computed using filtration process explained in Figure +2. In final step, a Sliced Wasserstein distance (SW) is obtained between the PDs of G+ and G− to get the KP score. However, +ranking metrics run the KGE methods over all the O(|E|2) triples as explained in bottom left part of the figure(red box). +the threshold weights are added in the complex. The filtration pro- +cess [15] results in the creation(birth) and destruction(death) of +components, holes, etc. Thus each structure is associated with a +birth-death pair (𝑎𝑖,𝑎𝑗) ∈ 𝑅2 with 𝑖 ≤ 𝑗. The persistence or life- +time of each component can then be given by 𝑎𝑗 − 𝑎𝑖. A persistence +diagram (PD) summarizes the (birth,death) pair of each object on +a 2D plot, with birth times on the x axis and death times on the y +axis. The points near the diagonal are shortlived components and +generally are considered noise (local topology), whereas the persis- +tent objects (global topology) are treated as features. We consider +local and global topology to compare two PDs (i.e., positive and +negative triple graphs in our case). +4 +PROBLEM STATEMENT AND METHOD +4.1 +Problem Setup +We define a KG as a tuple 𝐾𝐺 = (E, R, T +) where E denotes +the set of entities (vertices), R is the set of relations (edges), and +T + ⊆ E × R × E is a set of all triples. A triple τ = (𝑒ℎ,𝑟,𝑒𝑡) ∈ T + +indicates that, for the relation 𝑟 ∈ R, 𝑒ℎ is the head entity (origin +of the relation) while 𝑒𝑡 is the tail entity. Since 𝐾𝐺 is a multigraph; +𝑒ℎ = 𝑒𝑡 may hold and |{𝑟𝑒ℎ,𝑒𝑡 }| ≥ 0 for any two entities. The KG +completion task predicts the entity pairs ⟨𝑒𝑖,𝑒𝑗⟩ in the KG that have +a relation 𝑟𝑐 ∈ R between them. +4.2 +Proposed Method +In this section we describe our approach for evaluating KG embed- +ding methods using the theory of persistent homology (PH) . This +process is divided into three steps ( Figure 1), namely: (i) Graph con- +struction, (ii) Filtration process and (iii) Sliced Wasserstein distance +computation. The first step creates two graphs (one for positive +triples, another for negative triples) using sampling(O(V) triples), +with scores calculated by a KGE method as edge weights. The +second step considers these graphs and, using a process called "fil- +tration," converts to an equivalent lower dimension representation. +The last step calculates the distance between graphs to provide a +final metric score. We now detail the approach. +4.2.1 +Graph Construction. We envisioned KGE from the topolog- +ical lens while proposing an efficient solution for its evaluation. +Previous works such as [40] proposed a KGE metric only consider- +ing embedding space. However, we intend to preserve the topology +(graph structure and its topological feature) along with the KG em- +bedding features. We first construct graphs of positive and negative +triples. We denote a graph as (V, E) where V is the set of 𝑁 nodes +and E represents the edges between them. Consider a KG embed- +ding method M that takes as input the triple τ = (ℎ,𝑟,𝑡) ∈ T and +gives the score 𝑠τ of it being a right triple. We construct a weighted +directed graph G+ from positive triples τ ∈ T + in the train set, +with the entities as the nodes and the relations between them as the +edges having 𝑠τ as the edge weights. Here, 𝑠τ is the score calculated +by KGE method for a triple and we propose to use it as the edge +weights. Our idea is to capture topology of graph (G+) with repre- +sentation learned by a KG embedding method. We sample an order +of O(|E|) triples, |E| being the number of entities to keep compu- +tational time linear. Similarly, we construct a negative graph G− +by sampling the same number of unknown triples as the positive +samples. One question may arise if KP is robust to sampling, that +we answer theoretically in Theorem 4.4 and empirically in section +6. Note, here we do not take all the negative triples in the graphs +and consider only a fraction of what the ranking metrics need. This +is a fundamental difference with ranking metrics. Ranking metrics +use all the unlabeled triples as negatives for ranking, thus incurring +a computational cost of 𝑂(|E|2). +4.2.2 +Filtration Process. Having constructed the Graphs G+ and +G−, we now need some definition of a distance between them + +1. Graph Construction +2. Filtration Process +Winneror +gt +a=0 +Einstein(HAE) +Hans +Prize(NP) +Hans +KG Embedding +Einstein +GrandSonof +method +Alfred +Sonot +n SupervisedBy +r(A +a=2 +3Albert +Einstein +Homen +Alfred +Birth +3. Sliced Wasserstein +Distance Computation +D+ +O(E*)Graph with scores from the +KGE method onthe edges +HAE +KP(G+, G-) = SW(D+, D-) +0.3 +Ranking +Ranking +metric +Albert +Einstein +HE +AK +Hermann +Einstein +Birth +Ranking Metrics +D- +process +Sampled O(E) Graphs +with scores from the KGE +method on the edgesWWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Bastos, et al. +Figure 2: For a KGE method, the positive triple graph G+ is used as input (leftmost graph with edge weights) and filtration +process is applied on the edge weights (calculated by KGE method) for the graph. The filtration starts with only nodes as first +step, and based on the edge weights, edges are added to the nodes. The persistence diagram is given on the right with red dots +indicating 0-dimensional homology (components) and the blue dots indicating 1-dimensional homology (cycles). Persistent +Diagram generated from this filtration process is a condensed 2D representation of G+. A similar process is repeated for G−. +to define a metric. However, since the KGs could be large with +many entities and relations, directly comparing the graphs could +be computationally challenging. Therefore, we allude to the theory +of persistent homology (PH) to summarize the structures in the +graphs in the form of the persistence diagram (PD). Such summa- +rizing is obtained by a process known as filtration [64]. One can +imagine a PD as mapping of higher dimensional data to a 2D plane +upholding the representation of data points and we can then derive +computational efficiency for distance comparison between 2D repre- +sentations. Specifically, we compute the 0-dimensional topological +features (i.e., connected-nodes/components) for each graph (G− +and G+) to keep the computation time linear. We also experimented +using the 1-dimensional features without much empirical benefit. +Consider the positive triple graph G+ as input (cf., Figure 2). +We would need a scale (as pointed in section 3) for the filtration +process. Once the filtration process starts, initially, we have a graph +structure containing only the nodes (entities) and no edges of G+. +For capturing topological features at various scales, we define a +variable 𝑎 which varies from −∞ to +∞ and it is then compared +with edge weights (𝑠τ). A scale allows to capture topology at various +timesteps. Thus, we use the edge weights obtained from the scores +(𝑠τ) of the KGE methods for filtration. As the filtration proceeds, +the graph structures (components) are generated/removed. At a +given scale 𝑎, the graph structure ((G+ +𝑠𝑢𝑏)𝑎) contains those edges +(triples) for which 𝑠τ ≤ 𝑎. Formally, this is expressed as: +(G+ +𝑠𝑢𝑏)𝑎 = {(V, E+ +𝑎)|E+ +𝑎 ⊆ E,𝑠τ ≤ 𝑎 ∀τ ∈ E+ +𝑎 } +Alternatively, we add those edges for which score of the triple is +greater than or equal to the filtration value, i.e., 𝑠τ ≥ 𝑎 defined as +(G+ +𝑠𝑢𝑝𝑒𝑟)𝑎 = {(V, E𝑎+)|E𝑎+ ⊆ E,𝑠τ ≥ 𝑎 ∀τ ∈ E𝑎+} +One can imagine that for filtration, graph G+ is subdivided into +(G+ +𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 as the filtration adds/deletes edges for cap- +turing topological features. Hence, specific components in a sub- +graphs will appear and certain components will disappear at differ- +ent scale levels (timesteps) 𝑎 = 1, 3, 5 and so on. Please note, Figure 2 +explains creation of PD for (G+ +𝑠𝑢𝑏)𝑎. A similar process is repeated for +(G+𝑠𝑢𝑝𝑒𝑟)𝑎. This expansion/contraction process enables capturing +topology at different time-steps without worrying about defining an +optimal scale (similar to hyperparameter). Next step is the creation +of persistent diagrams of (G+ +𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 where the x-axis +and y-axis denotes the timesteps of appearance/disappearance of +components. For creating a 2D representation graph, components of +graphs which appear(disappear) during filtration process at 𝑎𝑥 (𝑎𝑦) +are plotted on (𝑎𝑥,𝑎𝑦). The persistence or lifetime of each compo- +nent can then be given by 𝑎𝑦 − 𝑎𝑥. At implementation level, one +can view PDs(∈ 𝑅𝑁×2) of (G+ +𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 as tensors which +are concatenated into one common tensor representing positive +triple graph G+. Hence, final PD of G+ is a concatenation of PDs +of (G+ +𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎. This final persistent diagram represents +a summary of the local and global topological features of the graph +G+. Following are the benefits of a persistent diagram against con- +sidering the whole graph: 1) a 2D summary of a higher dimensional +graph structure data is highly beneficial for large graphs in terms +of the computational efficiency. 2) The summary could contain +fewer data points than the original graph, preserving the topologi- +cal information. Similarly, the process is repeated for negative triple +graph G− for creating its persistence diagram. Now, the two newly +created PDs are used for calculating the proposed metric score. +4.2.3 +Sliced Wasserstein distance computation. To compare two +PDs, generally the Wasserstein distance between them is computed +[16]. As the Wasserstein distance could be computationally costly, +we find the sliced Wasserstein distance [13] between the PDs, which +we empirically observe to be eight times faster on average. The +Sliced Wasserstein distance(𝑆𝑊 ) between measures 𝜇 and 𝜈 is: +𝑆𝑊𝑝 (𝜇,𝜈) = +�∫ +𝑆𝑑−1 𝑊 𝑝 +𝑝 (𝑅𝜇 (.,𝜃), 𝑅𝜈 (.,𝜃)) +� 1 +𝑝 +where 𝑅𝜇 (.,𝜃) is the projection of 𝜇 along 𝜃,𝑊 is initial Wasserstein +distance. Generally a Monte Carlo average over 𝐿 samples is done +instead of the integral. The 𝑆𝑊 distance takes O(𝐿𝑁𝑑 +𝐿𝑁𝑙𝑜𝑔(𝑁)) +time which can be improved to linear time O(𝑁𝑑) for 𝑆𝑊2 (i.e., +Euclidean distance) as a closed form solution [30]. Thus, +KP(G+, G−) = 𝑆𝑊 (𝐷+, 𝐷−) +(1) +where 𝐷+, 𝐷− are the persistence diagrams for G+, G− respectively. +Since the metric is obtained by summarizing the Knowledge graph +using Persistence diagrams we term it as Knowledge Persistence(KP). +As KP correlates well with ranking metrics (sections 4.2.4 and 5), +higher KP signifies a better performance of the KGE method. +4.2.4 +Theoretical justification. This section briefly states the theo- +retical results justifying the proposed method to approximate the + +1Can Persistent Homology provide an efficient alternative +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +ranking metrics. We begin the analysis by assuming two distribu- +tions: One for the positive graph’s edge weights(scores) and the +other for the negative graph. We define a metric "PERM" (Figure 3), +that is a proxy to the ranking metrics while being continuous(for +the definition of integrals and derivatives) for ease of theoretical +analysis. The proof sketches are given in the appendix. +Figure 3: Figure gives an intuition of the metric PERM which +is designed to be a proxy to the ranking metrics for ease of +theoretical analysis. For a given positive triple 𝜏 with score +𝑥𝜏 the expected rank(𝐸𝑅(𝜏)) is defined as the area under the +curve of the negative distribution from 𝑥𝜏 to ∞(shown in the +shaded area above). PERM is then defined as the expectation +of the expected rank under the positive distribution. +Definition 4.1 (Expected Ranking(ER)). Consider the positive triples +to have the distribution 𝐷+ and the negative triples to have the +distribution 𝐷−. For a positive triple with score 𝑎 its expected rank- +ing(ER) is defined as, 𝐸𝑅(𝑎) = +∫ 𝑥=∞ +𝑥=𝑎 +𝐷−(𝑥)𝑑𝑥 +Definition 4.2 (PERM). Consider the positive triples to have the +distribution 𝐷+ and the negative triples to have the distribution 𝐷−. +The PERM metric is then defined as, 𝑃𝐸𝑅𝑀 = +∫ 𝑥=∞ +𝑥=−∞ 𝐷+(𝑥)𝐸𝑅(𝑥)𝑑𝑥 +It is easy to see that PERM has a monotone increasing corre- +spondence with the actual ranking metrics. That is, as many of +the negative triples get a higher score than the positive triples, the +distribution of the negative triples will shift further right of the pos- +itive distribution. Hence, the area under the curve would increase +for a given triple(x=a). We just established a monotone increasing +correspondence of PERM with the ranking metrics, we now need +show that there exists a one-one correspondence between PERM +and KP. For closed-form solutions, we work with normalised dis- +tributions (can be extended to other distributions using [39]) of +KGE score under the following mild consideration: As the KGE +method converges, the mean statistic(𝑚𝜈) of the scores of the posi- +tive triples consistently lies on one side of the half-plane formed +by the mean statistic(𝑚𝜇) of the negative triples, irrespective of the +data distribution. +Lemma 4.1. KP has a monotone increasing correspondence with +the Proxy of the Expected Ranking Metrics(PERM) under the above +stated considerations as 𝑚𝜈 deviates from 𝑚𝜇 +The above lemma shows that there is a one-one correspondence +between KP and PERM and by definition PERM has a one-one cor- +respondence with the ranking metrics. Therefore, the next theorem +follows as a natural consequence: +Theorem 4.3. KP has a one-one correspondence with the ranking +metrics under the above stated considerations. +The above theorem states that, with high probability, there exists +a correlation between KP and the ranking metrics under certain +considerations and proof details are in the appendix. In an ideal +case, we seek a linear relationship between the proposed mea- +sure and the ranking metric. This would help interpret whether +an increase/decrease in the measure would cause a corresponding +increase/decrease in the ranking metric we wish to simulate. Such +interpretation becomes essential when the proposed metric has +different behavior from the existing metric. While the correlation +could be high, for interpretability of the results, we would also like +the change in KP to be bounded for a change in the scores(ranking +metrics). The below theorem gives a sense for this bound. +Theorem 4.4. Under the considerations of theorem 4.3, the relative +change in KP on addition of random noise to the scores is bounded +by a function of the original and noise-induced covariance matrix +as ΔK P +K P ≤ 𝑚𝑎𝑥((1 − |Σ+1 +𝜇1 Σ−1 +𝜇2 | +3 +2 ), (1 − |Σ+1 +𝜈1 Σ−1 +𝜈2 | +3 +2 )), where Σ𝜇1 and +Σ𝜈1 are the covariance matrices of the positive and negative triples’ +scores respectively and Σ𝜇2 and Σ𝜈2 are that of the corrupted scores. +Theorem 4.4 gives a bound on the change in KP while inducing +noise in the KGE predictions. Ideally, the error/change would be 0, +and as the noise is increased(and the ranking changed), gradually, +the KP value also changes in a bounded manner as desired. +5 +EXPERIMENTAL SETUP +For de-facto KGC task (c.f., section 4.1), we use popular KG embed- +ding methods from its various categories: (1) Translation: TransE +[8], TransH [57], TransR [26] (2) Bilinear, Rotation, and Factoriza- +tion: RotatE [44] TuckER [3], and ComplEx [48], (3) Neural Network +based: ConvKB [32]. The method selection and evaluation choices +are similar to [28, 37] that propose new metrics for KG embeddings. +All methods run on a single P100 GPU machine for a maximum of +100 epochs each and evaluated every 5 epochs. For training/testing +the KG embedding methods we make use of the pykg2vec [61] +library and validation runs are executed 20 times on average. We +use the standard/best hyperparameters for these datasets that the +considered KGE methods reported [3, 8, 26, 44, 48, 57, 61]. +5.1 +Datasets +We use standard English KG completion datasets: WN18, WN18RR, +FB15k237, FB15k, YAGO3-10 [2, 44]. The WN18 dataset is obtained +from Wordnet [27] containing lexical relations between English +words. WN18RR removes the inverse relations in the WN18 dataset. +FB15k is obtained from the Freebase [7] knowledge graph, and +FB15k237 was created from FB15k by removing the inverse relations. +The dataset details are in the Table 1. For scaling experiment, we +rely on large scale YAGO3-10 dataset [2] and due to brevity, results +for Yago3-10 are in appendix ( cf., Figure 6 and table 9). +5.2 +Comparative Methods +Considering ours is the first work of its kind, we select some com- +petitive baselines as below and explain "why" we chose them. For +evaluation, we report correlation [14] between KP and baselines +with ranking metrics (Hits@N (N= 1,3,10), MRR and MR). +Conicity [40]: It finds the average cosine of the angle between an +embedding and the mean embedding vector. In a sense, it gives + +PERM =E.(ER(T) +Distribution +Distribution +of positive +ER(T) +of neqative +triples +triples +Score of a positive triple TWWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Bastos, et al. +spread of a KG embedding method in space. We would like to +observe instead of topology, if calculating geometric properties of +a KG embedding method be an alternative for ranking metric. +Average Vector Length: This metric was also proposed by Sharma +et al. [40] to study the geometry of the KG embedding methods. It +computes the average length of the embeddings. +Graph Kernel (GK): we use graph kernels to compare the two +graphs(G+, G−) obtained for our approach. The rationale is to check +if we could get some distance metric that correlates with the ranking +metrics without persistent homology. Hence, this baseline empha- +sizes a direct comparison for the validity of persistent homology in +our proposed method. As an implementation, we employ the widely +used shortest path kernel [10] to compare how the paths(edge +weights/scores) change between the two graphs. Since the method +is computationally expensive, we sample nodes [11] and apply the +kernel on the sampled graph, averaging multiple runs. +Table 1: (Open-Source)Benchmark Datasets for Experi- +ments. +Dataset +Triples +Entities +Relations +FB15K +592,213 +14.951 +1,345 +FB15K-237 +272,115 +14,541 +237 +WN18 +151,442 +40,943 +18 +WN18RR +93,003 +40,943 +11 +Yago3-10 +1,089,040 +123,182 +37 +6 +RESULTS AND DISCUSSION +We conduct our experiments in response to the following research +questions: RQ1: Is there a correlation between the proposed metric +and ranking metrics for popular KG embedding methods? RQ2: +Can the proposed metric be used to perform early stopping during +training? RQ3: What is the computational efficiency of proposed +metric wrt ranking metrics for KGE evaluation? +KP for faster prototyping of KGE methods: Our core hypoth- +esis in the paper is to develop an efficient alternative (proxy) to +the ranking metrics. Hence, for a fair evaluation, we use the triples +in the test set for computing KP. Ideally, this should be able to +simulate the evaluation of the ranking metrics on the same (test) +set. If true, there exists a high correlation between the two mea- +sures, namely the KP and the ranking metrics. Table 2 shows the +linear correlations between the ranking metrics and our method & +baselines. We report the linear(Pearson’s) correlation because we +would like a linear relationship between the proposed measure and +the ranking metric (for brevity, other correlations are in appendix +Tables 7, 8). This would help interpret whether an increase/decrease +in the measure would cause a corresponding increase/decrease in +the ranking metric that we wish to simulate. Specifically we train all +the KG embedding methods for a predefined number of epochs and +evaluate the finally obtained models to get the ranking metrics and +KP. The correlations are then computed between KP and each +of the ranking metrics. We observe that KP(test) configuration +(triples are sampled from the test set) achieves the highest correla- +tion coefficient value among all the existing geometric and kernel +baseline methods in most cases. For instance, on FB15K, KP(test) +reports high correlation value of 0.786 with Hits@1, whereas best +baseline for this dataset (AVL) has corresponding correlation value +as 0.339. Similarly for WN18RR, KP(test) has correlation value +of 0.482 compared to AVL with -0.272 correlation with Hits@1. +Conicity and AVL that provide geometric perspective shows mostly +low positive correlation with ranking metrics whereas the Graph +Kernel based method shows highly negative correlations, making +these methods unsuitable for direct applicability. It indicates that +the topology of the KG induced by the learnt representations seems +a good predictor of the performance on similar data distributions +with high correlation with ranking metric (answering RQ1). +Furthermore, the results also report a configuration KP(train) in +which we compute KP on the triples of the train set and find the +correlation with the ranking metrics obtained from the test set. +Here our rationale is to study whether the proposed metric would +be able to capture the generalizability of the unseen test (real world) +data that is of a similar distribution as the training data. Initial re- +sults in Table 2 are promising with high correlation of KP(train) +with ranking metric. Hence, it may enable the use of KP in settings +without test/validation data while using the available (possibly lim- +ited) data for training, for example, in few-shot scenarios. We leave +this promising direction of research for future. +6.1 +KP as a criterion for early stopping +Does KP hold correlation while early stopping? To know +when to stop the training process to prevent overfitting, we must +be able to estimate the variance of the model. This is generally done +by observing the validation/test set error. Thus, to use a method as +a criterion for early stopping, it should be able to predict this gen- +eralization error. Table 2 explains that KP(Train) can predict the +generalizability of methods on the last epoch, it remains to empiri- +cally verify that KP also predicts the performance at every interval +during the training process. Hence, we study the correlations of +the proposed method with the ranking metrics for individual KG +embedding methods in the intra-method setting. Specifically, for +a given method, we obtain the KP score and the ranking metrics +on the test set and compute the correlations at every evaluation +interval. Results in Table 3 suggest that KP has a decent correla- +tion in the intra-method setting. It indicates that KP could be used +in place of the ranking metrics for deciding a criterion on early +stopping if the score keeps persistently falling (answering RQ2). +What is the relative error of early stopping between KP +and Ranking Metric? To further cross-validate our response to +RQ2, we now compute the absolute relative error between the rank- +ing metrics of the best models selected by KP and the expected +ranking metrics. Ideally, we would expect the performance of the +model obtained using this process on unseen test data(preferably +of the same distribution) to be close to the best achievable result, +i.e., the relative error should be small. This is important as if we +were to use any metric for faster prototyping, it should also be +a good criterion for model selection(selecting a model with less +generalization error) and being efficient. Table 4 shows that the +relative error is marginal, of the order of 10−2, in most cases(with +few exceptions), indicating that KP could be used for early stop- +ping. The deviation is higher for some methods, such as ConvKB, + +Can Persistent Homology provide an efficient alternative +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Metrics +FB15K +FB15K237 +WN18 +WN18RR +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Conicity +-0.156 +-0.170 +-0.202 +0.085 +-0.183 +0.509 +0.379 +0.356 +-0.352 +0.424 +-0.052 +-0.096 +-0.123 +0.389 +-0.096 +-0.267 +-0.471 +-0.510 +0.266 +-0.448 +AVL +0.339 +0.325 +0.261 +-0.423 +0.308 +-0.527 +-0.149 +-0.158 +0.188 +-0.284 +0.805 +0.825 +0.856 +-0.884 +0.840 +-0.272 +-0.456 +-0.488 +0.303 +-0.438 +GK(train) +-0.825 +-0.852 +-0.815 +0.952 +-0.843 +-0.903 +-0.955 +-0.972 +0.970 +-0.965 +-0.645 +-0.648 +-0.669 +0.611 +-0.663 +-0.518 +-0.808 +-0.840 +0.591 +-0.779 +GK(test) +-0.285 +-0.318 +-0.247 +0.629 +-0.300 +-0.031 +-0.130 +-0.123 +0.101 +-0.095 +-0.579 +-0.565 +-0.569 +0.412 +-0.575 +-0.276 +-0.589 +-0.658 +0.470 +-0.549 +KP (Train) +0.482 +0.418 +0.449 +-0.072 +0.433 +0.773 +0.711 +0.702 +-0.714 +0.745 +0.769 +0.769 +0.782 +-0.682 +0.780 +0.500 +0.809 +0.852 +-0.755 +0.777 +KP (Test) +0.786 +0.731 +0.661 +-0.669 +0.721 +0.825 +0.870 +0.864 +-0.861 +0.871 +0.875 +0.887 +0.909 +-0.884 +0.899 +0.482 +0.816 +0.863 +-0.683 +0.776 +Table 2: Pearson’s linear correlation (𝑟) scores computed from the metric scores with respect to the ranking metrics on the +standard KG embedding datasets. The KG methods are evaluated after training. Green values are the best. +Datasets +FB15K237 +WN18RR +KG methods +r +𝜌 +𝜏 +r +𝜌 +𝜏 +TransE +0.955 +0.861 +0.709 +0.876 +0.833 +0.722 +TransH +0.688 +0.570 +0.409 +0.864 +0.717 +0.555 +TransR +0.975 +0.942 +0.811 +0.954 +0.967 +0.889 +Complex +0.938 +0.788 +0.610 +0.833 +0.933 +0.833 +RotatE +0.896 +0.735 +0.579 +0.774 +0.983 +0.944 +TuckER +0.906 +0.676 +0.527 +0.352 +0.25 +0.167 +ConvKB +0.086 +0.012 +0.007 +0.276 +0.569 +0.422 +Table 3: Correlation scores computed between KP and the +ranking metric(Hits@10) on the standard KG embedding +datasets with the methods evaluated at every interval as the +training progresses. Here, r: Pearson correlation co-efficient, +𝜌: Spearman’s correlation co-efficient, 𝜏: Kendall’s Tau. +which had convergence issues. We infer from observed behavior +that if the KG embedding method has not converged(to good re- +sults), the correlation and, thus, the early stopping prediction may +suffer. Despite a few outliers, the promising results shall encourage +the community to research, develop, and use KGE benchmarking +methods that are also computationally efficient. +Datasets +FB15K237 +WN18RR +KG methods +hits@1 +hits@10 +MRR +hits@1 +hits@10 +MRR +TransE +0.006 +0.006 +0.007 +0.000 +0.007 +0.004 +TransH +0.045 +0.015 +0.019 +0.130 +0.018 +0023 +TransR +0.074 +0.045 +0.053 +0.242 +0.062 +0.016 +Complex +0.001 +0.002 +0.003 +0.317 +0.021 +0.028 +RotatE +0.022 +0.009 +0.007 +0.017 +0.005 +0.009 +TuckER +0.008 +0.006 +0.002 +0.293 +0.022 +0.101 +ConvKB +0.000 +0.043 +0.043 +0.659 +0.453 +0.569 +Table 4: Early stopping using KP. The values depict the ab- +solute relative error between the metrics of the best models +selected using KP and ranking metrics. +6.2 +Timing analysis and carbon footprint +We now study the time taken for running the evaluation (including +evaluation at intervals) of the same methods as in section 6.1 on +the standard datasets. Table 5 shows the evaluation times (valida- +tion+test) and speedup for each method on the respective datasets. +The training time is constant for ranking metric and KP. In some +cases (ConvKB), we observe KP achieves a speedup of up to 2576 +times on model evaluation time drastically reducing evaluation time +from 18 hours to 27 seconds; the latter is even roughly equal to the +carbon footprint of making a cup of coffee2. Furthermore, Figure +2https://tinyurl.com/4w2xmwry +Figure 4: Figure shows a study on the carbon footprint on +WN18RR when using KP vs Hits@10. The x-axis shows the +the carbon footprint in g eq 𝐶𝑂2. +4 illustrates the carbon footprints [33, 59] of the overall process +(training + evaluation) for the methods when using KP vs ranking +metrics. Due to evaluation time drastically reduced by KP, it also +reduces overall carbon footprints. The promising results validate +our attempt to develop alternative method for faster prototyping +of KGE methods, thus saving carbon footprint (answering RQ3). +6.3 +Ablation Studies +We systematically provide several studies to support our evaluation +and characterize different properties of KP. +Robustness to noise induced by sampling: An important +property that makes persistent homology worthwhile is its stability +concerning perturbations of the filtration function. This means that +persistent homology is robust to noise and encodes the intrinsic +topological properties of the data [19]. However, in our applica- +tion of predicting the performance of KG embedding methods, one +source of noise is because of sampling the negative and positive +triples. It could cause perturbations in the graph topology due to the +addition and deletion of edges (cf., Figure 2). Therefore, we would +like the proposed metric to be stable concerning perturbations. To +understand the behavior of KP against this noise, we conduct a +study by incrementally adding samples to the graph and observing +the mean and standard deviation of the correlation at each stage. +In an ideal case, assuming the KG topology remains similar, the +mean correlations should be in a narrow range with slight standard +deviations. We observe a similar effect in Figure 5 where we report +the mean correlation at various fractions of triples sampled, with +the standard deviation(error bands). Here, the mean correlation +coefficients are within the range of 0.06(0.04), and the average stan- +dard deviations are about 0.02(0.02) for the FB15K237(WN18RR) + +Carbon Footprint of the Overall KGE prototyping process +Carbon Footprint using KP +Carbon Footprint using ranking metrics +TransE +TransH +TransR +Complex +RotatE +ConvKB +TuckER +0 +100 +200 +300 +400WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Bastos, et al. +Metrics +Hits@10 +KP +Speedup ↑ +Dataset +FB15K237 +WN18RR +FB15K237 +WN18RR +FB15K237 +WN18RR +split +val + test +val + test +val + test +val + test +Avg +Avg +TransE +103.6 +86.1 +0.337 +0.120 +x 322.8 +x 754.1 +TransH +37.1 +21.2 +0.333 +0.099 +x 117.0 +x 224.4 +TransR +192.0 +137.1 +0.352 +0.135 +x 572.0 +x 1066.4 +Complex +136.1 +151.4 +0.340 +0.142 +x 420.1 +x 1121.7 +RotatE +174.2 +155.2 +0.359 +0.142 +x 509.5 +x 1145.6 +TuckER +94.8 +22.1 +0.332 +0.098 +x 300.0 +x 241.9 +ConvKB +1106.0 +138.1 +0.451 +0.139 +x 2576.6 +x 1044.3 +Table 5: Evaluation Metric Comparison wrt Computing +Time (in minutes, for 100 epochs). Column 1 denotes +popular KGE methods. Depicted values denote evalua- +tion(validation+test) time for computing a metric and corre- +sponding speedup using KP. KP significantly reduces the +evaluation time (green). +dataset. This shows that KP inherits the robustness of the topo- +logical data analysis techniques, enabling linear time by sampling +from the graph for dense KGs while keeping it robust. +Figure 5: Effect of sample size on the correlation coefficient +between KP and the ranking metrics on FB15K237 (left dia- +gram) and WN18RR datasets. The correlations for the differ- +ent sampling fractions are comparable. Also, the standard +deviation is less, indicating the method’s robustness due to +changes in local topology while doing sampling. +Generalizability Study- Correlation with Stratified Rank- +ing Metric: Mohamed et al. [28] proposed a new stratified metric +(strat-metric) that can be tuned to focus on the unpopular entities, +unlike the standard ranking metrics, using certain hyperparameters +(𝛽𝑒 ∈ (−1, 1), 𝛽𝑟 ∈ (−1, 1)). Special cases of these hyperparame- +ters give the micro and macro ranking metrics. Goal here is to +study whether our method can predict strat-metric for the spe- +cial case of 𝛽𝑒 = 1, 𝛽𝑟 = 0, which estimates the performance for +unpopular(sparse) entities. Also, we aim to observe if KP holds +a correlation with variants of the ranking metric concerning its +generalization ability. The results (cf., Table 6) shows that KP has +a good correlation with each of the stratified ranking metrics which +indicate KP also takes into account the local geometry/topology +[1] of the sparse entities and relations. +6.4 +Summary of Results and Open Directions +To sum up, following are key observations gleaning from empirical +studies: 1) KP shows high correlation with ranking metrics (Ta- +ble 2) and its stratified version (Table 6). It paves the way for the +use of KP for faster prototyping of KGE methods. 2) KP holds +Datasets +FB15K237 +WN18RR +Metrics +r +𝜌 +𝜏 +r +𝜌 +𝜏 +Strat-Hits@1 (↑) +0.965 +0.857 +0.714 +0.513 +0.482 +0.411 +Strat-Hits@3 (↑) +0.898 +0.821 +0.619 +0.691 +0.714 +0.524 +Strat-Hits@10 (↑) +0.871 +0.821 +0.619 +0.870 +0.750 +0.619 +Strat-MR (↓) +-0.813 +-0.679 +-0.524 +-0.701 +-0.821 +-0.619 +Strat-MRR (↑) +0.806 +0.679 +0.524 +0.658 +0.714 +0.524 +Table 6: KP correlation with stratified ranking metrics as +proposed in [28]. +a high correlation at every interval during the training process +(Table 3) with marginal relative error; hence, it could be used for +early stopping of a KGE method. 3) KP inherits key properties of +persistent homology, i.e., it is robust to noise induced by sampling. +4) The overall carbon footprints of the evaluation cycle is drastically +reduced if KP is preferred over ranking metrics. +What’s Next? We show that topological data analysis based on +persistent homology can act as a proxy for ranking metrics with +conclusive empirical evidence and supporting theoretical founda- +tions. However, it is the first step toward a more extensive research +agenda. We believe substantial work is needed collectively in the +research community to develop strong foundations, solving scal- +ing issues (across embedding methods, datasets, KGs, etc.) until +persistent homology-based methods are widely adopted. +For example, there could be methods/datasets where the correla- +tion turns out to be a small positive value or even negative, in which +case we may not be able to use KP in the existing form to simu- +late the ranking metrics for these methods/datasets. In those cases, +some alteration may exist for the same and seek further exploration +similar to what stratified ranking metric [28] does by fixing issues +encountered in the ranking metric. Furthermore, theorem 4.4 would +be a key to understand error bounds when interpreting limited per- +formance (e.g., when the correlation is a small positive). However, +this does not limit the use of KP for KGE methods as it captures and +contrasts the topology of the positive and negative sampled graphs +learned from these methods, which could be a useful metric by +itself. In this paper, the emphasis is on the need for evaluation and +benchmarking methods that are computationally efficient rather +than providing an exhaustive one method fits all metric. We believe +that there is much scope for future research in this direction. Some +promising directions include 1) better sampling techniques(instead +of the random sampling used in this paper), 2) rigorous theoretical +analysis drawing the boundaries on the abilities/limitations across +settings (zero-shot, few-shot, etc.), 3) using KP (and related metrics) +in continuous spaces, that could be differentiable and approximate +the ranking metrics, in the optimization process of KGE methods. +7 +CONCLUSION +We propose Knowledge Persistence (KP), first work that uses tech- +niques from topological data analysis, as a predictor of the ranking +metrics to efficiently evaluate the performance of KG embedding +approaches. With theoretical and empirical evidences, our work +brings efficiency at center stage in the evaluation of KG embedding +methods along with traditional way of reporting their performance. +Finally, with efficiency as crucial criteria for evaluation, we hope + +Hits@10 +0.90 +MRR +0.88 +0.86 +0.84 +0.82 +0.80 +0.78 +0.2 +0.4 +0.6 +0.8 +1.00.95 +Hits@10 +MRR +0.90 +0.85 +0.80 +0.75 +0.2 +0.4 +0.6 +0.8 +1.0Can Persistent Homology provide an efficient alternative +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +KGE research becomes more inclusive and accessible to the broader +research community with limited computing resources. +Acknowledgment This work was partly supported by JSPS +KAKENHI Grant Number JP21K21280. +REFERENCES +[1] Henry Adams and Michael Moy. 2021. Topology Applied to Machine +Learning: From Global to Local. Frontiers in Artificial Intelligence 4 +(2021), 54. +[2] Mehdi Ali, Max Berrendorf, Charles Tapley Hoyt, Laurent Vermue, +Mikhail Galkin, Sahand Sharifzadeh, Asja Fischer, Volker Tresp, and +Jens Lehmann. 2021. Bringing light into the dark: A large-scale evalua- +tion of knowledge graph embedding models under a unified framework. +IEEE Transactions on Pattern Analysis and Machine Intelligence (2021). +[3] Ivana Balažević, Carl Allen, and Timothy Hospedales. 2019. TuckER: +Tensor Factorization for Knowledge Graph Completion. In Proceedings +of the 2019 Conference on Empirical Methods in Natural Language Pro- +cessing and the 9th International Joint Conference on Natural Language +Processing (EMNLP-IJCNLP). 5185–5194. +[4] Iti Bansal, Sudhanshu Tiwari, and Carlos R Rivero. 2020. The impact +of negative triple generation strategies and anomalies on knowledge +graph completion. In Proceedings of the 29th ACM International Con- +ference on Information & Knowledge Management. 45–54. +[5] Anson Bastos, Kuldeep Singh, Abhishek Nadgeri, Saeedeh Shekarpour, +Isaiah Onando Mulang, and Johannes Hoffart. 2021. Hopfe: Knowl- +edge graph representation learning using inverse hopf fibrations. In +Proceedings of the 30th ACM International Conference on Information & +Knowledge Management. 89–99. +[6] Max Berrendorf, Evgeniy Faerman, Laurent Vermue, and Volker Tresp. +2020. Interpretable and Fair Comparison of Link Prediction or Entity +Alignment Methods. In 2020 IEEE/WIC/ACM International Joint Con- +ference on Web Intelligence and Intelligent Agent Technology (WI-IAT). +IEEE, 371–374. +[7] Kurt D. Bollacker, Robert P. Cook, and Patrick Tufts. 2007. Freebase: A +Shared Database of Structured General Human Knowledge. In AAAI. +[8] Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, +and Oksana Yakhnenko. 2013. Translating embeddings for modeling +multi-relational data. In NeurlPS. 1–9. +[9] Antoine Bordes, Jason Weston, Ronan Collobert, and Yoshua Bengio. +2011. Learning structured embeddings of knowledge bases. In Twenty- +fifth AAAI conference on artificial intelligence. +[10] Karsten M Borgwardt and Hans-Peter Kriegel. 2005. Shortest-path +kernels on graphs. In Fifth IEEE international conference on data mining +(ICDM’05). IEEE, 8–pp. +[11] Karsten M. Borgwardt, Tobias Petri, S. V. N. Vishwanathan, and Hans- +Peter Kriegel. 2007. An Efficient Sampling Scheme For Comparison of +Large Graphs. In Mining and Learning with Graphs, MLG. +[12] Andrei Z Broder, Marc Najork, and Janet L Wiener. 2003. Efficient +URL caching for world wide web crawling. In Proceedings of the 12th +international conference on World Wide Web. 679–689. +[13] Mathieu Carrière, Marco Cuturi, and Steve Oudot. 2017. Sliced Wasser- +stein Kernel for Persistence Diagrams. In Proceedings of the 34th In- +ternational Conference on Machine Learning (Proceedings of Machine +Learning Research), Vol. 70. PMLR, 664–673. +[14] Nian Shong Chok. 2010. Pearson’s versus Spearman’s and Kendall’s cor- +relation coefficients for continuous data. Ph.D. Dissertation. University +of Pittsburgh. +[15] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. 2000. +Topological persistence and simplification. In Proceedings 41st annual +symposium on foundations of computer science. IEEE, 454–463. +[16] Brittany Fasy, Yu Qin, Brian Summa, and Carola Wenk. 2020. Compar- +ing Distance Metrics on Vectorized Persistence Summaries. In NeurIPS +2020 Workshop on Topological Data Analysis and Beyond. +[17] Luis Galárraga, Katja Hose, and Ralf Schenkel. 2014. Partout: a dis- +tributed engine for efficient RDF processing. In Proceedings of the 23rd +International Conference on World Wide Web. 267–268. +[18] Genet Asefa Gesese, Russa Biswas, Mehwish Alam, and Harald Sack. +2019. A survey on knowledge graph embeddings with literals: Which +model links better literal-ly? Semantic Web Preprint (2019), 1–31. +[19] Felix Hensel, Michael Moor, and Bastian Rieck. 2021. A survey of topo- +logical machine learning methods. Frontiers in Artificial Intelligence 4 +(2021), 52. +[20] Nitisha Jain, Jan-Christoph Kalo, Wolf-Tilo Balke, and Ralf Krestel. +2021. Do Embeddings Actually Capture Knowledge Graph Semantics?. +In European Semantic Web Conference. Springer, 143–159. +[21] Prachi Jain, Sushant Rathi, Soumen Chakrabarti, et al. 2020. Knowl- +edge base completion: Baseline strikes back (again). arXiv preprint +arXiv:2005.00804 (2020). +[22] Shaoxiong Ji, Shirui Pan, Erik Cambria, Pekka Marttinen, and Philip S +Yu. 2021. A survey on knowledge graphs: Representation, acquisition +and applications. EEE Transactions on Neural Networks and Learning +Systems (2021). +[23] Rudolf Kadlec, Ondřej Bajgar, and Jan Kleindienst. 2017. Knowledge +Base Completion: Baselines Strike Back. In Proceedings of the 2nd +Workshop on Representation Learning for NLP. 69–74. +[24] Zelong Li, Jianchao Ji, Zuohui Fu, Yingqiang Ge, Shuyuan Xu, Chong +Chen, and Yongfeng Zhang. 2021. Efficient non-sampling knowledge +graph embedding. In Proceedings of the Web Conference 2021. 1727– +1736. +[25] Lipyeow Lim, Min Wang, Sriram Padmanabhan, Jeffrey Scott Vitter, +and Ramesh Agarwal. 2003. Dynamic maintenance of web indexes +using landmarks. In Proceedings of the 12th international conference on +World Wide Web. 102–111. +[26] Yankai Lin, Zhiyuan Liu, Maosong Sun, Yang Liu, and Xuan Zhu. 2015. +Learning entity and relation embeddings for knowledge graph com- +pletion. In Proceedings of the AAAI Conference on Artificial Intelligence, +Vol. 29. +[27] George A Miller. 1995. WordNet: a lexical database for English. Com- +mun. ACM 38, 11 (1995), 39–41. +[28] Aisha Mohamed, Shameem Parambath, Zoi Kaoudi, and Ashraf Aboul- +naga. 2020. Popularity agnostic evaluation of knowledge graph em- +beddings. In Conference on Uncertainty in Artificial Intelligence. PMLR, +1059–1068. +[29] Michael Moor, Max Horn, Bastian Rieck, and Karsten Borgwardt. 2020. +Topological autoencoders. In International conference on machine learn- +ing. PMLR, 7045–7054. +[30] Kimia Nadjahi, Alain Durmus, Pierre E Jacob, Roland Badeau, and +Umut Simsekli. 2021. Fast Approximation of the Sliced-Wasserstein +Distance Using Concentration of Random Projections. Advances in +Neural Information Processing Systems 34 (2021). +[31] Mojtaba +Nayyeri, +Chengjin +Xu, +Yadollah +Yaghoobzadeh, +Hamed Shariat Yazdi, and Jens Lehmann. 2019. +Toward Un- +derstanding The Effect Of Loss function On Then Performance Of +Knowledge Graph Embedding. arXiv preprint arXiv:1909.00519 (2019). +[32] Tu Dinh Nguyen, Dat Quoc Nguyen, Dinh Phung, et al. 2018. A +Novel Embedding Model for Knowledge Base Completion Based on +Convolutional Neural Network. In NAACL. 327–333. +[33] David Patterson, Joseph Gonzalez, Urs Hölzle, Quoc Le, Chen Liang, +Lluis-Miquel Munguia, Daniel Rothchild, David So, Maud Texier, and +Jeff Dean. 2022. The Carbon Footprint of Machine Learning Training +Will Plateau, Then Shrink. arXiv preprint arXiv:2204.05149 (2022). +[34] Xutan Peng, Guanyi Chen, Chenghua Lin, and Mark Stevenson. 2021. +Highly Efficient Knowledge Graph Embedding Learning with Orthog- +onal Procrustes Analysis. In NAACL. 2364–2375. +[35] Pouya Pezeshkpour, Yifan Tian, and Sameer Singh. 2020. Revisit- +ing evaluation of knowledge base completion models. In Automated +Knowledge Base Construction. +[36] Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max +Horn, Thomas Gumbsch, and Karsten Borgwardt. 2018. Neural Per- +sistence: A Complexity Measure for Deep Neural Networks Using +Algebraic Topology. In International Conference on Learning Represen- +tations. +[37] Wiem Ben Rim, Carolin Lawrence, Kiril Gashteovski, Mathias Niepert, +and Naoaki Okazaki. 2021. Behavioral Testing of Knowledge Graph +Embedding Models for Link Prediction. In 3rd Conference on Automated +Knowledge Base Construction. +[38] Tara Safavi and Danai Koutra. 2020. CoDEx: A Comprehensive Knowl- +edge Graph Completion Benchmark. In Proceedings of the 2020 Confer- +ence on Empirical Methods in Natural Language Processing (EMNLP). +8328–8350. +[39] Remi M Sakia. 1992. The Box-Cox transformation technique: a review. +Journal of the Royal Statistical Society: Series D (The Statistician) 41, 2 +(1992), 169–178. +[40] Aditya Sharma, Partha Talukdar, et al. 2018. Towards understanding +the geometry of knowledge graph embeddings. In Proceedings of the +56th Annual Meeting of the Association for Computational Linguistics +(Volume 1: Long Papers). 122–131. + +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Bastos, et al. +[41] Kuldeep Singh, Arun Sethupat Radhakrishna, Andreas Both, Saeedeh +Shekarpour, Ioanna Lytra, Ricardo Usbeck, Akhilesh Vyas, Akmal +Khikmatullaev, Dharmen Punjani, Christoph Lange, et al. 2018. Why +reinvent the wheel: Let’s build question answering systems together. +In Proceedings of the 2018 world wide web conference. 1247–1256. +[42] C. Spearman. 1907. Demonstration of Formulæ for True Measurement +of Correlation. The American Journal of Psychology 18, 2 (1907), 161– +169. http://www.jstor.org/stable/1412408 +[43] Marina Speranskaya, Martin Schmitt, and Benjamin Roth. 2020. Rank- +ing vs. Classifying: Measuring Knowledge Base Completion Quality. +In Automated Knowledge Base Construction. +[44] Zhiqing Sun, Zhi-Hong Deng, Jian-Yun Nie, and Jian Tang. 2018. Ro- +tatE: Knowledge Graph Embedding by Relational Rotation in Complex +Space. In International Conference on Learning Representations. +[45] Zhiqing Sun, Shikhar Vashishth, Soumya Sanyal, Partha Talukdar, and +Yiming Yang. 2020. A Re-evaluation of Knowledge Graph Completion +Methods. In Proceedings of the 58th Annual Meeting of the Association +for Computational Linguistics. 5516–5522. +[46] Pedro Tabacof and Luca Costabello. 2019. Probability Calibration for +Knowledge Graph Embedding Models. In International Conference on +Learning Representations. +[47] Sudhanshu Tiwari, Iti Bansal, and Carlos R Rivero. 2021. Revisiting +the evaluation protocol of knowledge graph completion methods for +link prediction. In Proceedings of the Web Conference 2021. 809–820. +[48] Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier, and +Guillaume Bouchard. 2016. +Complex embeddings for simple link +prediction. In International Conference on Machine Learning. PMLR, +2071–2080. +[49] Ke Tu, Jianxin Ma, Peng Cui, Jian Pei, and Wenwu Zhu. 2019. Au- +tone: Hyperparameter optimization for massive network embedding. +In Proceedings of the 25th ACM SIGKDD International Conference on +Knowledge Discovery & Data Mining. 216–225. +[50] Renata Turkeš, Guido Montúfar, and Nina Otter. 2022. On the effec- +tiveness of persistent homology. https://doi.org/10.48550/ARXIV.2206. +10551 +[51] C. Villani. 2009. Optimal transport. Grundlehren der Mathematischen +Wissenschaften [Fundamental Principles of Mathematical Sciences] 338 +(2009). +[52] Haoyu Wang, Yaqing Wang, Defu Lian, and Jing Gao. 2021. A light- +weight knowledge graph embedding framework for efficient inference +and storage. In Proceedings of the 30th ACM International Conference +on Information & Knowledge Management. 1909–1918. +[53] Kai Wang, Yu Liu, Qian Ma, and Quan Z Sheng. 2021. Mulde: Multi- +teacher knowledge distillation for low-dimensional knowledge graph +embeddings. In Proceedings of the Web Conference 2021. 1716–1726. +[54] Kai Wang, Yu Liu, and Quan Z Sheng. 2022. Swift and Sure: Hardness- +aware Contrastive Learning for Low-dimensional Knowledge Graph +Embeddings. In Proceedings of the ACM Web Conference 2022. 838–849. +[55] Quan Wang, Zhendong Mao, Bin Wang, and Li Guo. 2017. Knowledge +graph embedding: A survey of approaches and applications. IEEE +Transactions on Knowledge and Data Engineering 29, 12 (2017), 2724– +2743. +[56] Xin Wang, Shuyi Fan, Kun Kuang, and Wenwu Zhu. 2021. Explain- +able automated graph representation learning with hyperparameter +importance. In International Conference on Machine Learning. PMLR, +10727–10737. +[57] Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen. 2014. +Knowledge graph embedding by translating on hyperplanes. In Pro- +ceedings of the AAAI Conference on Artificial Intelligence, Vol. 28. +[58] Larry Wasserman. 2018. Topological data analysis. Annual Review of +Statistics and Its Application 5 (2018), 501–532. +[59] Carole-Jean Wu, Ramya Raghavendra, Udit Gupta, Bilge Acun, New- +sha Ardalani, Kiwan Maeng, Gloria Chang, Fiona Aga, Jinshi Huang, +Charles Bai, et al. 2022. Sustainable ai: Environmental implications, +challenges and opportunities. Proceedings of Machine Learning and +Systems 4 (2022), 795–813. +[60] Bishan Yang, Wen-tau Yih, Xiaodong He, Jianfeng Gao, and Li Deng. +2015. Embedding Entities and Relations for Learning and Inference in +Knowledge Bases. In 3rd International Conference on Learning Repre- +sentations, ICLR. +[61] Shih-Yuan Yu, Sujit Rokka Chhetri, Arquimedes Canedo, Palash Goyal, +and Mohammad Abdullah Al Faruque. 2021. Pykg2vec: A Python +Library for Knowledge Graph Embedding. J. Mach. Learn. Res. 22 +(2021), 16–1. +[62] Yufeng Zhang, Weiqing Wang, Wei Chen, Jiajie Xu, An Liu, and Lei +Zhao. 2021. Meta-Learning Based Hyper-Relation Feature Modeling for +Out-of-Knowledge-Base Embedding. In Proceedings of the 30th ACM +International Conference on Information & Knowledge Management. +2637–2646. +[63] Yongqi Zhang, Zhanke Zhou, Quanming Yao, and Yong Li. 2022. KG- +Tuner: Efficient Hyper-parameter Search for Knowledge Graph Learn- +ing. CoRR abs/2205.02460 (2022). +[64] Afra Zomorodian and Gunnar Carlsson. 2005. Computing persistent +homology. Discrete & Computational Geometry 33, 2 (2005), 249–274. +8 +APPENDIX +Figure 6: Study on the carbon footprint of the evaluation +phase of the KGE methods on YAGO3-10 when using KP vs +Hits@10. The x-axis shows the the carbon footprint in g eq +𝐶𝑂2 in log scale. +Figure 7: Study on the carbon footprint of the evaluation +phase of the KGE methods on Wikidata when using KP vs +Hits@10. The x-axis shows the the carbon footprint in g eq +𝐶𝑂2 in log scale. +8.1 +Extended Evaluation +Effect of KP on Efficient KGE Methods Evaluation: The re- +search community has recently proposed several KGE methods +to improve training efficiency [34, 52, 54]. Our idea in this experi- +ment is to perceive if efficient KGE methods improve their overall +carbon footprint using KP. For the same, we selected state-of-the- +art efficient KGE methods: Procrustes [34] and HalE [54]. Figure +8 illustrates that using KP for evaluation drastically reduces the +carbon footprints of already efficient KGE methods. For instance, +the carbon footprint of HalE is reduced from 110g (using hits@10) +to 20g of CO2 (using KP). + +Carbon Footprint of the Evaluation phase of the KGE +prototyping process (YAGO3_10) +Carbon Footprint using Kp +Carbon Footprint using ranking metrics +TransE +TransH +TransR +Method +Complex +RotatE +ConvKB +TuckER +0.5 +L0Carbon Footprint of the Evaluation phase of the KGE +prototyping process (Wikidata) +Carbon Footprint using KP +Carbon Footprint using ranking metrics +TransE +TransH +TransR +Method +Complex +RotatE +ConvkB +TuckER +LOCan Persistent Homology provide an efficient alternative +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Metrics +FB15K +FB15K237 +WN18 +WN18RR +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Conicity +0.071 +-0.071 +0.000 +-0.036 +-0.214 +0.393 +0.250 +-0.071 +0.036 +0.250 +0.600 +0.600 +0.600 +-0.143 +0.600 +-0.393 +-0.607 +-0.607 +0.357 +-0.607 +AVL +0.607 +0.214 +0.250 +-0.679 +0.179 +-0.107 +-0.321 +-0.429 +0.393 +-0.250 +0.886 +0.886 +0.886 +-0.771 +0.886 +0.321 +-0.143 +-0.464 +0.607 +-0.143 +Graph Kernel (Train) +-0.536 +-0.321 +-0.357 +0.964 +-0.393 +-0.929 +-0.714 +-0.607 +0.643 +-0.821 +-0.943 +-0.943 +-0.943 +0.714 +-0.943 +-0.357 +-0.786 +-0.607 +0.571 +-0.786 +Graph Kernel (Test) +-0.107 +0.107 +0.000 +0.893 +0.036 +-0.429 +-0.607 +-0.679 +0.464 +-0.607 +-0.657 +-0.657 +-0.657 +0.086 +-0.657 +-0.393 +-0.714 +-0.821 +0.786 +-0.714 +KP (Train) +0.214 +0.536 +0.750 +0.000 +0.607 +0.893 +0.750 +0.643 +-0.679 +0.786 +0.829 +0.829 +0.829 +-0.600 +0.829 +0.286 +0.714 +0.643 +-0.750 +0.714 +KP (Test) +0.964 +0.750 +0.750 +-0.536 +0.714 +0.714 +0.821 +0.857 +-0.750 +0.857 +0.943 +0.943 +0.943 +-0.829 +0.943 +0.286 +0.714 +0.643 +-0.643 +0.714 +Table 7: Spearman’s ranked correlation (𝜌) scores computed from the metric scores with respect to the ranking metrics on the +standard KG embedding datasets. The KG methods are evaluated after training. +Metrics +FB15K +FB15K237 +WN18 +WN18RR +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Hits1(↑) +Hits3(↑) +Hits10(↑) +MR(↓) +MRR(↑) +Conicity +0.048 +-0.048 +-0.048 +0.048 +-0.143 +0.238 +0.143 +-0.048 +-0.048 +0.143 +0.467 +0.467 +0.467 +-0.333 +0.467 +-0.238 +-0.333 +-0.429 +0.238 +-0.333 +AVL +0.429 +0.143 +0.143 +-0.524 +0.048 +-0.143 +-0.238 +-0.429 +0.333 +-0.238 +0.733 +0.733 +0.733 +-0.600 +0.733 +0.238 +-0.048 +-0.333 +0.524 +-0.048 +Graph Kernel (Train) +-0.429 +-0.143 +-0.143 +0.905 +-0.238 +-0.810 +-0.524 +-0.333 +0.429 +-0.714 +-0.867 +-0.867 +-0.867 +0.467 +-0.867 +-0.333 +-0.619 +-0.333 +0.333 +-0.619 +Graph Kernel (Test) +-0.143 +0.143 +-0.048 +0.810 +0.048 +-0.429 +-0.524 +-0.524 +0.429 +-0.524 +-0.600 +-0.600 +-0.600 +0.200 +-0.600 +-0.238 +-0.524 +-0.619 +0.619 +-0.524 +KP (Train) +0.143 +0.429 +0.619 +-0.048 +0.524 +0.714 +0.619 +0.429 +-0.524 +0.619 +0.600 +0.600 +0.600 +-0.467 +0.600 +0.238 +0.524 +0.429 +-0.619 +0.524 +KP (Test) +0.905 +0.619 +0.619 +-0.429 +0.524 +0.619 +0.714 +0.714 +-0.619 +0.714 +0.867 +0.867 +0.867 +-0.733 +0.867 +0.238 +0.524 +0.429 +-0.429 +0.524 +Table 8: Kendall’s tau (𝜏) scores computed from the metric scores with respect to the ranking metrics on the standard KG +embedding datasets. The KG methods are evaluated after training. +Figure 8: Study on efficient KGE methods and their the car- +bon footprint on WN18RR when using KP vs Hits@10. The +x-axis shows the the carbon footprint in g eq 𝐶𝑂2. +Robustness and Efficiency on large KGs: This ablation study +aims to gauge the correlation behavior of KP and ranking metric +on a large-scale KG. For the experiment, we use Yago3-10 dataset. +A key reason to select the Yago-based dataset is that besides being +large-scale, it has rich semantics. Results in Table 9 illustrate KP +shows a stable and high correlation with the ranking metric, con- +firming the robustness of KP. We show carbon footprint results in +Figure 6 for the yago dataset. Further we also study the efficiency +of KP on the wikidata dataset in Figure 7 which reaffirms that KP +maintains its efficiency on large scale datasets. +Efficiency comparison of Sliced Wasserstein vs Wasserstein +as distance metric in KP: In this study we empirically provide +a rationale for using sliced wasserstein as a distance metric over +the wasserstein distance in KP. The results are in table 10. We see +that KP using sliced wasserstein distance provides a significant +computational advantage over wasserstein distance, while having +a good performance as seen in the previous experiments. Thus +we need an efficient approximation such as the sliced wasserstein +distance as the distance metric in place of wasserstein distance in +KP. +Metrics +Hits@1(↑) +Hits@3(↑) +Hits@10(↑) +MR(↓) +MRR(↑) +r +0.657 +0.594 +0.414 +-0.920 +0.572 +𝜌 +0.679 +0.679 +0.5 +-0.714 +0.643 +𝜏 +0.524 +0.524 +0.333 +-0.524 +0.429 +Table 9: KP correlations on the YAGO dataset. +Metrics +KP(W) +KP(SW) +Speedup ↑ +Dataset +FB15K237 +WN18RR +FB15K237 +WN18RR +FB15K237 +WN18RR +split +val + test +val + test +val + test +val + test +Avg +Avg +TransE +1136.766 +9.655 +0.321 +0.114 +x 3540.3 +x 84.6 +TransH +2943.869 +7.549 +0.317 +0.095 +x 9278.5 +x 79.8 +TransR +1734.576 +4.423 +0.336 +0.129 +x 5168.3 +x 34.4 +Complex +1054.721 +13.089 +0.324 +0.135 +x 3255.3 +x 97.0 +RotatE +865.417 +12.783 +0.342 +0.136 +x 2531.1 +x 94.3 +TuckER +1021.649 +3.840 +0.316 +0.098 +x 3230.0 +x 39.1 +ConvKB +719.310 +5.154 +0.429 +0.132 +x 1675.7 +x 39.0 +Table 10: Evaluation Metric Comparison wrt Computing +Time (in minutes, for 100 epochs). Column 1 denotes +popular KGE methods. Depicted values denote evalua- +tion(validation+test) time for computing a metric and cor- +responding speedup using KP(𝑆𝑊 ). KP(𝑆𝑊 ) with sliced +wasserstein as the distance metric significantly reduces the +evaluation time (green) in comparison with KP(𝑊 ) which +uses the wasserstein distance. +8.2 +Theoretical Proof Sketches +We work under the following considerations: As the KGE method +converges the mean statistic(𝑚𝜈) of the scores of the positive triples +consistently lies on one side of the half plane formed by the mean +statistic(𝑚𝜇) of the negative triples, irrespective of the data distri- +bution. The detail proofs are here. +Lemma 8.1. KP has a monotone increasing correspondence with +the Proxy of the Expected Ranking Metrics(PERM) under the above +stated considerations as 𝑚𝜈 deviates from 𝑚𝜇 + +Carbon Footprint of the Overall KGE prototyping process using +methods that save on training time(WN18RR) +Carbon Footprint using KP + Carbon Footprint using ranking metrics +HaLE +Method +ProcrustEs +0 +25 +50 +75 +100 +125WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Bastos, et al. +Proof Sketch. Considering the 0-dimensional PD as used by +KP and a normal distribution for the edge weights (can be extended +to other distributions using techniques like [39]) of the graph(scores +of the triples), we have univariate gaussian measures [40] 𝜇 and 𝜈 +for the positive and negative distributions respectively. Denote by +𝑚𝜇 and 𝑚𝜈 the means of the distributions 𝜇 and 𝜈 respectively and +by Σ𝜇, Σ𝜈 the respective covariance matrices. +𝑊 2 +2 (𝜇,𝜈) = ∥𝜇 − 𝜈∥2 + 𝐵(Σ𝜇, Σ𝜈)2 +(2) +where 𝐵(Σ𝜇, Σ𝜈)2 = 𝑡𝑟 (Σ𝜇 + Σ𝜈 − 2(Σ +1 +2𝜇 Σ𝜈Σ +1 +2𝜇 ) +1 +2 ). +Next we see how that changing the means of the distribution(and +also variance) changes PERM and KP. We can show that, +𝑃 = +∫ 𝑥=∞ +𝑥=−∞ +𝐷+(𝑥) +�∫ 𝑦=∞ +𝑦=𝑥 +𝐷−(𝑥)𝑑𝑦 +� +𝑑𝑥 +𝜕𝑃 +𝜕𝑚𝜈 += +∫ 𝑥=∞ +𝑥=−∞ +𝐷+(𝑥) +�∫ 𝑦=∞ +𝑦=𝑥 +𝜕𝐷−(𝑥) +𝜕𝑚𝜈 +𝑑𝑦 +� +𝑑𝑥 +≥ 0 +𝜕𝑃 +𝜕Σ𝜈 += +∫ 𝑥=∞ +𝑥=−∞ +𝐷+(𝑥) +�∫ 𝑦=∞ +𝑦=𝑥 +𝜕𝐷−(𝑦) +𝜕Σ𝜈 +𝑑𝑦 +� +𝑑𝑥 +≤ 0 +𝜕𝑃 +𝜕Σ𝜇 += +∫ 𝑥=∞ +𝑥=−∞ +𝜕𝐷+(𝑥) +𝜕Σ𝜈 +�∫ 𝑦=∞ +𝑦=𝑥 +𝐷−(𝑦)𝑑𝑦 +� +𝑑𝑥 +Since KP is the (sliced) wasserstein distance between PDs we can +show the respective gradients are as below, +𝜕𝑊 2 +2 (𝜇,𝜈) +𝜕𝑚𝜈 += 2|𝑚𝜇 − 𝑚𝜈 | +≥ 0 +𝜕𝑊 2 +2 (𝜇,𝜈) +𝜕Σ𝜈 += 𝐼 − Σ +1 +2𝜇 (Σ +1 +2𝜇 Σ𝜈Σ +1 +2𝜇 ) +−1 +2 Σ +1 +2𝜇 +As the generating process of the scores changes the gradient of +PERM along the direction (𝑑𝑚𝜈,𝑑𝜎𝜇,𝑑𝜎𝜈) can be shown to be the +following +� +(𝑑𝑚𝜈,𝑑𝜎,𝑑𝜎) , +� 𝜕𝑃𝐸𝑅𝑀 +𝜕𝑚𝜈 +, 𝜕𝑃𝐸𝑅𝑀 +𝜕Σ𝜇 +, 𝜕𝑃𝐸𝑅𝑀 +𝜕Σ𝜈 +�� +≥ 0 +Similarly the gradient of KP along the direction (𝑑𝑚𝜇,𝑑𝜎𝜇,𝑑𝜎𝜈) +is +� +(𝑑𝑚𝜈,𝑑𝜎,𝑑𝜎), ( +𝜕𝑊 2 +2 (𝜇,𝜈) +𝜕𝑚𝜈 +, +𝜕𝑊 2 +2 (𝜇,𝜈) +𝜕Σ𝜇 +, +𝜕𝑊 2 +2 (𝜇,𝜈) +𝜕Σ𝜈 +) +� +≥ 0 +Since both PERM and and KP vary in the same manner as the +distribution changes, the two have a one-one correspondence [42]. +□ +The above lemma shows that there is a one-one correspondence +between KP and PERM and by definition PERM has a one-one cor- +respondence with the ranking metrics. Therefore, the next theorem +follows as a natural consequence +Theorem 8.1. KP has a one-one correspondence with the Ranking +Metrics under the above stated considerations +Theorem 8.2. Under the considerations of theorem 8.1, the relative +change in KP on addition of random noise to the scores is bounded +by a function of the original and noise-induced covariance matrix +as ΔK P +K P ≤ 𝑚𝑎𝑥((1 − |Σ+1 +𝜇1 Σ−1 +𝜇2 | +3 +2 ), (1 − |Σ+1 +𝜈1 Σ−1 +𝜈2 | +3 +2 )), where Σ𝜇1 and +Σ𝜈1 are the covariance matrices of the positive and negative triples’ +scores respectively and Σ𝜇2 and Σ𝜈2 are that of the corrupted scores. +Proof Sketch. Consider a zero mean random noise to simulate +the process of varying the distribution of the scores of the KGE +method. Let 𝑚𝜇1 and 𝑚𝜈1 be the means of the positive and negative +triples’ scores of the original method and Σ𝜇1, Σ𝜈1 be the respective +covariance matrices. Let 𝑚𝜇2 and 𝑚𝜈2 be the means of the positive +and negative triples’ scores of the corrupted method and Σ𝜇2, Σ𝜈2 +be the respective covariance matrices. Considering the kantorovich +duality [51] and taking the difference between the two measures +we have +KP1 − KP2 += +𝑖𝑛𝑓 +𝛾1∈Π(𝑥,𝑦) +∫ +𝛾1 +𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑥,𝑦)𝑑𝛾1(𝑥,𝑦) +− +𝑖𝑛𝑓 +𝛾2∈Π(𝑥,𝑦) +∫ +𝛾2 +𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑥,𝑦)𝑑𝛾2(𝑥,𝑦) +≤ 𝑠𝑢𝑝 +Φ,Ψ +∫ +𝑥 +Φ(𝑥)𝑑𝜇1(𝑥) + +∫ +𝑦 +Ψ(𝑦)𝑑𝜈1(𝑦) +− +∫ +𝑥 +Φ(𝑥)𝑑𝜇2(𝑥) − +∫ +𝑦 +Ψ(𝑦)𝑑𝜈2(𝑦) +≤ 𝑠𝑢𝑝 +Φ,Ψ +∫ +𝑥 +Φ(𝑥)(𝑑𝜇1(𝑥) − 𝑑𝜇2(𝑥)) + +∫ +𝑦 +Ψ(𝑦)(𝑑𝜈1(𝑦) − 𝑑𝜈2(𝑦)) +Now by definition of the measure 𝜇1 we have +𝜕𝜇1 +𝜕𝑥 = −𝜇1Σ−1 +𝜇1 (𝑥 − 𝑚𝜇1) +𝑑𝜇1(𝑥𝑖) = −(𝜇1Σ−1 +𝜇1 (𝑥 − 𝑚𝜇1))[𝑖]𝑑𝑥𝑖 +∴ 𝑑𝜇1(𝑥) = 𝑑𝑒𝑡(𝑑𝑖𝑎𝑔(−𝜇1Σ−1 +𝜇1 (𝑥 − 𝑚𝜇1)))𝑑𝑥 +From the above results we can show the following +KP1 − KP2 +≤ 𝑚𝑎𝑥((1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1 +𝜇2 ) +𝑛 +2 +1), (1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1 +𝜈2 ) +𝑛 +2 +1))KP1 +∴ ΔKP +KP +≤ 𝑚𝑎𝑥 +�� +1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1 +𝜇2 ) +𝑛 +2 +1� +, +� +1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1 +𝜈2 ) +𝑛 +2 +1�� +In our case as we work in the univariate setting 𝑛 = 1 and thus we +have ΔK P +K P ≤ 𝑚𝑎𝑥 +�� +1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1 +𝜇2 ) +3 +2 +� +, +� +1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1 +𝜈2 ) +3 +2 +�� +, as +required. +□ + +Can Persistent Homology provide an efficient alternative +WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA +Theorem 8.2 shows that as noise is induced gradually, the KP +value changes in a bounded manner as desired. + diff --git a/KNFOT4oBgHgl3EQfyzQ_/content/tmp_files/load_file.txt b/KNFOT4oBgHgl3EQfyzQ_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3161390bcbd4d21e485598c22ba46896a2271b33 --- /dev/null +++ b/KNFOT4oBgHgl3EQfyzQ_/content/tmp_files/load_file.txt @@ -0,0 +1,1459 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf,len=1458 +page_content='Can Persistent Homology provide an efficient alternative for Evaluation of Knowledge Graph Completion Methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Anson Bastos cs20resch11002@iith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='in IIT, Hyderabad India Kuldeep Singh kuldeep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='singh1@cerence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='com Zerotha Research and Cerence GmbH Germany Abhishek Nadgeri abhishek22596@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='com Zerotha Research and RWTH Aachen Germany Johannes Hoffart johannes@hoffart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='ai SAP Germany Toyotaro Suzumura suzumura@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='org The University of Tokyo Japan Manish Singh msingh@cse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='iith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='in IIT Hyderabad India ABSTRACT In this paper we present a novel method, Knowledge Persistence (KP), for faster evaluation of Knowledge Graph (KG) completion approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Current ranking-based evaluation is quadratic in the size of the KG, leading to long evaluation times and consequently a high carbon footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP addresses this by representing the topol- ogy of the KG completion methods through the lens of topological data analysis, concretely using persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The character- istics of persistent homology allow KP to evaluate the quality of the KG completion looking only at a fraction of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Experi- mental results on standard datasets show that the proposed metric is highly correlated with ranking metrics (Hits@N, MR, MRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Per- formance evaluation shows that KP is computationally efficient: In some cases, the evaluation time (validation+test) of a KG com- pletion method has been reduced from 18 hours (using Hits@10) to 27 seconds (using KP), and on average (across methods & data) reduces the evaluation time (validation+test) by ≈ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' ACM Reference Format: Anson Bastos, Kuldeep Singh, Abhishek Nadgeri, Johannes Hoffart, Toyotaro Suzumura, and Manish Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Can Persistent Homology provide an efficient alternative for Evaluation of Knowledge Graph Completion Methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='. In Proceedings of the Web Conference 2023 (WWW ’23), APRIL 30 - MAY 4, 2023, Texas, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' WWW, Texas, USA, 13 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' XXXXX/YYYYY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3449917 1 INTRODUCTION Publicly available Knowledge Graphs (KGs) find broad applicability in several downstream tasks such as entity linking, relation extrac- tion, fact-checking, and question answering [22, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' These KGs are large graph databases used to express facts in the form of relations between real-world entities and store these facts as triples (subject, Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Copyrights for third- party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' ACM ISBN 978-Y-4500-YYYY-7/21/04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='XXXXX/YYYYY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3449917 relation, object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KGs must be continuously updated because new en- tities might emerge or facts about entities are extended or updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Knowledge Graph Completion (KGC) task aims to fill the missing piece of information into an incomplete triple of KG [5, 18, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Several Knowledge Graph Embedding (KGE) approaches have been proposed to model entities and relations in vector space for missing link prediction in a KG [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KGE methods infer the connec- tivity patterns (symmetry, asymmetry, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=') in the KGs by defining a scoring function to calculate the plausibility of a knowledge graph triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' While calculating plausibility of a KG triple τ = (𝑒ℎ,𝑟,𝑒𝑡), the predicted score by scoring function affirms the confidence of a model that entities 𝑒𝑡 and 𝑒ℎ are linked by 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For evaluating KGE methods, ranking metrics have been widely used [22] which is based on the following criteria: given a KG triple with a missing head or tail entity, what is the ability of the KGE method to rank candidate entities averaged over triples in a held- out test set [28]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' These ranking metrics are useful as they intend to gauge the behavior of the methods in real world applications of KG completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Since 2019, over 100 KGE articles have been published in various leading conferences and journals that use ranking metrics as evaluation protocol1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Limitations of Ranking-based Evaluation: The key challenge while computing ranking metrics for model evaluation is the time taken to obtain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Since the (most of) KGE models aim to rank all the negative triples that are not present in the KG [8, 9], comput- ing these metrics takes a quadratic time in the number of entities in the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Moreover, the problem gets alleviated in the case of hyper-relations [62] where more than two entities participate, lead- ing to exponential computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For instance, Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [2] spent 24,804 GPU hours of computation time while performing a large-scale benchmarking of KGE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' There are two issues with high model evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Firstly, efficiency at evaluation time is not a widely-adapted criterion for assessing KGE models alongside accuracy and related measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' There are efforts to make KGE methods efficient at training time [52, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, these methods also use ranking-based protocols resulting in high evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Secondly, the need for signifi- cant computational resources for the KG completion task excludes a large group of researchers in universities/labs with restricted GPU 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='com/xinguoxia/KGE#papers arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='12929v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='LG] 30 Jan 2023 WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Bastos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Such preliminary exclusion implicitly challenges the ba- sic notion of various diversity and inclusion initiatives for making the Web and its related research accessible to a wider community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In past, researchers have worked extensively towards efficient Web- related technologies such as Web Crawling [12], Web Indexing [25], RDF processing [17], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, for the KG completion task, similar to other efficient Web-based research, there is a necessity to develop alternative evaluation protocols to reduce the computation com- plexity, a crucial research gap in available KGE scientific literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Another critical issue in ranking metrics is that they are biased towards popular entities and such popularity bias is not captured by current evaluation metrics [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, we need a metric which is efficient than popular ranking metrics and also omits such biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Motivation and Contribution: In this work, we focus on ad- dressing above-mentioned key research gaps and aim for the first study to make KGE evaluation more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We introduce Knowl- edge Persistence(KP), a method for characterizing the topology of the learnt KG representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' It builds upon Topological Data Analysis [58] based on the concepts from Persistent Homology(PH) [15], which has been proven beneficial for analyzing deep networks [29, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' PH is able to effectively capture the geometry of the mani- fold on which the representations reside whilst requiring fraction of data [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This property allows to reduce the quadratic complexity of considering all the data points (KG triples in our case) for rank- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Another crucial fact that makes PH useful is its stability with respect to perturbations making KP robust to noise [19] mitigating the issues due to the open-world problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Thus we use PH due to its effectiveness for limited resources and noise [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Concretely, the following are our key contributions: (1) We propose (KP), a novel approach along with its theoreti- cal foundations to estimate the performance of KGE models through the lens of topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This allows us to drastically reduce the computation factor from order of O(|E|2) to O(|E|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The code is here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' (2) We run extensive experiments on families of KGE methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', Translation, Rotation, Bi-Linear, Factorization, Neural Network methods) using standard benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The experiments show that KP correlates well with the stan- dard ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, KP could be used for faster prototyping of KGE methods and paves the way for efficient evaluation methods in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In the remainder of the paper, related work is in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Section 3 briefly explains the concept of persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Section 4 describes the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Later, section 5 shows associated empirical results and we conclude in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2 RELATED WORK Broadly, KG embeddings are classified into translation and semantic matching models [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Translation methods such as TransE [8], TransH [57], TransR [26] use distance-based scoring functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Whereas semantic matching models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', ComplEx [48], Distmult [60], RotatE [44]) use similarity-based scoring functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Kadlec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [23] first pointed limitations of KGE evaluation and its dependency on hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [45] with exhaus- tive evaluation (using ranking metrics) showed issues of scoring functions of KGE methods whereas [31] studied the effect of loss function of KGE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [20] studied if KGE meth- ods capture KG semantic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Work in [35] provides a new dataset that allows the study of calibration results for KGE mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Speranskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [43] used precision and recall rather than rankings to measure the quality of completion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Authors pro- posed a new dataset containing triples such that their completion is both possible and impossible based on queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, queries were build by creating a tight dependency on such queries for the evaluation as pointed by [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Rim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [37] proposed a capability- based evaluation where the focus is to evaluate KGE methods on various dimensions such as relation symmetry, entity hierarchy, entity disambiguation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Mohamed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [28] fixed the popularity bias of ranking metrics by introducing modified ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The geometric perspective of KGE methods was introduced by [40] and its correlation with task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Berrendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [6] sug- gested the adjusted mean rank to improve reciprocal rank, which is an ordinal scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Authors do not consider the effect of negative triples available for a given triple under evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [47] propose to balances the number of negatives per triple to improve rank- ing metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Authors suggested the preparation of training/testing splits by maintaining the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Work in [24] proposes efficient non-sampling techniques for KG embedding training, few other initiatives improve efficiency of KGE training time [52–54], and hyperparameter search efficiency of embedding models [49, 56, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Overall, the literature is rich with evaluations of knowledge graph completion methods [4, 21, 38, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, to the best of our knowledge, extensive attempts have not been made to im- prove KG evaluation protocols’ efficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', to reduce run-time of widely-used ranking metrics for faster prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We position our work orthogonal to existing attempts such as [40], [47], [28], and [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In contrast with these attempts, our approach provides a topological perspective of the learned KG embeddings and focuses on improving the efficiency of KGE evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 3 PRELIMINARIES We now briefly describe concepts used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Ranking metrics have been used for evaluating KG embedding methods since the inception of the KG completion task [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' These metrics include the Mean Rank (MR), Mean Reciprocal Rank (MRR) and the cut-off hit ratio (Hits@N (N=1,3,10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' MR reports the average predicted rank of all the labeled triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' MRR is the average of the inverse rank of the labelled triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hits@N evaluates the fraction of the labeled triples that are present in the top N predicted results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Persistent Homology (PH) [15, 19]: studies the topological features such as components in 0-dimension (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', a node), holes in 1-dimension (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', a void area bounded by triangle edges) and so on, spread over a scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Thus, one need not choose a scale before- hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The number(rank) of these topological features(homology group) in every dimension at a particular scale can be used for downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Consider the simplicial complex ( e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', point is a 0-simplex, an edge is a 1-simplex, a triangle is a 2-simplex ) 𝐶 with weights 𝑎0 ≤ 𝑎1 ≤ 𝑎2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝑎𝑚−1, which could represent the edge weights, for example, the triple score from the KG embed- ding method in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' One can then define a Filtration process [15], which refers to generating a nested sequence of complexes 𝜙 ⊆ 𝐶1 ⊆ 𝐶2 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝐶𝑚 = 𝐶 in time/scale as the simplices below Can Persistent Homology provide an efficient alternative WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Figure 1: Calculating Knowledge Persistence(KP) score from the given KG and KG embedding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The KG is sampled for positive(G+) and negative(G−) triples (step one), keeping the order O(|E|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The edge weights represent the score obtained from the KG embedding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In step two, the persistence diagram (PD) is computed using filtration process explained in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In final step, a Sliced Wasserstein distance (SW) is obtained between the PDs of G+ and G− to get the KP score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, ranking metrics run the KGE methods over all the O(|E|2) triples as explained in bottom left part of the figure(red box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' the threshold weights are added in the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The filtration pro- cess [15] results in the creation(birth) and destruction(death) of components, holes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Thus each structure is associated with a birth-death pair (𝑎𝑖,𝑎𝑗) ∈ 𝑅2 with 𝑖 ≤ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The persistence or life- time of each component can then be given by 𝑎𝑗 − 𝑎𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A persistence diagram (PD) summarizes the (birth,death) pair of each object on a 2D plot, with birth times on the x axis and death times on the y axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The points near the diagonal are shortlived components and generally are considered noise (local topology), whereas the persis- tent objects (global topology) are treated as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We consider local and global topology to compare two PDs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', positive and negative triple graphs in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4 PROBLEM STATEMENT AND METHOD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 Problem Setup We define a KG as a tuple 𝐾𝐺 = (E, R, T +) where E denotes the set of entities (vertices), R is the set of relations (edges), and T + ⊆ E × R × E is a set of all triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A triple τ = (𝑒ℎ,𝑟,𝑒𝑡) ∈ T + indicates that, for the relation 𝑟 ∈ R, 𝑒ℎ is the head entity (origin of the relation) while 𝑒𝑡 is the tail entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Since 𝐾𝐺 is a multigraph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝑒ℎ = 𝑒𝑡 may hold and |{𝑟𝑒ℎ,𝑒𝑡 }| ≥ 0 for any two entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The KG completion task predicts the entity pairs ⟨𝑒𝑖,𝑒𝑗⟩ in the KG that have a relation 𝑟𝑐 ∈ R between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 Proposed Method In this section we describe our approach for evaluating KG embed- ding methods using the theory of persistent homology (PH) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This process is divided into three steps ( Figure 1), namely: (i) Graph con- struction, (ii) Filtration process and (iii) Sliced Wasserstein distance computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The first step creates two graphs (one for positive triples, another for negative triples) using sampling(O(V) triples), with scores calculated by a KGE method as edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The second step considers these graphs and, using a process called "fil- tration," converts to an equivalent lower dimension representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The last step calculates the distance between graphs to provide a final metric score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We now detail the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 Graph Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We envisioned KGE from the topolog- ical lens while proposing an efficient solution for its evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Previous works such as [40] proposed a KGE metric only consider- ing embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, we intend to preserve the topology (graph structure and its topological feature) along with the KG em- bedding features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We first construct graphs of positive and negative triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We denote a graph as (V, E) where V is the set of 𝑁 nodes and E represents the edges between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Consider a KG embed- ding method M that takes as input the triple τ = (ℎ,𝑟,𝑡) ∈ T and gives the score 𝑠τ of it being a right triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We construct a weighted directed graph G+ from positive triples τ ∈ T + in the train set, with the entities as the nodes and the relations between them as the edges having 𝑠τ as the edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Here, 𝑠τ is the score calculated by KGE method for a triple and we propose to use it as the edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Our idea is to capture topology of graph (G+) with repre- sentation learned by a KG embedding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We sample an order of O(|E|) triples, |E| being the number of entities to keep compu- tational time linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Similarly, we construct a negative graph G− by sampling the same number of unknown triples as the positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' One question may arise if KP is robust to sampling, that we answer theoretically in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 and empirically in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Note, here we do not take all the negative triples in the graphs and consider only a fraction of what the ranking metrics need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This is a fundamental difference with ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Ranking metrics use all the unlabeled triples as negatives for ranking, thus incurring a computational cost of 𝑂(|E|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 Filtration Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Having constructed the Graphs G+ and G−, we now need some definition of a distance between them 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Graph Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Filtration Process Winneror gt a=0 Einstein(HAE) Hans Prize(NP) Hans KG Embedding Einstein GrandSonof method Alfred Sonot n SupervisedBy r(A a=2 3Albert Einstein Homen Alfred Birth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Sliced Wasserstein Distance Computation D+ O(E*)Graph with scores from the KGE method onthe edges HAE KP(G+, G-) = SW(D+, D-) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3 Ranking Ranking metric Albert Einstein HE AK Hermann Einstein Birth Ranking Metrics D- process Sampled O(E) Graphs with scores from the KGE method on the edgesWWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Bastos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Figure 2: For a KGE method, the positive triple graph G+ is used as input (leftmost graph with edge weights) and filtration process is applied on the edge weights (calculated by KGE method) for the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The filtration starts with only nodes as first step, and based on the edge weights, edges are added to the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The persistence diagram is given on the right with red dots indicating 0-dimensional homology (components) and the blue dots indicating 1-dimensional homology (cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Persistent Diagram generated from this filtration process is a condensed 2D representation of G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A similar process is repeated for G−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' to define a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, since the KGs could be large with many entities and relations, directly comparing the graphs could be computationally challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Therefore, we allude to the theory of persistent homology (PH) to summarize the structures in the graphs in the form of the persistence diagram (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Such summa- rizing is obtained by a process known as filtration [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' One can imagine a PD as mapping of higher dimensional data to a 2D plane upholding the representation of data points and we can then derive computational efficiency for distance comparison between 2D repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Specifically, we compute the 0-dimensional topological features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', connected-nodes/components) for each graph (G− and G+) to keep the computation time linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We also experimented using the 1-dimensional features without much empirical benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Consider the positive triple graph G+ as input (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We would need a scale (as pointed in section 3) for the filtration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Once the filtration process starts, initially, we have a graph structure containing only the nodes (entities) and no edges of G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For capturing topological features at various scales, we define a variable 𝑎 which varies from −∞ to +∞ and it is then compared with edge weights (𝑠τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A scale allows to capture topology at various timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Thus, we use the edge weights obtained from the scores (𝑠τ) of the KGE methods for filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' As the filtration proceeds, the graph structures (components) are generated/removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' At a given scale 𝑎, the graph structure ((G+ 𝑠𝑢𝑏)𝑎) contains those edges (triples) for which 𝑠τ ≤ 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Formally, this is expressed as: (G+ 𝑠𝑢𝑏)𝑎 = {(V, E+ 𝑎)|E+ 𝑎 ⊆ E,𝑠τ ≤ 𝑎 ∀τ ∈ E+ 𝑎 } Alternatively, we add those edges for which score of the triple is greater than or equal to the filtration value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', 𝑠τ ≥ 𝑎 defined as (G+ 𝑠𝑢𝑝𝑒𝑟)𝑎 = {(V, E𝑎+)|E𝑎+ ⊆ E,𝑠τ ≥ 𝑎 ∀τ ∈ E𝑎+} One can imagine that for filtration, graph G+ is subdivided into (G+ 𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 as the filtration adds/deletes edges for cap- turing topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, specific components in a sub- graphs will appear and certain components will disappear at differ- ent scale levels (timesteps) 𝑎 = 1, 3, 5 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Please note, Figure 2 explains creation of PD for (G+ 𝑠𝑢𝑏)𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A similar process is repeated for (G+𝑠𝑢𝑝𝑒𝑟)𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This expansion/contraction process enables capturing topology at different time-steps without worrying about defining an optimal scale (similar to hyperparameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Next step is the creation of persistent diagrams of (G+ 𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 where the x-axis and y-axis denotes the timesteps of appearance/disappearance of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For creating a 2D representation graph, components of graphs which appear(disappear) during filtration process at 𝑎𝑥 (𝑎𝑦) are plotted on (𝑎𝑥,𝑎𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The persistence or lifetime of each compo- nent can then be given by 𝑎𝑦 − 𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' At implementation level, one can view PDs(∈ 𝑅𝑁×2) of (G+ 𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 as tensors which are concatenated into one common tensor representing positive triple graph G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, final PD of G+ is a concatenation of PDs of (G+ 𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This final persistent diagram represents a summary of the local and global topological features of the graph G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Following are the benefits of a persistent diagram against con- sidering the whole graph: 1) a 2D summary of a higher dimensional graph structure data is highly beneficial for large graphs in terms of the computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2) The summary could contain fewer data points than the original graph, preserving the topologi- cal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Similarly, the process is repeated for negative triple graph G− for creating its persistence diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Now, the two newly created PDs are used for calculating the proposed metric score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3 Sliced Wasserstein distance computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' To compare two PDs, generally the Wasserstein distance between them is computed [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' As the Wasserstein distance could be computationally costly, we find the sliced Wasserstein distance [13] between the PDs, which we empirically observe to be eight times faster on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The Sliced Wasserstein distance(𝑆𝑊 ) between measures 𝜇 and 𝜈 is: 𝑆𝑊𝑝 (𝜇,𝜈) = �∫ 𝑆𝑑−1 𝑊 𝑝 𝑝 (𝑅𝜇 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=',𝜃), 𝑅𝜈 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=',𝜃)) � 1 𝑝 where 𝑅𝜇 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=',𝜃) is the projection of 𝜇 along 𝜃,𝑊 is initial Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Generally a Monte Carlo average over 𝐿 samples is done instead of the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The 𝑆𝑊 distance takes O(𝐿𝑁𝑑 +𝐿𝑁𝑙𝑜𝑔(𝑁)) time which can be improved to linear time O(𝑁𝑑) for 𝑆𝑊2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', Euclidean distance) as a closed form solution [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Thus, KP(G+, G−) = 𝑆𝑊 (𝐷+, 𝐷−) (1) where 𝐷+, 𝐷− are the persistence diagrams for G+, G− respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Since the metric is obtained by summarizing the Knowledge graph using Persistence diagrams we term it as Knowledge Persistence(KP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' As KP correlates well with ranking metrics (sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 and 5), higher KP signifies a better performance of the KGE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 Theoretical justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This section briefly states the theo- retical results justifying the proposed method to approximate the 1Can Persistent Homology provide an efficient alternative WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We begin the analysis by assuming two distribu- tions: One for the positive graph’s edge weights(scores) and the other for the negative graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We define a metric "PERM" (Figure 3), that is a proxy to the ranking metrics while being continuous(for the definition of integrals and derivatives) for ease of theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The proof sketches are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Figure 3: Figure gives an intuition of the metric PERM which is designed to be a proxy to the ranking metrics for ease of theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For a given positive triple 𝜏 with score 𝑥𝜏 the expected rank(𝐸𝑅(𝜏)) is defined as the area under the curve of the negative distribution from 𝑥𝜏 to ∞(shown in the shaded area above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' PERM is then defined as the expectation of the expected rank under the positive distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 (Expected Ranking(ER)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Consider the positive triples to have the distribution 𝐷+ and the negative triples to have the distribution 𝐷−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For a positive triple with score 𝑎 its expected rank- ing(ER) is defined as, 𝐸𝑅(𝑎) = ∫ 𝑥=∞ 𝑥=𝑎 𝐷−(𝑥)𝑑𝑥 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 (PERM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Consider the positive triples to have the distribution 𝐷+ and the negative triples to have the distribution 𝐷−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The PERM metric is then defined as, 𝑃𝐸𝑅𝑀 = ∫ 𝑥=∞ 𝑥=−∞ 𝐷+(𝑥)𝐸𝑅(𝑥)𝑑𝑥 It is easy to see that PERM has a monotone increasing corre- spondence with the actual ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' That is, as many of the negative triples get a higher score than the positive triples, the distribution of the negative triples will shift further right of the pos- itive distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, the area under the curve would increase for a given triple(x=a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We just established a monotone increasing correspondence of PERM with the ranking metrics, we now need show that there exists a one-one correspondence between PERM and KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For closed-form solutions, we work with normalised dis- tributions (can be extended to other distributions using [39]) of KGE score under the following mild consideration: As the KGE method converges, the mean statistic(𝑚𝜈) of the scores of the posi- tive triples consistently lies on one side of the half-plane formed by the mean statistic(𝑚𝜇) of the negative triples, irrespective of the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP has a monotone increasing correspondence with the Proxy of the Expected Ranking Metrics(PERM) under the above stated considerations as 𝑚𝜈 deviates from 𝑚𝜇 The above lemma shows that there is a one-one correspondence between KP and PERM and by definition PERM has a one-one cor- respondence with the ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Therefore, the next theorem follows as a natural consequence: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP has a one-one correspondence with the ranking metrics under the above stated considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The above theorem states that, with high probability, there exists a correlation between KP and the ranking metrics under certain considerations and proof details are in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In an ideal case, we seek a linear relationship between the proposed mea- sure and the ranking metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This would help interpret whether an increase/decrease in the measure would cause a corresponding increase/decrease in the ranking metric we wish to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Such interpretation becomes essential when the proposed metric has different behavior from the existing metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' While the correlation could be high, for interpretability of the results, we would also like the change in KP to be bounded for a change in the scores(ranking metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The below theorem gives a sense for this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Under the considerations of theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3, the relative change in KP on addition of random noise to the scores is bounded by a function of the original and noise-induced covariance matrix as ΔK P K P ≤ 𝑚𝑎𝑥((1 − |Σ+1 𝜇1 Σ−1 𝜇2 | 3 2 ), (1 − |Σ+1 𝜈1 Σ−1 𝜈2 | 3 2 )), where Σ𝜇1 and Σ𝜈1 are the covariance matrices of the positive and negative triples’ scores respectively and Σ𝜇2 and Σ𝜈2 are that of the corrupted scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 gives a bound on the change in KP while inducing noise in the KGE predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Ideally, the error/change would be 0, and as the noise is increased(and the ranking changed), gradually, the KP value also changes in a bounded manner as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 5 EXPERIMENTAL SETUP For de-facto KGC task (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1), we use popular KG embed- ding methods from its various categories: (1) Translation: TransE [8], TransH [57], TransR [26] (2) Bilinear, Rotation, and Factoriza- tion: RotatE [44] TuckER [3], and ComplEx [48], (3) Neural Network based: ConvKB [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The method selection and evaluation choices are similar to [28, 37] that propose new metrics for KG embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' All methods run on a single P100 GPU machine for a maximum of 100 epochs each and evaluated every 5 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For training/testing the KG embedding methods we make use of the pykg2vec [61] library and validation runs are executed 20 times on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We use the standard/best hyperparameters for these datasets that the considered KGE methods reported [3, 8, 26, 44, 48, 57, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 Datasets We use standard English KG completion datasets: WN18, WN18RR, FB15k237, FB15k, YAGO3-10 [2, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The WN18 dataset is obtained from Wordnet [27] containing lexical relations between English words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' WN18RR removes the inverse relations in the WN18 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' FB15k is obtained from the Freebase [7] knowledge graph, and FB15k237 was created from FB15k by removing the inverse relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The dataset details are in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For scaling experiment, we rely on large scale YAGO3-10 dataset [2] and due to brevity, results for Yago3-10 are in appendix ( cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', Figure 6 and table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 Comparative Methods Considering ours is the first work of its kind, we select some com- petitive baselines as below and explain "why" we chose them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For evaluation, we report correlation [14] between KP and baselines with ranking metrics (Hits@N (N= 1,3,10), MRR and MR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Conicity [40]: It finds the average cosine of the angle between an embedding and the mean embedding vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In a sense, it gives PERM =E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' (ER(T) Distribution Distribution of positive ER(T) of neqative triples triples Score of a positive triple TWWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Bastos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' spread of a KG embedding method in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We would like to observe instead of topology, if calculating geometric properties of a KG embedding method be an alternative for ranking metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Average Vector Length: This metric was also proposed by Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [40] to study the geometry of the KG embedding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' It computes the average length of the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Graph Kernel (GK): we use graph kernels to compare the two graphs(G+, G−) obtained for our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The rationale is to check if we could get some distance metric that correlates with the ranking metrics without persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, this baseline empha- sizes a direct comparison for the validity of persistent homology in our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' As an implementation, we employ the widely used shortest path kernel [10] to compare how the paths(edge weights/scores) change between the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Since the method is computationally expensive, we sample nodes [11] and apply the kernel on the sampled graph, averaging multiple runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Table 1: (Open-Source)Benchmark Datasets for Experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Dataset Triples Entities Relations FB15K 592,213 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='951 1,345 FB15K-237 272,115 14,541 237 WN18 151,442 40,943 18 WN18RR 93,003 40,943 11 Yago3-10 1,089,040 123,182 37 6 RESULTS AND DISCUSSION We conduct our experiments in response to the following research questions: RQ1: Is there a correlation between the proposed metric and ranking metrics for popular KG embedding methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' RQ2: Can the proposed metric be used to perform early stopping during training?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' RQ3: What is the computational efficiency of proposed metric wrt ranking metrics for KGE evaluation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP for faster prototyping of KGE methods: Our core hypoth- esis in the paper is to develop an efficient alternative (proxy) to the ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, for a fair evaluation, we use the triples in the test set for computing KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Ideally, this should be able to simulate the evaluation of the ranking metrics on the same (test) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' If true, there exists a high correlation between the two mea- sures, namely the KP and the ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Table 2 shows the linear correlations between the ranking metrics and our method & baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We report the linear(Pearson’s) correlation because we would like a linear relationship between the proposed measure and the ranking metric (for brevity, other correlations are in appendix Tables 7, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This would help interpret whether an increase/decrease in the measure would cause a corresponding increase/decrease in the ranking metric that we wish to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Specifically we train all the KG embedding methods for a predefined number of epochs and evaluate the finally obtained models to get the ranking metrics and KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The correlations are then computed between KP and each of the ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We observe that KP(test) configuration (triples are sampled from the test set) achieves the highest correla- tion coefficient value among all the existing geometric and kernel baseline methods in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For instance, on FB15K, KP(test) reports high correlation value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='786 with Hits@1, whereas best baseline for this dataset (AVL) has corresponding correlation value as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Similarly for WN18RR, KP(test) has correlation value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='482 compared to AVL with -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='272 correlation with Hits@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Conicity and AVL that provide geometric perspective shows mostly low positive correlation with ranking metrics whereas the Graph Kernel based method shows highly negative correlations, making these methods unsuitable for direct applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' It indicates that the topology of the KG induced by the learnt representations seems a good predictor of the performance on similar data distributions with high correlation with ranking metric (answering RQ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Furthermore, the results also report a configuration KP(train) in which we compute KP on the triples of the train set and find the correlation with the ranking metrics obtained from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Here our rationale is to study whether the proposed metric would be able to capture the generalizability of the unseen test (real world) data that is of a similar distribution as the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Initial re- sults in Table 2 are promising with high correlation of KP(train) with ranking metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, it may enable the use of KP in settings without test/validation data while using the available (possibly lim- ited) data for training, for example, in few-shot scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We leave this promising direction of research for future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 KP as a criterion for early stopping Does KP hold correlation while early stopping?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' To know when to stop the training process to prevent overfitting, we must be able to estimate the variance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This is generally done by observing the validation/test set error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Thus, to use a method as a criterion for early stopping, it should be able to predict this gen- eralization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Table 2 explains that KP(Train) can predict the generalizability of methods on the last epoch, it remains to empiri- cally verify that KP also predicts the performance at every interval during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hence, we study the correlations of the proposed method with the ranking metrics for individual KG embedding methods in the intra-method setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Specifically, for a given method, we obtain the KP score and the ranking metrics on the test set and compute the correlations at every evaluation interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Results in Table 3 suggest that KP has a decent correla- tion in the intra-method setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' It indicates that KP could be used in place of the ranking metrics for deciding a criterion on early stopping if the score keeps persistently falling (answering RQ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' What is the relative error of early stopping between KP and Ranking Metric?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' To further cross-validate our response to RQ2, we now compute the absolute relative error between the rank- ing metrics of the best models selected by KP and the expected ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Ideally, we would expect the performance of the model obtained using this process on unseen test data(preferably of the same distribution) to be close to the best achievable result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', the relative error should be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This is important as if we were to use any metric for faster prototyping, it should also be a good criterion for model selection(selecting a model with less generalization error) and being efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Table 4 shows that the relative error is marginal, of the order of 10−2, in most cases(with few exceptions), indicating that KP could be used for early stop- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The deviation is higher for some methods, such as ConvKB, Can Persistent Homology provide an efficient alternative WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Metrics FB15K FB15K237 WN18 WN18RR Hits1(↑) Hits3(↑) Hits10(↑) MR(↓) MRR(↑) Hits1(↑) Hits3(↑) Hits10(↑) MR(↓) MRR(↑) Hits1(↑) Hits3(↑) Hits10(↑) MR(↓) MRR(↑) Hits1(↑) Hits3(↑) Hits10(↑) MR(↓) MRR(↑) Conicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='156 0.' metadata={'source': 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scores computed from the metric scores with respect to the ranking metrics on the standard KG embedding datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The KG methods are evaluated after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Green values are the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Datasets FB15K237 WN18RR KG methods r 𝜌 𝜏 r 𝜌 𝜏 TransE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='861 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='422 Table 3: Correlation scores computed between KP and the ranking metric(Hits@10) on the standard KG embedding datasets with the methods evaluated at every interval as the training progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Here, r: Pearson correlation co-efficient, 𝜌: Spearman’s correlation co-efficient, 𝜏: Kendall’s Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' which had convergence issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We infer from observed behavior that if the KG embedding method has not converged(to good re- sults), the correlation and, thus, the early stopping prediction may suffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Despite a few outliers, the promising results shall encourage the community to research, develop, and use KGE benchmarking methods that are also computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Datasets FB15K237 WN18RR KG methods hits@1 hits@10 MRR hits@1 hits@10 MRR TransE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='004 TransH 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='101 ConvKB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='569 Table 4: Early stopping using KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The values depict the ab- solute relative error between the metrics of the best models selected using KP and ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 Timing analysis and carbon footprint We now study the time taken for running the evaluation (including evaluation at intervals) of the same methods as in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 on the standard datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Table 5 shows the evaluation times (valida- tion+test) and speedup for each method on the respective datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The training time is constant for ranking metric and KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In some cases (ConvKB), we observe KP achieves a speedup of up to 2576 times on model evaluation time drastically reducing evaluation time from 18 hours to 27 seconds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' the latter is even roughly equal to the carbon footprint of making a cup of coffee2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Furthermore, Figure 2https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='com/4w2xmwry Figure 4: Figure shows a study on the carbon footprint on WN18RR when using KP vs Hits@10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The x-axis shows the the carbon footprint in g eq 𝐶𝑂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4 illustrates the carbon footprints [33, 59] of the overall process (training + evaluation) for the methods when using KP vs ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Due to evaluation time drastically reduced by KP, it also reduces overall carbon footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The promising results validate our attempt to develop alternative method for faster prototyping of KGE methods, thus saving carbon footprint (answering RQ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3 Ablation Studies We systematically provide several studies to support our evaluation and characterize different properties of KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Robustness to noise induced by sampling: An important property that makes persistent homology worthwhile is its stability concerning perturbations of the filtration function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This means that persistent homology is robust to noise and encodes the intrinsic topological properties of the data [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, in our applica- tion of predicting the performance of KG embedding methods, one source of noise is because of sampling the negative and positive triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' It could cause perturbations in the graph topology due to the addition and deletion of edges (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Therefore, we would like the proposed metric to be stable concerning perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' To understand the behavior of KP against this noise, we conduct a study by incrementally adding samples to the graph and observing the mean and standard deviation of the correlation at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In an ideal case, assuming the KG topology remains similar, the mean correlations should be in a narrow range with slight standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We observe a similar effect in Figure 5 where we report the mean correlation at various fractions of triples sampled, with the standard deviation(error bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Here, the mean correlation coefficients are within the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='04), and the average stan- dard deviations are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='02(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='02) for the FB15K237(WN18RR) Carbon Footprint of the Overall KGE prototyping process Carbon Footprint using KP Carbon Footprint using ranking metrics TransE TransH TransR Complex RotatE ConvKB TuckER 0 100 200 300 400WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Bastos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Metrics Hits@10 KP Speedup ↑ Dataset FB15K237 WN18RR FB15K237 WN18RR FB15K237 WN18RR split val + test val + test val + test val + test Avg Avg TransE 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='120 x 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='8 x 754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 TransH 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='099 x 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 x 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 TransR 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='135 x 572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 x 1066.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='142 x 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='5 x 1145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='6 TuckER 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='098 x 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 x 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='9 ConvKB 1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='139 x 2576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='6 x 1044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3 Table 5: Evaluation Metric Comparison wrt Computing Time (in minutes, for 100 epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Column 1 denotes popular KGE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Depicted values denote evalua- tion(validation+test) time for computing a metric and corre- sponding speedup using KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP significantly reduces the evaluation time (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' This shows that KP inherits the robustness of the topo- logical data analysis techniques, enabling linear time by sampling from the graph for dense KGs while keeping it robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Figure 5: Effect of sample size on the correlation coefficient between KP and the ranking metrics on FB15K237 (left dia- gram) and WN18RR datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The correlations for the differ- ent sampling fractions are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Also, the standard deviation is less, indicating the method’s robustness due to changes in local topology while doing sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Generalizability Study- Correlation with Stratified Rank- ing Metric: Mohamed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [28] proposed a new stratified metric (strat-metric) that can be tuned to focus on the unpopular entities, unlike the standard ranking metrics, using certain hyperparameters (𝛽𝑒 ∈ (−1, 1), 𝛽𝑟 ∈ (−1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Special cases of these hyperparame- ters give the micro and macro ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Goal here is to study whether our method can predict strat-metric for the spe- cial case of 𝛽𝑒 = 1, 𝛽𝑟 = 0, which estimates the performance for unpopular(sparse) entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Also, we aim to observe if KP holds a correlation with variants of the ranking metric concerning its generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The results (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', Table 6) shows that KP has a good correlation with each of the stratified ranking metrics which indicate KP also takes into account the local geometry/topology [1] of the sparse entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 Summary of Results and Open Directions To sum up, following are key observations gleaning from empirical studies: 1) KP shows high correlation with ranking metrics (Ta- ble 2) and its stratified version (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' It paves the way for the use of KP for faster prototyping of KGE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2) KP holds Datasets FB15K237 WN18RR Metrics r 𝜌 𝜏 r 𝜌 𝜏 Strat-Hits@1 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='411 Strat-Hits@3 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 Strat-Hits@10 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='619 Strat-MR (↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='619 Strat-MRR (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 Table 6: KP correlation with stratified ranking metrics as proposed in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' a high correlation at every interval during the training process (Table 3) with marginal relative error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' hence, it could be used for early stopping of a KGE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 3) KP inherits key properties of persistent homology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', it is robust to noise induced by sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 4) The overall carbon footprints of the evaluation cycle is drastically reduced if KP is preferred over ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' What’s Next?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We show that topological data analysis based on persistent homology can act as a proxy for ranking metrics with conclusive empirical evidence and supporting theoretical founda- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, it is the first step toward a more extensive research agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We believe substantial work is needed collectively in the research community to develop strong foundations, solving scal- ing issues (across embedding methods, datasets, KGs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=') until persistent homology-based methods are widely adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For example, there could be methods/datasets where the correla- tion turns out to be a small positive value or even negative, in which case we may not be able to use KP in the existing form to simu- late the ranking metrics for these methods/datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In those cases, some alteration may exist for the same and seek further exploration similar to what stratified ranking metric [28] does by fixing issues encountered in the ranking metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Furthermore, theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 would be a key to understand error bounds when interpreting limited per- formance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=', when the correlation is a small positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' However, this does not limit the use of KP for KGE methods as it captures and contrasts the topology of the positive and negative sampled graphs learned from these methods, which could be a useful metric by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In this paper, the emphasis is on the need for evaluation and benchmarking methods that are computationally efficient rather than providing an exhaustive one method fits all metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We believe that there is much scope for future research in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Some promising directions include 1) better sampling techniques(instead of the random sampling used in this paper), 2) rigorous theoretical analysis drawing the boundaries on the abilities/limitations across settings (zero-shot, few-shot, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' ), 3) using KP (and related metrics) in continuous spaces, that could be differentiable and approximate the ranking metrics, in the optimization process of KGE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 7 CONCLUSION We propose Knowledge Persistence (KP), first work that uses tech- niques from topological data analysis, as a predictor of the ranking metrics to efficiently evaluate the performance of KG embedding approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' With theoretical and empirical evidences, our work brings efficiency at center stage in the evaluation of KG embedding methods along with traditional way of reporting their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Finally, with efficiency as crucial criteria for evaluation, we hope Hits@10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='90 MRR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='95 Hits@10 MRR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0Can Persistent Homology provide an efficient alternative WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA KGE research becomes more inclusive and accessible to the broader research community with limited computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Acknowledgment This work was partly supported by JSPS KAKENHI Grant Number JP21K21280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' REFERENCES [1] Henry Adams and Michael Moy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Topology Applied to Machine Learning: From Global to Local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Frontiers in Artificial Intelligence 4 (2021), 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [2] Mehdi Ali, Max Berrendorf, Charles Tapley Hoyt, Laurent Vermue, Mikhail Galkin, Sahand Sharifzadeh, Asja Fischer, Volker Tresp, and Jens Lehmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Bringing light into the dark: A large-scale evalua- tion of knowledge graph embedding models under a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [3] Ivana Balažević, Carl Allen, and Timothy Hospedales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' TuckER: Tensor Factorization for Knowledge Graph Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Pro- cessing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 5185–5194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [4] Iti Bansal, Sudhanshu Tiwari, and Carlos R Rivero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The impact of negative triple generation strategies and anomalies on knowledge graph completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 29th ACM International Con- ference on Information & Knowledge Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 45–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [5] Anson Bastos, Kuldeep Singh, Abhishek Nadgeri, Saeedeh Shekarpour, Isaiah Onando Mulang, and Johannes Hoffart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Hopfe: Knowl- edge graph representation learning using inverse hopf fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 30th ACM International Conference on Information & Knowledge Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 89–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [6] Max Berrendorf, Evgeniy Faerman, Laurent Vermue, and Volker Tresp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Interpretable and Fair Comparison of Link Prediction or Entity Alignment Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In 2020 IEEE/WIC/ACM International Joint Con- ference on Web Intelligence and Intelligent Agent Technology (WI-IAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' IEEE, 371–374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [7] Kurt D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Bollacker, Robert P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Cook, and Patrick Tufts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Freebase: A Shared Database of Structured General Human Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [8] Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, and Oksana Yakhnenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Translating embeddings for modeling multi-relational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In NeurlPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [9] Antoine Bordes, Jason Weston, Ronan Collobert, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Learning structured embeddings of knowledge bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Twenty- fifth AAAI conference on artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [10] Karsten M Borgwardt and Hans-Peter Kriegel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Shortest-path kernels on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Fifth IEEE international conference on data mining (ICDM’05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' IEEE, 8–pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [11] Karsten M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Borgwardt, Tobias Petri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Vishwanathan, and Hans- Peter Kriegel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' An Efficient Sampling Scheme For Comparison of Large Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Mining and Learning with Graphs, MLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [12] Andrei Z Broder, Marc Najork, and Janet L Wiener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Efficient URL caching for world wide web crawling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 12th international conference on World Wide Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 679–689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [13] Mathieu Carrière, Marco Cuturi, and Steve Oudot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Sliced Wasser- stein Kernel for Persistence Diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 34th In- ternational Conference on Machine Learning (Proceedings of Machine Learning Research), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' PMLR, 664–673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [14] Nian Shong Chok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Pearson’s versus Spearman’s and Kendall’s cor- relation coefficients for continuous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' University of Pittsburgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [15] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Topological persistence and simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings 41st annual symposium on foundations of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' IEEE, 454–463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [16] Brittany Fasy, Yu Qin, Brian Summa, and Carola Wenk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Compar- ing Distance Metrics on Vectorized Persistence Summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In NeurIPS 2020 Workshop on Topological Data Analysis and Beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [17] Luis Galárraga, Katja Hose, and Ralf Schenkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Partout: a dis- tributed engine for efficient RDF processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 23rd International Conference on World Wide Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 267–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [18] Genet Asefa Gesese, Russa Biswas, Mehwish Alam, and Harald Sack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A survey on knowledge graph embeddings with literals: Which model links better literal-ly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Semantic Web Preprint (2019), 1–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [19] Felix Hensel, Michael Moor, and Bastian Rieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A survey of topo- logical machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Frontiers in Artificial Intelligence 4 (2021), 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [20] Nitisha Jain, Jan-Christoph Kalo, Wolf-Tilo Balke, and Ralf Krestel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Do Embeddings Actually Capture Knowledge Graph Semantics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='. In European Semantic Web Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Springer, 143–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [21] Prachi Jain, Sushant Rathi, Soumen Chakrabarti, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Knowl- edge base completion: Baseline strikes back (again).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='00804 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [22] Shaoxiong Ji, Shirui Pan, Erik Cambria, Pekka Marttinen, and Philip S Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A survey on knowledge graphs: Representation, acquisition and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' EEE Transactions on Neural Networks and Learning Systems (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [23] Rudolf Kadlec, Ondřej Bajgar, and Jan Kleindienst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Knowledge Base Completion: Baselines Strike Back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 2nd Workshop on Representation Learning for NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 69–74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [24] Zelong Li, Jianchao Ji, Zuohui Fu, Yingqiang Ge, Shuyuan Xu, Chong Chen, and Yongfeng Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Efficient non-sampling knowledge graph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the Web Conference 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1727– 1736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [25] Lipyeow Lim, Min Wang, Sriram Padmanabhan, Jeffrey Scott Vitter, and Ramesh Agarwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Dynamic maintenance of web indexes using landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 12th international conference on World Wide Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 102–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [26] Yankai Lin, Zhiyuan Liu, Maosong Sun, Yang Liu, and Xuan Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Learning entity and relation embeddings for knowledge graph com- pletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [27] George A Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' WordNet: a lexical database for English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' ACM 38, 11 (1995), 39–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [28] Aisha Mohamed, Shameem Parambath, Zoi Kaoudi, and Ashraf Aboul- naga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Popularity agnostic evaluation of knowledge graph em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Conference on Uncertainty in Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' PMLR, 1059–1068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [29] Michael Moor, Max Horn, Bastian Rieck, and Karsten Borgwardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Topological autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In International conference on machine learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' PMLR, 7045–7054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [30] Kimia Nadjahi, Alain Durmus, Pierre E Jacob, Roland Badeau, and Umut Simsekli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 34 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [31] Mojtaba Nayyeri, Chengjin Xu, Yadollah Yaghoobzadeh, Hamed Shariat Yazdi, and Jens Lehmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Toward Un- derstanding The Effect Of Loss function On Then Performance Of Knowledge Graph Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' arXiv preprint arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='00519 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [32] Tu Dinh Nguyen, Dat Quoc Nguyen, Dinh Phung, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A Novel Embedding Model for Knowledge Base Completion Based on Convolutional Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 327–333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [33] David Patterson, Joseph Gonzalez, Urs Hölzle, Quoc Le, Chen Liang, Lluis-Miquel Munguia, Daniel Rothchild, David So, Maud Texier, and Jeff Dean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The Carbon Footprint of Machine Learning Training Will Plateau, Then Shrink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='05149 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [34] Xutan Peng, Guanyi Chen, Chenghua Lin, and Mark Stevenson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Highly Efficient Knowledge Graph Embedding Learning with Orthog- onal Procrustes Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2364–2375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [35] Pouya Pezeshkpour, Yifan Tian, and Sameer Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Revisit- ing evaluation of knowledge base completion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Automated Knowledge Base Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [36] Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max Horn, Thomas Gumbsch, and Karsten Borgwardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Neural Per- sistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In International Conference on Learning Represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [37] Wiem Ben Rim, Carolin Lawrence, Kiril Gashteovski, Mathias Niepert, and Naoaki Okazaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Behavioral Testing of Knowledge Graph Embedding Models for Link Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In 3rd Conference on Automated Knowledge Base Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [38] Tara Safavi and Danai Koutra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' CoDEx: A Comprehensive Knowl- edge Graph Completion Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 2020 Confer- ence on Empirical Methods in Natural Language Processing (EMNLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 8328–8350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [39] Remi M Sakia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The Box-Cox transformation technique: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series D (The Statistician) 41, 2 (1992), 169–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [40] Aditya Sharma, Partha Talukdar, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Towards understanding the geometry of knowledge graph embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 122–131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Bastos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [41] Kuldeep Singh, Arun Sethupat Radhakrishna, Andreas Both, Saeedeh Shekarpour, Ioanna Lytra, Ricardo Usbeck, Akhilesh Vyas, Akmal Khikmatullaev, Dharmen Punjani, Christoph Lange, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Why reinvent the wheel: Let’s build question answering systems together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 2018 world wide web conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1247–1256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [42] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Spearman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Demonstration of Formulæ for True Measurement of Correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The American Journal of Psychology 18, 2 (1907), 161– 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='org/stable/1412408 [43] Marina Speranskaya, Martin Schmitt, and Benjamin Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Rank- ing vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Classifying: Measuring Knowledge Base Completion Quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Automated Knowledge Base Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [44] Zhiqing Sun, Zhi-Hong Deng, Jian-Yun Nie, and Jian Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Ro- tatE: Knowledge Graph Embedding by Relational Rotation in Complex Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [45] Zhiqing Sun, Shikhar Vashishth, Soumya Sanyal, Partha Talukdar, and Yiming Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A Re-evaluation of Knowledge Graph Completion Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 5516–5522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [46] Pedro Tabacof and Luca Costabello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Probability Calibration for Knowledge Graph Embedding Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [47] Sudhanshu Tiwari, Iti Bansal, and Carlos R Rivero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Revisiting the evaluation protocol of knowledge graph completion methods for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the Web Conference 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 809–820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [48] Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier, and Guillaume Bouchard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Complex embeddings for simple link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' PMLR, 2071–2080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [49] Ke Tu, Jianxin Ma, Peng Cui, Jian Pei, and Wenwu Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Au- tone: Hyperparameter optimization for massive network embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 216–225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [50] Renata Turkeš, Guido Montúfar, and Nina Otter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' On the effec- tiveness of persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 10551 [51] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [52] Haoyu Wang, Yaqing Wang, Defu Lian, and Jing Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A light- weight knowledge graph embedding framework for efficient inference and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 30th ACM International Conference on Information & Knowledge Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1909–1918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [53] Kai Wang, Yu Liu, Qian Ma, and Quan Z Sheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Mulde: Multi- teacher knowledge distillation for low-dimensional knowledge graph embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the Web Conference 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 1716–1726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [54] Kai Wang, Yu Liu, and Quan Z Sheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Swift and Sure: Hardness- aware Contrastive Learning for Low-dimensional Knowledge Graph Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the ACM Web Conference 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 838–849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [55] Quan Wang, Zhendong Mao, Bin Wang, and Li Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Knowledge graph embedding: A survey of approaches and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' IEEE Transactions on Knowledge and Data Engineering 29, 12 (2017), 2724– 2743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [56] Xin Wang, Shuyi Fan, Kun Kuang, and Wenwu Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Explain- able automated graph representation learning with hyperparameter importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' PMLR, 10727–10737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [57] Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Knowledge graph embedding by translating on hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Pro- ceedings of the AAAI Conference on Artificial Intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [58] Larry Wasserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Annual Review of Statistics and Its Application 5 (2018), 501–532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [59] Carole-Jean Wu, Ramya Raghavendra, Udit Gupta, Bilge Acun, New- sha Ardalani, Kiwan Maeng, Gloria Chang, Fiona Aga, Jinshi Huang, Charles Bai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Sustainable ai: Environmental implications, challenges and opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Proceedings of Machine Learning and Systems 4 (2022), 795–813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [60] Bishan Yang, Wen-tau Yih, Xiaodong He, Jianfeng Gao, and Li Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Embedding Entities and Relations for Learning and Inference in Knowledge Bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In 3rd International Conference on Learning Repre- sentations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [61] Shih-Yuan Yu, Sujit Rokka Chhetri, Arquimedes Canedo, Palash Goyal, and Mohammad Abdullah Al Faruque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Pykg2vec: A Python Library for Knowledge Graph Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 22 (2021), 16–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [62] Yufeng Zhang, Weiqing Wang, Wei Chen, Jiajie Xu, An Liu, and Lei Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Meta-Learning Based Hyper-Relation Feature Modeling for Out-of-Knowledge-Base Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' In Proceedings of the 30th ACM International Conference on Information & Knowledge Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2637–2646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [63] Yongqi Zhang, Zhanke Zhou, Quanming Yao, and Yong Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KG- Tuner: Efficient Hyper-parameter Search for Knowledge Graph Learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' CoRR abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='02460 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' [64] Afra Zomorodian and Gunnar Carlsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Computing persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Discrete & Computational Geometry 33, 2 (2005), 249–274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 8 APPENDIX Figure 6: Study on the carbon footprint of the evaluation phase of the KGE methods on YAGO3-10 when using KP vs Hits@10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The x-axis shows the the carbon footprint in g eq 𝐶𝑂2 in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Figure 7: Study on the carbon footprint of the evaluation phase of the KGE methods on Wikidata when using KP vs Hits@10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The x-axis shows the the carbon footprint in g eq 𝐶𝑂2 in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 Extended Evaluation Effect of KP on Efficient KGE Methods Evaluation: The re- search community has recently proposed several KGE methods to improve training efficiency [34, 52, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Our idea in this experi- ment is to perceive if efficient KGE methods improve their overall carbon footprint using KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For the same, we selected state-of-the- art efficient KGE methods: Procrustes [34] and HalE [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Figure 8 illustrates that using KP for evaluation drastically reduces the carbon footprints of already efficient KGE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For instance, the carbon footprint of HalE is reduced from 110g (using hits@10) to 20g of CO2 (using KP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Carbon Footprint of the Evaluation phase of the KGE prototyping process (YAGO3_10) Carbon Footprint using Kp Carbon Footprint using ranking metrics TransE TransH TransR Method Complex RotatE ConvKB TuckER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='5 L0Carbon Footprint of the Evaluation phase of the KGE prototyping process (Wikidata) Carbon Footprint using KP Carbon Footprint using ranking metrics TransE TransH TransR Method Complex RotatE ConvkB TuckER LOCan Persistent Homology provide an efficient alternative WWW ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' APRIL 30 - MAY 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Texas,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 Table 8: Kendall’s tau (𝜏) scores computed from the metric scores with respect to the ranking metrics on the standard KG embedding datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The KG methods are evaluated after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Figure 8: Study on efficient KGE methods and their the car- bon footprint on WN18RR when using KP vs Hits@10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The x-axis shows the the carbon footprint in g eq 𝐶𝑂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Robustness and Efficiency on large KGs: This ablation study aims to gauge the correlation behavior of KP and ranking metric on a large-scale KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' For the experiment, we use Yago3-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' A key reason to select the Yago-based dataset is that besides being large-scale, it has rich semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Results in Table 9 illustrate KP shows a stable and high correlation with the ranking metric, con- firming the robustness of KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We show carbon footprint results in Figure 6 for the yago dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Further we also study the efficiency of KP on the wikidata dataset in Figure 7 which reaffirms that KP maintains its efficiency on large scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Efficiency comparison of Sliced Wasserstein vs Wasserstein as distance metric in KP: In this study we empirically provide a rationale for using sliced wasserstein as a distance metric over the wasserstein distance in KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The results are in table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We see that KP using sliced wasserstein distance provides a significant computational advantage over wasserstein distance, while having a good performance as seen in the previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Thus we need an efficient approximation such as the sliced wasserstein distance as the distance metric in place of wasserstein distance in KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Metrics Hits@1(↑) Hits@3(↑) Hits@10(↑) MR(↓) MRR(↑) r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='572 𝜌 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='643 𝜏 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='429 Table 9: KP correlations on the YAGO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Metrics KP(W) KP(SW) Speedup ↑ Dataset FB15K237 WN18RR FB15K237 WN18RR FB15K237 WN18RR split val + test val + test val + test val + test Avg Avg TransE 1136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='766 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='114 x 3540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3 x 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='6 TransH 2943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='869 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='095 x 9278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='5 x 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='8 TransR 1734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='576 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='129 x 5168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3 x 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='4 Complex 1054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='721 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='135 x 3255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='3 x 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 RotatE 865.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='098 x 3230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 x 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1 ConvKB 719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='310 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='132 x 1675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='7 x 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='0 Table 10: Evaluation Metric Comparison wrt Computing Time (in minutes, for 100 epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Column 1 denotes popular KGE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Depicted values denote evalua- tion(validation+test) time for computing a metric and cor- responding speedup using KP(𝑆𝑊 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP(𝑆𝑊 ) with sliced wasserstein as the distance metric significantly reduces the evaluation time (green) in comparison with KP(𝑊 ) which uses the wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 Theoretical Proof Sketches We work under the following considerations: As the KGE method converges the mean statistic(𝑚𝜈) of the scores of the positive triples consistently lies on one side of the half plane formed by the mean statistic(𝑚𝜇) of the negative triples, irrespective of the data distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' The detail proofs are here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP has a monotone increasing correspondence with the Proxy of the Expected Ranking Metrics(PERM) under the above stated considerations as 𝑚𝜈 deviates from 𝑚𝜇 Carbon Footprint of the Overall KGE prototyping process using methods that save on training time(WN18RR) Carbon Footprint using KP Carbon Footprint using ranking metrics HaLE Method ProcrustEs 0 25 50 75 100 125WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Bastos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Considering the 0-dimensional PD as used by KP and a normal distribution for the edge weights (can be extended to other distributions using techniques like [39]) of the graph(scores of the triples), we have univariate gaussian measures [40] 𝜇 and 𝜈 for the positive and negative distributions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Denote by 𝑚𝜇 and 𝑚𝜈 the means of the distributions 𝜇 and 𝜈 respectively and by Σ𝜇, Σ𝜈 the respective covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝑊 2 2 (𝜇,𝜈) = ∥𝜇 − 𝜈∥2 + 𝐵(Σ𝜇, Σ𝜈)2 (2) where 𝐵(Σ𝜇, Σ𝜈)2 = 𝑡𝑟 (Σ𝜇 + Σ𝜈 − 2(Σ 1 2𝜇 Σ𝜈Σ 1 2𝜇 ) 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Next we see how that changing the means of the distribution(and also variance) changes PERM and KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' We can show that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝑃 = ∫ 𝑥=∞ 𝑥=−∞ 𝐷+(𝑥) �∫ 𝑦=∞ 𝑦=𝑥 𝐷−(𝑥)𝑑𝑦 � 𝑑𝑥 𝜕𝑃 𝜕𝑚𝜈 = ∫ 𝑥=∞ 𝑥=−∞ 𝐷+(𝑥) �∫ 𝑦=∞ 𝑦=𝑥 𝜕𝐷−(𝑥) 𝜕𝑚𝜈 𝑑𝑦 � 𝑑𝑥 ≥ 0 𝜕𝑃 𝜕Σ𝜈 = ∫ 𝑥=∞ 𝑥=−∞ 𝐷+(𝑥) �∫ 𝑦=∞ 𝑦=𝑥 𝜕𝐷−(𝑦) 𝜕Σ𝜈 𝑑𝑦 � 𝑑𝑥 ≤ 0 𝜕𝑃 𝜕Σ𝜇 = ∫ 𝑥=∞ 𝑥=−∞ 𝜕𝐷+(𝑥) 𝜕Σ𝜈 �∫ 𝑦=∞ 𝑦=𝑥 𝐷−(𝑦)𝑑𝑦 � 𝑑𝑥 Since KP is the (sliced) wasserstein distance between PDs we can show the respective gradients are as below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝜕𝑊 2 2 (𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝜈) 𝜕𝑚𝜈 = 2|𝑚𝜇 − 𝑚𝜈 | ≥ 0 𝜕𝑊 2 2 (𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝜈) 𝜕Σ𝜈 = 𝐼 − Σ 1 2𝜇 (Σ 1 2𝜇 Σ𝜈Σ 1 2𝜇 ) −1 2 Σ 1 2𝜇 As the generating process of the scores changes the gradient of PERM along the direction (𝑑𝑚𝜈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎𝜈) can be shown to be the following � (𝑑𝑚𝜈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' � 𝜕𝑃𝐸𝑅𝑀 𝜕𝑚𝜈 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝜕𝑃𝐸𝑅𝑀 𝜕Σ𝜇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝜕𝑃𝐸𝑅𝑀 𝜕Σ𝜈 �� ≥ 0 Similarly the gradient of KP along the direction (𝑑𝑚𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎𝜈) is � (𝑑𝑚𝜈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑑𝜎),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' ( 𝜕𝑊 2 2 (𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝜈) 𝜕𝑚𝜈 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝜕𝑊 2 2 (𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝜈) 𝜕Σ𝜇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' 𝜕𝑊 2 2 (𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝜈) 𝜕Σ𝜈 ) � ≥ 0 Since both PERM and and KP vary in the same manner as the distribution changes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' the two have a one-one correspondence [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' □ The above lemma shows that there is a one-one correspondence between KP and PERM and by definition PERM has a one-one cor- respondence with the ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Therefore, the next theorem follows as a natural consequence Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' KP has a one-one correspondence with the Ranking Metrics under the above stated considerations Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Under the considerations of theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='1, the relative change in KP on addition of random noise to the scores is bounded by a function of the original and noise-induced covariance matrix as ΔK P K P ≤ 𝑚𝑎𝑥((1 − |Σ+1 𝜇1 Σ−1 𝜇2 | 3 2 ), (1 − |Σ+1 𝜈1 Σ−1 𝜈2 | 3 2 )), where Σ𝜇1 and Σ𝜈1 are the covariance matrices of the positive and negative triples’ scores respectively and Σ𝜇2 and Σ𝜈2 are that of the corrupted scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Consider a zero mean random noise to simulate the process of varying the distribution of the scores of the KGE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Let 𝑚𝜇1 and 𝑚𝜈1 be the means of the positive and negative triples’ scores of the original method and Σ𝜇1, Σ𝜈1 be the respective covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Let 𝑚𝜇2 and 𝑚𝜈2 be the means of the positive and negative triples’ scores of the corrupted method and Σ𝜇2, Σ𝜈2 be the respective covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' Considering the kantorovich duality [51] and taking the difference between the two measures we have KP1 − KP2 = 𝑖𝑛𝑓 𝛾1∈Π(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑦) ∫ 𝛾1 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑦)𝑑𝛾1(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑦) − 𝑖𝑛𝑓 𝛾2∈Π(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑦) ∫ 𝛾2 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑦)𝑑𝛾2(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='𝑦) ≤ 𝑠𝑢𝑝 Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='Ψ ∫ 𝑥 Φ(𝑥)𝑑𝜇1(𝑥) + ∫ 𝑦 Ψ(𝑦)𝑑𝜈1(𝑦) − ∫ 𝑥 Φ(𝑥)𝑑𝜇2(𝑥) − ∫ 𝑦 Ψ(𝑦)𝑑𝜈2(𝑦) ≤ 𝑠𝑢𝑝 Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='Ψ ∫ 𝑥 Φ(𝑥)(𝑑𝜇1(𝑥) − 𝑑𝜇2(𝑥)) + ∫ 𝑦 Ψ(𝑦)(𝑑𝜈1(𝑦) − 𝑑𝜈2(𝑦)) Now by definition of the measure 𝜇1 we have 𝜕𝜇1 𝜕𝑥 = −𝜇1Σ−1 𝜇1 (𝑥 − 𝑚𝜇1) 𝑑𝜇1(𝑥𝑖) = −(𝜇1Σ−1 𝜇1 (𝑥 − 𝑚𝜇1))[𝑖]𝑑𝑥𝑖 ∴ 𝑑𝜇1(𝑥) = 𝑑𝑒𝑡(𝑑𝑖𝑎𝑔(−𝜇1Σ−1 𝜇1 (𝑥 − 𝑚𝜇1)))𝑑𝑥 From the above results we can show the following KP1 − KP2 ≤ 𝑚𝑎𝑥((1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1 𝜇2 ) 𝑛 2 +1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' (1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1 𝜈2 ) 𝑛 2 +1))KP1 ∴ ΔKP KP ≤ 𝑚𝑎𝑥 �� 1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1 𝜇2 ) 𝑛 2 +1� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' � 1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1 𝜈2 ) 𝑛 2 +1�� In our case as we work in the univariate setting 𝑛 = 1 and thus we have ΔK P K P ≤ 𝑚𝑎𝑥 �� 1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1 𝜇2 ) 3 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' � 1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1 𝜈2 ) 3 2 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content=' □ Can Persistent Homology provide an efficient alternative WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} +page_content='2 shows that as noise is induced gradually, the KP value changes in a bounded manner as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFOT4oBgHgl3EQfyzQ_/content/2301.12929v1.pdf'} diff --git a/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf b/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..14a3d40a5c6d438ba5f72ba39e60a6584287a01a --- /dev/null +++ b/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1d6f7c670e5c633f9f71c37ba3746432d676eb4405bf50a906d239dd4a5c14fb +size 1036651 diff --git a/KtE4T4oBgHgl3EQf7w7r/vector_store/index.pkl 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0000000000000000000000000000000000000000..626625b473614980653d869fa93003b242199da6 --- /dev/null +++ b/LtE0T4oBgHgl3EQfSgD7/content/tmp_files/2301.02225v1.pdf.txt @@ -0,0 +1,1638 @@ +arXiv:2301.02225v1 [stat.ML] 4 Jan 2023 +Submitted to the Annals of Applied Statistics +L1−2 GLASSO: L1−2 REGULARIZED MULTI-TASK GRAPHICAL LASSO +FOR JOINT ESTIMATION OF EQTL MAPPING AND GENE NETWORK +BY WEI MIAO1,a, LAN YAO2,b +1College of Mathematics, Hunan University, amiaow@hnu.edu.cn +2College of Mathematics, Hunan University, byao@hnu.edu.cn +A critical problem in genetics is to discover how gene expression is reg- +ulated within cells. Two major tasks of regulatory association learning are +: (i) identifying SNP-gene relationships, known as eQTL mapping, and (ii) +determining gene-gene relationships, known as gene network estimation. To +share information between these two tasks, we focus on the unified model for +joint estimation of eQTL mapping and gene network, and propose a L1−2 +regularized multi-task graphical lasso, named L1−2 GLasso. Numerical ex- +periments on artificial datasets demonstrate the competitive performance of +L1−2 GLasso on capturing the true sparse structure of eQTL mapping and +gene network. L1−2 GLasso is further applied to real dataset of ADNI-1 and +experimental results show that L1−2 GLasso can obtain sparser and more +accurate solutions than other commonly-used methods. +1. Introduction. +Developments in sequencing technology allow us to obtain more and +more genomic data since the publication of the first human genome sequence. Computational +techniques can help us to mine meaningful information from raw data and understand how +gene expression is regulated in cells. In general, these problems include identifying cancer +gene co-expression (co-expression: simultaneous expression of two or more genes) mod- +ules, determining SNP-gene relationships through eQTL (expression quantitative trait locus) +mapping and determining gene-gene relationships by estimating gene network structure, etc +(Rockman and Kruglyak, 2006; Gardner and Faith, 2005). Given a dataset containing single +nucleotide polymorphisms (SNPs) and mRNA expression, the problem is to understand the +SNP-gene and gene-gene relationships. For example, assuming SNPs x = (x1,...,xp) and +genes y = (y1,...,yq), the SNP-gene relationships in eQTL mapping are determined by a +regression coefficient matrix and the gene-gene relationships in gene network estimation are +captured by output structure. +There have been many types of research on eQTL mapping and gene network estimation. +The traditional method of eQTL mapping is to determine whether there is an association +between a gene and an SNP. Later, multivariate models have been developed to determine +relationships between multiple SNPs and a gene (Michaelson et al., 2010). More recently, +several models have been proposed to determine relationships between multiple SNPs and +multiple genes (Kim and Xing, 2012). +As for gene network estimation, the traditional method is to construct a graph and connect +two related genes with an edge. To be specific, many previous studies inferred gene-gene re- +lationships from gene expression data. For example, in Gaussian Graphical Models (GGM) +framework, graphical models use graphs to represent dependencies between random vari- +ables (Schäfer and Strimmer, 2005; Segal et al., 2005; Li and Gui, 2006; Peng et al., 2009). +In GGM, multivariate vectors follow a multivariate normal distribution and have a specific +structure of the covariance matrix. The inverse of the covariance matrix is called the con- +centration matrix. GGM assumes that the expression variation pattern of a given gene can +Keywords and phrases: Multivariate regression, structured sparsity, difference of l1 and l2 norms. +1 + +2 +be predicted by a small subset of other genes (Meinshausen and Bühlmann, 2006). The as- +sumption leads to the sparsity (i.e., multiple zeros) in the concentration matrix and reduces +the problem to a well-known neighborhood selection or covariance selection problem. In +the concentration map modeling framework, the key idea is to use a partial correlation as +a measure of the independence of any two genes, thereby directly distinguishing between +direct and indirect interactions. In other approaches, Bayesian Networks are also utilized to +establish the structure between genes (Marbach et al., 2010). +The Multi-task regression model can be used to jointly estimate the regression coefficient +matrix and the output structure. One challenge to be faced is the high-dimensional data disas- +ter which is very common in genetic data. In previous studies, sparse learning is a good way +to deal with this problem and has attracted wide attention due to its advantages of sparse solu- +tions, strong interpretability, and convenient computation (Bertsimas and Van Parys, 2020). +Furthermore, to enhance the expression ability, researchers have proposed various structured +sparse models which combine sparse learning with structured regularization. In various fields +of computing and engineering, it is an important research topic to construct a structured +sparse model based on the prior assumption of sparsity and the specific structural character- +istics of the problem. +Many models which are based on structured sparsity regularization have been reviewed +(Vinga, 2021). Group Lasso, which encourages related exit groups to have nonzero coef- +ficients for the same subset of inputs, has been studied extensively (Yuan and Lin, 2006). +A computationally efficient way was provided to perform Lasso-regularized estimation of +sparse concentration matrices (Friedman, Hastie and Tibshirani, 2008). Graph-Guide Fused +Lasso encourages pairs of outputs linked in a graph to have similar coefficient values +(Kim and Xing, 2009). Conditional Gaussian Graphical Models (CGGM) have been devel- +oped to estimate both the output structure and the regression coefficients with structured +sparsity at the same time (Yin and Li, 2011; Li, Chun and Zhao, 2012; Chun et al., 2013). +However, these models all require us to have prior knowledge of relationships between the +output y. Another class of models, which focuses on estimating the conditional covariance of +y|x rather than the covariance structure of the output y, has been developed to learn both the +regression coefficient matrix and the output structure (Rothman, Levina and Zhu, 2010). Un- +der the influence of noisy data, these models may not end up with the true structure between +outputs. Recently, a novel approach called Inverse-Covariance-Fused Lasso (ICLasso) which +focuses on the covariance structure of the output y, can also jointly estimate the regression +coefficient matrix and the output structure (Marchetti-Bowick et al., 2019). The structured +sparsity regularization penalty is formed by the l1 norm in ICLasso. +In addition, many other regularization penalties have also been studied. The projection +operators that can enforce both l1 and l2 norms have been developed for encouraging sparsity +in structured sparse models (Hoyer, 2004). One of the regularization penalties that has been +studied a lot is the difference of l1 and l2 norms. The penalty is considered robust and can +help select sparse solutions (Yin, Esser and Xin, 2014). It has been used in nonnegative least +squares (NNLS) and orthogonal matching pursuit. The comparisons with l1 minimization for +imaging data can be found in (Esser, Lou and Xin, 2013). Some researchers have also applied +the difference of l1 and l2 norms in sparse signal reconstruction problems to approximate the +original l0-norm-based sparseness (Liu et al., 2016). It can be seen in other areas such as +compressed sensing, seismic inversions, etc (Yin et al., 2015; Wang et al., 2019). +Motivated by these studies, we propose a new model based on difference of l1 and l2 +norms. Our model makes some important new contributions: (i) We introduce a new regular- +ization penalty into the model inducing a better approximation and use a faster algorithm to +solve the optimization problem. (ii) Under the same parameter setting, the solved regression +coefficient matrix is sparser compared with existing methods such as MRCE, ICLasso, etc. +(iii) Our model outperforms other baseline methods in the recovery of the output structure. + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +3 +In Section 2, we give an introduction to several baseline methods. In Section 3, we describe +our new model with a different penalty and the optimization algorithm in detail. In sections +4 and 5, we evaluate the effectiveness of our method on simulated and real datasets. Finally, +we summarize the article in section 6. +2. Background. +We assume that X ∈ Rn×p is a matrix of SNP genotypes and Y ∈ Rn×q +is a matrix of gene expression values. Here, n represents the number of samples, q represents +the number of genes, and p represents the number of SNPs. We show the exact matrix form +as: +(1) + + +y11 y12 ··· y1q +y21 y22 ··· y2q +... +... +... +... +yn1 yn2 ··· ynq + + +� +�� +� +Y += + + +x11 x12 ··· x1p +x21 x22 ··· x2p +... +... +... +... +xn1 xn2 ··· xnp + + +� +�� +� +X +× + + +β11 β12 ··· β1q +β21 β22 ··· β2q +... +... +... +... +βp1 βp2 ··· βpq + + +� +�� +� +B +2.1. Multi-Task Lasso. +Multi-Task Lasso can be used for statistical tests to detect SNPs +that are associated with genes (Tibshirani, 1996). Given X and Y , the multivariate linear +regression model is given by +(2) +yk = Xβk + ǫk,k = 1,...,q, +where βk = [β1k,...,βpk]T represents regression coefficients. It can be used to detect SNPs +that are significantly associated with genes. We assume ǫk ∼ N(0,σ2) and the mathematical +expression in matrix form is: +(3) +min +B +1 +n∥Y − XB∥2 +F + λ∥B∥1, +where B ∈ Rp×q represents the regression coefficient matrix, λ is the regularization parame- +ter, which is used to control the degree of sparsity. +2.2. Multivariate regression with covariance estimation. +Multivariate regression with +covariance estimation (MRCE) is a method that can jointly estimate the regression coeffi- +cient matrix and the output structure (Rothman, Levina and Zhu, 2010). It assumes that X +has the linear relationship with Y : Y = XB + E, in which E ∼ N +� +0,Ω−1� +is a Gaussian +noise matrix. We can calculate that Y | X ∼ N +� +XB,Ω−1� +. MRCE can be expressed as +follows: +(4) +min +B,Ω +1 +n tr +� +(Y − XB)T (Y − XB)Ω +� +− log det(Ω) + λ1∥B∥1 + λ2∥Ω∥1, +where Ω represents the conditional inverse covariance of Y |X rather than the inverse covari- +ance of Y . Based on previous assumptions, Ω is related to the Gaussian noise matrix E and +it can not capture the exact relationship between the regression coefficient matrix B and the +output structure in Y . + +4 +2.3. Inverse-Covariance-Fused Lasso. +Marchetti-Bowick et al. (2019) introduced a new +model called Inverse-Covariance-Fused Lasso. The model can also jointly estimate regres- +sion coefficients and the output structure. Compared with previous studies, the method cap- +tures the marginal inverse covariance of Y rather than the conditional inverse covariance of +Y |X. +ICLasso (Inverse-Covariance-Fused Lasso) begins with two core modeling assumptions: +x ∼ N(0,T) +y|x ∼ N(xT B,E), +where T represents the covariance of x and E represents the conditional covariance of y|x, +Θ represents the marginal inverse covariance of Y , which is different from Ω in (4). +With these assumptions, we can derive the marginal distribution of y. Based on the fact +that the marginal distribution p(y) follows the Gaussian distribution, then use the law of total +expectation and the law of total variance to derive the mean and covariance of y, as follows: +(5) +Ey(y) = Ex +� +Ey|x(y | x) +� += Ex +� +xT B) +� += 0 +Covy(y) = Ex +� +Covy|x(y | x) +� ++ Covx +� +Ey|x(y | x) +� += Ex (E) + Covx +� +xT B +� += E + BT TB. +We can calculate the distribution of y : +(6) +y ∼ N +� +0,Θ−1� +, +where Θ−1 = E + BTTB is the marginal covariance of y. This is a connection between the +inverse covariance of y and the regression coefficient matrix B. For simplicity, we assume +T = τ 2Ip×p and E = ε2Iq×q and then Θ−1 ∝ BTB. +Given i.i.d. observations of SNPs x ∈ Rp and genes y ∈ Rq, in order to jointly estimate +the regression coefficient matrix B ∈ Rp×q and the inverse covariance matrix Θ ∈ Rq×q, the +inverse-covariance-fused lasso optimization problem can be written as: +(7) +min +B,Θ +1 +n∥Y − XB∥2 +F + 1 +n tr +� +Y T Y Θ +� +− log det(Θ) ++ λ1∥B∥1 + λ2∥Θ∥1 ++ γ +� +(k,m) +|θkm| · ∥β.k + sgn(θkm)β.m∥1 . +This objective effectively boils down to a combination of problems: Multi-Task Lasso and +Sparse Inverse Covariance Estimation including a graph-guided fusion penalty form. The role +of the penalty � +(k,m) |θkm| · ∥β.k + sgn(θkm)β.m∥1 is to encourage structural information +sharing between B and Θ. +2.4. Estimating Model Parameters with a Fusion Penalty. +Previous studies provide us +with an idea to estimate these parameters and also encourage information sharing between B +and Θ. To do this, the model is formulated as a convex optimization problem, whose objective +function is: +(8) +lossy|x(B) + lossy(Θ) + penalty(B,Θ), + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +5 +where we can see: +• lossy|x(B) can be derived from the negative log-likelihood of y|x; +• lossy(Θ) can be derived from the negative marginal log-likelihood of y; +• penalty(B,Θ) is a penalty term that encourages shared information between the regres- +sion coefficient matrix B and the output structure Θ. +The l1 norm penalty ∥B∥1 and ∥Θ∥1 induce sparsity in the estimates of B and Θ, which +make the model feasible even on high dimensional data. To obtain a sparser solution, we can +naturally generalize the l1 norm penalty. Based on this framework, we propose our model in +the next section. +3. The l1−2 Graphical Lasso. +Since the sparsity of B is reflected by the number of its +nonzero terms, it is equivalent to the so-called l0 norm penalty. ICLasso replaces l0 norm +penalty with l1 norm penalty. A reconstruction framework based on the difference l1 and l2 +norms was proposed (Esser, Lou and Xin, 2013). It can be reformulated as follows: +(9) +minx λ∥x∥1 − τ∥x∥2 +s.t. +y = Ax. +It was proved that as long as the selection of appropriate λ and τ, the solution can be close +to the solution of the problem with l0 norm penalty. To get sparser solutions, we propose the +l1−2 Graphical Lasso. Our model can be described as: +(10) +lossy|x(B) + lossy(Θ) + penalty(B,Θ) + penalty(B) + penalty(Θ). +Specifically, the expression is as followed: +(11) +min +B,Θ +1 +n∥Y − XB∥2 +F + 1 +n tr +� +Y T Y Θ +� +− log det(Θ) ++ λ1||B||1 − τ||B||2,1 + λ2∥Θ∥1 ++ γ +� +(k,m) +|θkm| · ∥β.k + sgn(θkm)β.m∥1 . +We define: +(12) +g12(B) = 1 +n||Y − XB||2 +F + λ1||B||1 − τ||B||2,1 +(13) +h(Θ) = 1 +ntr(Y T Y Θ) − log det(Θ) + λ2||Θ||1 +(14) +GFL(B,−Θ) = +q +� +k=1 +q +� +m=1 +|θkm| · ∥β·k + sgn(θkm)β·m∥1 +The term 1 +n∥Y −XB∥2 +F , 1 +n tr +� +Y T Y Θ +� +−log det(Θ) and γ � +(k,m) |θkm|·∥βk + sgn(θkm)β.m∥1 +are derived from Equation (8) respectively. We describe the role of each item in detail: +• lossy|x(B): 1 +n∥Y − XB∥2 +F . According to y | x ∼ N +� +xT B,ε2I +� +, we can derive its ex- +pression by Maximum Likelihood Estimation. The role of this term is to encourage the re- +gression coefficient matrix B. +• lossy(Θ): 1 +n tr +� +Y T Y Θ +� +− log det(Θ). According to y ∼ N +� +0,Θ−1� +, we can derive its +expression by Maximum Likelihood Estimation. The role of this term is to encourage the +inverse covariance Θ to reflect the correlations among the outputs. + +6 +• penalty(B,Θ): γ � +(k,m) |θkm| · ∥β·k + sgn(θkm)β·m∥1 is a graph-guided fusion +penalty. It encourages the regression coefficients of closely related outputs to be similar. +When yk is partially positively correlated with ym and βjk ̸= βjm for any j, it imposes a +penalty proportional to θkm, and when yk and ym have a negative partial correlation for any +j, it imposes a penalty proportional to −θkm. +• penalty(B): λ1||B||1 − τ||B||2,1 = λ1 +� +j,k |βjk| + τ2 +�p +j=1 +��q +k=1 β2 +jk is an l1−2 +norm penalty over the regression coefficient matrix that induces sparsity in B. Compared +to l1 norm, the difference of l1 and l2 norms is closer to the l0 norm. +• penalty(Θ): λ2∥Θ∥1 = λ2 +� +k,m |θkm| is an l1 norm penalty that induces sparsity in Θ. +3.1. Relationship to ICLasso. +l1-2-GLasso also implicitly assumes two underlying mod- +eling assumptions: x ∼ N(0,T) and y|x ∼ N(xT B,E). The difference between l1-2-GLasso +and ICLasso is that we use difference of l1 and l2 norms to get sparse solutions that are closer +to the real world. The experimental results are shown in the next section. We find that l1-2- +GLasso can obtain sparser solutions than other models while maintaining the regression error. +3.2. Optimization. +Previous work on the problem (11) propose some off-the-shelf algo- +rithms to solve the above optimization problem. Based on these studies, we use the alternating +minimization strategy to solve the l1-2-GLasso. Pay attention to the GFL(B,−Θ) term, it is +clear that this term is a bi-convex function. Thus, upon defining +(15) +g12(B) = 1 +n∥Y − XB∥2 +F + λ1∥B∥1 − τ||B||2,1 +h(Θ) = 1 +n tr +� +Y T Y Θ +� +− log det(Θ) + λ2∥Θ∥1. +The original objective can be rewritten as +(16) +min +B,Θ g12(B) + h(Θ) + GFL(B,−Θ). +Compared with ICLasso, l1-2-GLasso can obtain sparser regression coefficient matrix by +difference of l1 and l2 norms. The difficulty of solving the problem also lies in this penalty. It +can be seen that the term GFL(B,−Θ) is bi-convex, so we can use an alternating minimiza- +tion strategy developed by Marchetti-Bowick et al. (2019) to solve the problem (15). +First, we fix Θ, so the problem becomes: +(17) +fΘ(B) = g12(B) + GFL(B,−Θ). +Although our model introduces a new regularization term λ1||B||1 − τ||B||2,1, it can be +decomposed into the form of a differentiable function and a non-differentiable function. For +the matrix B = [β1,β2,...,βp]T , βiT is the ith row in B. According to the definition: +(18) +||B||2,1 = +p +� +i=1 +(βiT βi) +1 +2 , +we define the Σ and the derivative of ||B||2,1 as: +(19) +Σ := + + +1 +||β1||2 +0 +··· +0 +0 +1 +||β2||2 ··· +0 +... +... +... +... +0 +0 +··· +1 +||βp||2 + + +, + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +7 +(20) +∂||B||2,1 +∂B += (∂ �p +i=1(βiT βi) +1 +2 +∂βi +)p×1 = + + +1 +||β1||2 +0 +··· +0 +0 +1 +||β2||2 ··· +0 +... +... +... +... +0 +0 +··· +1 +||βp||2 + + +B = ΣB. +When we use a small τ, the term ∥Y −XB∥2 +F −τ||B||2,1 is considered convex. This prob- +lem can be solved by the proximal-average proximal gradient descent (PA-PG) algorithm Yu +(2013). Compared to the proximal gradient descent, PA-PG converges consistently faster. It +was proved that with a suitable stepsize, the subgradient method converges in at most O(1/ǫ) +steps for any accuracy ǫ > 0. First, the derivation of ∥Y − XB∥2 +F − τ||B||2,1 is XT (XB − +Y ) − τΣ, then we take a gradient step of the form B − ν(XT (XB − Y ) − τΣ) and some +small step size ν are used. Other optimization procedures can follow Marchetti-Bowick et al. +(2019). Finally, this sub-problem can be written as +(21) +�βjk, �βjm = arg min +βjk,βjm +1 +2ν (βjk − zjk)2 + (βjm − zjm)2 + |βjk + sgn(θkm)βjm|. +We can find a closed-form solution for β. The solution to this sub-problem is relatively +simple, and the convergence speed of the algorithm can also be found in Yu (2013). +Then fix B, and the problem becomes: +(22) +min +Θ +h(Θ) + GFL(B,−Θ). +This problem can be solved by adapting the block coordinate descent (BCD) algorithm +(Friedman, Hastie and Tibshirani, 2008). Finally, this sub-problem can be written as: +(23) +min +α +1 +2αT �Hjα + uT α+ − lT α−. +We solve each coordinate using the coordinate descent method and applying a variant of +the soft threshold operator. +4. Simulation Study. +In this section, we compare different models on synthetic data +with known values of B and Θ, so that we can directly measure how well the true parameter +values are recovered. Models include Graph-Guided Fused Lasso (GFLasso), Sparse Mul- +tivariate Regression with Covariance Estimation (MRCE), Inverse-Covariance-Fused Lasso +(ICLasso), and l1-2-GLasso. For each model, we select hyperparameter values by minimizing +the error on a held-out validation set. +We evaluate each model from two dimensions: (i) Recovery of sparse structures. (ii) Re- +gression error of B and Θ. +(i) Recovery of sparse structures: To evaluate how well each model can estimate the +sparsity structure of B and Θ, we calculate the F1 score for recovering the elements of B +and Θ. To do this, we choose a threshold at which each element value is considered “zero” or +“nonzero” and then we calculate the Precision (P) and Recall (R) for different synthetic data. +(ii) Regression error of B and Θ: We examine the prediction error of each model when +using ˆB to predict Y from X. +For this analysis, we use a block-structured network over the outputs. The outputs are +divided into non-overlapping groups, and each group forms a fully connected subgraph in +the network. + +8 +Here we describe our procedure for generating synthetic data. At a high level, we first +fix the sparse structure of the underlying components of the model, then generate coefficient +values, and lastly sample X and Y according to our model. +Given the number of samples n, the number of genes q, and the number of SNPs p, firstly +we determine the module size in the gene network and fix the number of SNPs s associated +with each gene. Next, we randomly assign each gene to a module and select the set of s SNPs +associated with each module. For each module, we randomly assign a major gene in this +module and for each SNP xj associated with the primary gene yk, we generate its association +strengths according to βjk ∼ Uniform(0,1). For the other genes ym in this module, we +generate the association strengths according to βjm ∼ Uniform(βjk,ρ2), where ρ = 0.1 +(we can change this parameter for different synthetic data). +Then, we consider four settings of the covariance matrices E and T to generate the simu- +lated datasets. These are case 1: T = Ipp and E = Iqq. case 2: T = 0.6|j−k| and E = Iqq. case +3: T = Ipp and E = 0.6|j−k|. case 4: T = 0.6|j−k| and E = 0.6|j−k|. Finally, based on (5), +we generate Θ = (E + BT TB)−1. For each case, we average our results over 15 synthetic +datasets. +The comparison of results on a single synthetic dataset with n = 120, p = 60, q = 60 can +be seen in Figure 1, 2. The detailed description is as follows: we identify 20 groups, and each +group has 3 genes. In each group, y1 is related to x1,x2,x3; y4 is related to x4,x5,x6 and so +on. Let E = Iqq and T ̸= Ipp. +The real B and Θ are both given in the upper left corner of Figure 1, 2, the right side of +Figure 1, 2 show the results of GFLasso and MRCE and the bottom row shows the results of +ICLasso and l1-2-GLasso. All models can find the structure of B, but GFLasso’s estimates +are subject to significant error. By comparing with other models, it can be found that the +regression coefficient matrix calculated by l1-2-GLasso is closer to the real B. +(a) True +(b) GFLasso +(c) MRCE +(d) ICLasso +(e) l1-2-GLasso +FIG 1. The comparison of B with different models on single synthetic dataset (p = 60 and q = 60). The real B is +given in the upper left corner of the figure. + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +9 +(a) True +(b) GFLasso +(c) MRCE +(d) ICLasso +(e) l1-2-GLasso +FIG 2. The comparison of Θ with different models on single synthetic dataset (p = 60 and q = 60). The real Θ is +given in the upper left corner of the figure. +As can be seen from the Figure 2, only l1-2-GLasso and ICLasso can recover Θ more +accurately. The main results of our synthetic experiments are shown in Figure 3, 4, 5. We +evaluate our approach according to three metrics: (1) F1 score on B (2) F1 score on Θ (3) +Regression error on Y . It can be seen that l1-2-GLasso is closer to the real data in terms of +sparsity. Comparing the size of different input dimensions, l1-2-GLasso is also superior to +other models. +60 +120 +480 +720 +The numbers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score on B +S mulat on case 1 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(a) case 1: T = Ipp and E = Iqq +60 +120 +480 +720 +The numbers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score on B +S mulat on case 2 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(b) case 2: T = 0.6|j−k| and E = Iqq +60 +120 +480 +720 +The numbers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score on B +Simulation case 3 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(c) case 3: T = Ipp and E = 0.6|j−k| +60 +120 +480 +720 +The numbers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score on B +S mulat on case 4 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(d) case 4: T = 0.6|j−k| and E = 0.6|j−k| +FIG 3. The comparison of B on 15 synthetic datasets generated with different types of covariance structures (T , +E) and with different numbers of SNPs p. + +10 +60 +120 +480 +720 +The umbers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score o Θ +Simulatio case 1 +l +1 +− +2 +-GLasso +ICLasso +(a) case 1: T = Ipp and E = Iqq +60 +120 +480 +720 +The umbers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score o Θ +Simulatio case 2 +l +1 +− +2 +-GLasso +ICLasso +(b) case 2: T = 0.6|j−k| and E = Iqq +60 +120 +480 +720 +The nu bers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score on Θ +Si ulation case 3 +l +1 +− +2 +-GLasso +ICLasso +(c) case 3: T = Ipp and E = 0.6|j−k| +60 +120 +480 +720 +The umbers of SNPs p +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 score o Θ +Simulatio case 4 +l +1 +− +2 +-GLasso +ICLasso +(d) case 4: T = 0.6|j−k| and E = 0.6|j−k| +FIG 4. The comparison of Θ on 15 synthetic datasets generated with different types of covariance structures (T , +E) and with different numbers of SNPs p. +60 +120 +480 +720 +The numbers of SNPs p +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Regress on error +S mulat on case 1 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(a) case 1: T = Ipp and E = Iqq +60 +120 +480 +720 +The numbers of SNPs p +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Regress on error +S mulat on case 2 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(b) case 2: T = 0.6|j−k| and E = Iqq +60 +120 +480 +720 +The numbers of SNPs p +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Regress on error +S mulat on case 3 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(c) case 3: T = Ipp and E = 0.6|j−k| +60 +120 +480 +720 +The numbers of SNPs p +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Regress on error +S mulat on case 4 +l +1 +− +2 +-GLasso +ICLasso +MRCE +GFLasso +(d) case 4: T = 0.6|j−k| and E = 0.6|j−k| +FIG 5. The comparison of Regression Error on 15 synthetic datasets generated with different types of covariance +structures (T , E) and with different numbers of SNPs p. +In Figure 3, 4, we present the F1 score in the recovery of B and Θ. To some extent, the +F1 score reflects the ability of each model to learn the regression coefficient matrix and the + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +11 +output structure. In Figure 5, we show the regression error of Y . It can be seen that in terms of +regression error, l1-2-GLasso can be a little better than ICLasso. As shown in these Figures, +our results clearly show that l1-2-GLasso outperforms other baselines in the four cases we +considered. +0 +3 +6 +10 +20 +50 +100 +The ratio λ +1 +/τ +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Regression error +The regression error with different ratio +l +1 +− +2 +-GLasso +FIG 6. The regression error with different ratio λ1 +τ +Finally, because l1-2-GLasso introduces a new hyperparameter τ, we also explore the in- +fluence of different hyperparameters on the effect of our model. We use the dataset shown in +Figure 1, and the regression error with different ratios λ1 +τ is presented in Figure 6. As we can +see, when λ1 +τ < 3, the regression error is minimized. A larger ratio leads to a more satisfying +solution. When λ1 +τ ≥ 3, the regression error becomes more and more stable, so we finally +select λ1 +τ = 10 to conduct the following experiments. +5. Real dataset From eQTL Mapping. +Greenlaw et al. (2017) analyze a dataset ob- +tained from the ADNI-1 database. We compare l1-2-GLasso with other models on this dataset. +The genes used in our analysis are listed in Table 1. + +12 +TABLE 1 +The Gene ID in the ADNI-1 database +Gene ID +Measurement +Region of interest +Left_AmygVol +Volume +Amygdala +Left_CerebCtx +Volume +Cerebral cortex +Left_CerebWM +Volume +Cerebral white matter +Left_HippVol +Volume +Hippocampus +Left_InfLatVent +Volume +Inferior lateral ventricle +Left_LatVent +Volume +Lateral ventricle +Left_EntCtx +Thickness +Entorhinal cortex +Left_Fusiform +Thickness +Fusiform gyrus +Left_InfParietal +Thickness +Inferior parietal gyrus +Left_InfTemporal +Thickness +Inferior temporal gyrus +Left_MidTemporal +Thickness +Middle temporal gyrus +Left_Parahipp +Thickness +Parahippocampal gyrus +Left_PostCing +Thickness +Posterior cingulate +Left_Postcentral +Thickness +Postcentral gyrus +Left_Precentral +Thickness +Precentral gyurs +The dataset is available for n = 632 subjects, and among all possible SNPs, we include +only those SNPs belonging to the top 15 candidate genes listed on the AlzGene database. The +dataset presented here is queried from the most recent genome build as of December 2014, +from the ADNI-1 database. After quality control and imputation steps, the genetic dataset +used for this study includes p = 486 SNPs from q = 15 genes. +We apply l1-2-GLasso and other models to discover SNPs that influence the expression +levels of genes. This type of study is widely known as an expression quantitative trait locus +(eQTL) mapping in the genetics community. It is generally believed that when a genetic vari- +ation in the genome such as an SNP perturbs the expression of a gene, the effect propagates +through the gene network to influence the expressions of genes in downstream of the pathway. +First, we estimate the regression coefficient matrix using 480 samples and then compute the +regression error on the remaining 152 samples. As shown in Figure 7, l1-2-GLasso produces +the smallest regression error. It should be noted that we randomly select 480 samples each +time and repeat the sample 10 times. We show the median (red) and the discrete distribution +of the regression error on these 10 different datasets. + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +13 +GFLasso +MRCE +ICLasso +l +1 +− +2 +-GLasso +4 +5 +6 +7 +8 +9 +Regression error +Regression errors on the ADNI-1 database +FIG 7. Each box shows the discrete distribution of the regression error on these 10 different datasets. The median +is shown by the red line. +In the original methodology of Wang et al. (2012), based on biological experiments, 24 +SNPs that are highly correlated with 15 genes have been detected. According to the results of +biological experiments, we also apply the proposed model to find related SNPs. l1-2-GLasso +finds 22 SNPs among the determined 24 SNPs and these 22 SNPs are highlighted in Table +5. In addition, we also compared the GFLasso, MRCE, and ICLasso on the dataset with our +model in Table 2, 3, 4. The SNPs in bold are selected by each model. It can be seen in Table +4, 5 that l1-2-GLasso and ICLasso can both estimate the structure of two larger subnetworks +in the gene networks. This has been demonstrated by biological experiments. They show that +two SNPs: rs405509 and rs10787010 stand out as being potentially associated with the largest +number of ROIs. In Table 6, we show the SNPs associated with genes identified by different +models. l1-2-GLasso is better than the other models. To show the superiority of l1-2-GLasso, +we can see from Table 2, 3, 4, GFLasso picks out 14 of the determined 24 SNPs , MRCE +picks out 16 of the determined 24 SNPs and ICLasso picks out 17 of the determined SNPs. +The experimental results are in line with our assumptions about l1-2-GLasso. Compared with +other models, l1-2-GLasso can more accurately determine the SNPs related to genes. +Back to the sparsity, we compare the regression coefficient matrix B calculated by l1-2- +GLasso with ICLasso. As we can see in Figure 8, the proposed model can get sparser solu- +tions. For example, we set our sights on SNP rs4311. In the real dataset, the SNP rs4311 was +detected to have an association with the gene InfParietal (L) (Wang et al., 2012). ICLasso +not only finds an association between the SNP rs4311 and the gene InfParietal (L) but also +detects that the SNP rs4311 is associated with gene AmygVol (L), CerebCtx (L), LatVent (L), +EntCtx (L), Fusiform (L). But in our model, l1-2-GLasso calculates the coefficients between +SNP rs4311 and gene AmygVol (L), CerebCtx (L), LatVent (L), EntCtx (L), Fusiform (L) +that are 0.019947, 0, 0, 0, 0. For genes that are not correlated with an SNP, most regression +coefficients calculated by l1-2-GLasso are 0. The result shows that our model achieves sparser +solutions compared with ICLasso. + +14 +(a) B calculated by ICLasso +(b) B calculated by l1-2-GLasso +FIG 8. We show the regression coefficient matrix calculated by ICLasso and l1-2-GLasso. l1-2-GLasso can get a +sparser and more accurate regression coefficient matrix B +. +TABLE 2 +The 24 SNPs identified in biological experiments contain 14 SNPs chosen using GFLasso +SNP +Number of Phenotype +Phenotype ID(Hemisphere) +rs4311 +1 +InfParietal (L) +rs405509 +8 +AmygVol (L), CerebWM (L), Fusiform (L), HippVol (L), +InfParietal (L), InfTemporal (L), MidTemporal (L), Postcentral (L) +rs666004 +1 +InfTemporal (L) +rs1433099 +1 +CerebCtx (L) +rs1473180 +4 +CerebCtx (L) ,EntCtx (L), Fusiform (L), PostCing (L) +rs1475345 +1 +Parahipp (L) +rs1568400 +1 +Precentral (L) +rs2149196 +1 +Postcentral (L) +rs2418811 +1 +CerebWM (L) +rs4935774 +1 +CerebWM (L) +rs6107516 +1 +MidTemporal (L) +rs6584307 +1 +Parahipp (L) +rs11191692 +1 +EntCtx (L) +rs12209631 +2 +CerebCtx (L), HippVol (L) +rs16924159 +1 +PostCing (L) +rs1023024 +1 +Precentral (L) +rs1269918 +3 +CerebCtx (L), CerebWM (L), InfLatVent (L) +rs2327389 +1 +AmygVol (L) +rs2418811 +1 +CerebWM (L) +rs2756271 +4 +EntCtx (L), HippVol (L), InfTemporal (L), Parahipp (L) +rs7219773 +1 +Precentral (L) +rs9314349 +1 +Parahipp (L) +rs10787010 +7 +AmygVol (L), EntCtx (L), Fusiform (L), HippVol (L), +InfLatVent (L), InfTemporal (L), Precentral (L) +rs10787011 +1 +EntCtx (L) +rs11601726 +2 +CerebWM (L), LatVent (L) + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +15 +TABLE 3 +The 24 SNPs identified in biological experiments contain 16 SNPs chosen using MRCE +SNP +Number of Phenotype +Phenotype ID(Hemisphere) +rs4311 +1 +InfParietal (L) +rs405509 +3 +AmygVol (L), CerebWM (L), HippVol (L) +rs666004 +1 +InfTemporal (L) +rs1433099 +1 +CerebCtx (L) +rs1473180 +4 +CerebCtx (L) ,EntCtx (L), Fusiform (L), PostCing (L) +rs1475345 +1 +Parahipp (L) +rs1568400 +1 +Precentral (L) +rs2149196 +1 +Postcentral (L) +rs2327389 +1 +AmygVol (L) +rs4935774 +1 +CerebWM (L) +rs6107516 +1 +MidTemporal (L) +rs6584307 +1 +Parahipp (L) +rs9314349 +1 +Parahipp (L) +rs10787010 +2 +InfLatVent (L), Precentral (L) +rs11191692 +1 +EntCtx (L) +rs12209631 +2 +CerebCtx (L), HippVol (L) +rs1023024 +1 +Precentral (L) +rs1269918 +3 +CerebCtx (L), CerebWM (L), InfLatVent (L) +rs2418811 +1 +CerebWM (L) +rs2756271 +4 +EntCtx (L), HippVol (L), InfTemporal (L), Parahipp (L) +rs7219773 +1 +Precentral (L) +rs10787011 +1 +EntCtx (L) +rs11601726 +2 +CerebWM (L), LatVent (L) +rs16924159 +1 +PostCing (L) + +16 +TABLE 4 +The 24 SNPs identified in biological experiments contain 17 SNPs chosen using ICLasso +SNP +Number of Phenotype +Phenotype ID(Hemisphere) +rs4311 +1 +InfParietal (L) +rs405509 +8 +AmygVol (L), CerebWM (L), Fusiform (L), HippVol (L), +InfParietal (L), InfTemporal (L), MidTemporal (L), Postcentral (L) +rs1433099 +1 +CerebCtx (L) +rs1473180 +4 +CerebCtx (L) ,EntCtx (L), Fusiform (L), PostCing (L) +rs1475345 +1 +Parahipp (L) +rs1568400 +1 +Precentral (L) +rs2149196 +1 +Postcentral (L) +rs2418811 +1 +CerebWM (L) +rs4935774 +1 +CerebWM (L) +rs6107516 +1 +MidTemporal (L) +rs6584307 +1 +Parahipp (L) +rs9314349 +1 +Parahipp (L) +rs10787010 +7 +AmygVol (L), EntCtx (L), Fusiform (L), HippVol (L), +InfLatVent (L), InfTemporal (L), Precentral (L) +rs10787011 +1 +EntCtx (L) +rs11191692 +1 +EntCtx (L) +rs12209631 +2 +CerebCtx (L), HippVol (L) +rs16924159 +1 +PostCing (L) +rs666004 +1 +InfTemporal (L) +rs1023024 +1 +Precentral (L) +rs1269918 +3 +CerebCtx (L), CerebWM (L), InfLatVent (L) +rs2327389 +1 +AmygVol (L) +rs2756271 +4 +EntCtx (L), HippVol (L), InfTemporal (L), Parahipp (L) +rs7219773 +1 +Precentral (L) +rs11601726 +2 +CerebWM (L), LatVent (L) + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +17 +TABLE 5 +The 24 SNPs identified in biological experiments contain 22 SNPs chosen using l1-2-GLasso +SNP +Number of phenotype +Phenotype ID(Hemisphere) +rs4311 +1 +InfParietal (L) +rs405509 +8 +AmygVol (L), CerebWM (L), Fusiform (L), HippVol (L), +InfParietal (L), InfTemporal (L), MidTemporal (L), Postcentral (L) +rs666004 +1 +InfTemporal (L) +rs1023024 +1 +Precentral (L) +rs1269918 +3 +CerebCtx (L), CerebWM (L), InfLatVent (L) +rs1433099 +1 +CerebCtx (L) +rs1473180 +4 +CerebCtx (L) ,EntCtx (L), Fusiform (L), PostCing (L) +rs1475345 +1 +Parahipp (L) +rs1568400 +1 +Precentral (L) +rs2149196 +1 +Postcentral (L) +rs2418811 +1 +CerebWM (L) +rs2756271 +4 +EntCtx (L), HippVol (L), InfTemporal (L), Parahipp (L) +rs4935774 +1 +CerebWM (L) +rs6107516 +1 +MidTemporal (L) +rs6584307 +1 +Parahipp (L) +rs9314349 +1 +Parahipp (L) +rs10787010 +7 +AmygVol (L), EntCtx (L), Fusiform (L), HippVol (L), +InfLatVent (L), InfTemporal (L), Precentral (L) +rs10787011 +1 +EntCtx (L) +rs11191692 +1 +EntCtx (L) +rs11601726 +2 +CerebWM (L), LatVent (L) +rs12209631 +2 +CerebCtx (L), HippVol (L) +rs16924159 +1 +PostCing (L) +rs2327389 +1 +AmygVol (L) +rs7219773 +1 +Precentral (L) + +18 +TABLE 6 +SNPs identified by different models +SNP +GFLasso +MRCE +ICLasso +l1-2-GLasso +rs4311 +✘ +✘ +✘ +✘ +rs405509 +✘ +✘ +✘ +✘ +rs666004 +✘ +✘ +rs1023024 +✘ +rs1269918 +✘ +rs1433099 +✘ +✘ +✘ +✘ +rs1473180 +✘ +✘ +✘ +✘ +rs1475345 +✘ +✘ +✘ +✘ +rs1568400 +✘ +✘ +✘ +✘ +rs2149196 +✘ +✘ +✘ +✘ +rs2327389 +✘ +rs2418811 +✘ +✘ +✘ +rs2756271 +✘ +rs4935774 +✘ +✘ +✘ +rs6107516 +✘ +✘ +✘ +✘ +rs6584307 +✘ +✘ +✘ +✘ +rs7219773 +rs9314349 +✘ +✘ +✘ +rs10787010 +✘ +✘ +✘ +rs10787011 +✘ +✘ +✘ +rs11191692 +✘ +✘ +✘ +✘ +rs11601726 +✘ +rs12209631 +✘ +✘ +✘ +✘ +rs16924159 +✘ +✘ +✘ +Count +14 +16 +17 +22 +6. Conclusion. +In this paper, we propose the Graphical Lasso based on difference of l1 +and l2 norms, which introduce a new penalty for the sparsity of B. Based on two important +assumptions, we jointly estimate the regression coefficient matrix B and the output structure +Θ. Similar to ICLasso, the optimization problem can be solved based on existing algorithms. +Through the synthetic dataset, we demonstrate that l1-2-GLasso outperforms other models +in the recovery of the eQTL associations and the gene network structure. The results on real +datasets also show the superiority of l1-2-GLasso and confirm the assumptions of the model. +Also, we can use other penalty norm for B and Θ, like l 1 +2 penalty which is closer to the l0 +penalty. Future work will seek higher efficiency of the solution algorithm and decomposi- +tion of large-scale problems. The application domain of this model can also be extended to +financial data. +Data availability. +Data analyzed in this article are available in the R-package "bgsmtr". +With the R console: 1.data(bgsmtr-example-data) 2.str(bgsmtr-example-data) 3.SNP <- +t(bgsmtr-example-data-SNP-data) 4.BM <- t(bgsmtr-example-data-Brain-Measures) +REFERENCES +BERTSIMAS, D. and VAN PARYS, B. (2020). Sparse high-dimensional regression: Exact scalable algorithms and +phase transitions. The Annals of Statistics 48 300–323. +CHUN, H., CHEN, M., LI, B. and ZHAO, H. (2013). Joint conditional Gaussian graphical models with multiple +sources of genomic data. Frontiers in Genetics 4 294. +ESSER, E., LOU, Y. and XIN, J. (2013). A method for finding structured sparse solutions to nonnegative least +squares problems with applications. SIAM Journal on Imaging Sciences 6 2010–2046. +FRIEDMAN, J., HASTIE, T. and TIBSHIRANI, R. (2008). Sparse inverse covariance estimation with the graphical +lasso. Biostatistics 9 432–441. + +THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS +19 +GARDNER, T. S. and FAITH, J. J. (2005). Reverse-engineering transcription control networks. Physics of Life +Reviews 2 65–88. +GREENLAW, K., SZEFER, E., GRAHAM, J., LESPERANCE, M., NATHOO, F. S. and INITIATIVE, A. D. N. +(2017). A Bayesian group sparse multi-task regression model for imaging genetics. Bioinformatics 33 2513– +2522. +HOYER, P. O. (2004). Non-negative matrix factorization with sparseness constraints. Journal of Machine Learn- +ing Research 5. +KIM, S. and XING, E. P. (2009). Statistical estimation of correlated genome associations to a quantitative trait +network. PLoS Genetics 5 e1000587. +KIM, S. and XING, E. P. (2012). Tree-guided group lasso for multi-response regression with structured sparsity, +with an application to eQTL mapping. The Annals of Applied Statistics 6 1095–1117. +LI, B., CHUN, H. and ZHAO, H. (2012). Sparse estimation of conditional graphical models with application to +gene networks. Journal of the American Statistical Association 107 152–167. +LI, H. and GUI, J. (2006). Gradient directed regularization for sparse Gaussian concentration graphs, with appli- +cations to inference of genetic networks. Biostatistics 7 302–317. +LIU, C., CHEN, Q., ZHOU, B. and LI, H. (2016). L1- and L2-Norm Joint Regularization Based Sparse Signal +Reconstruction Scheme. Mathematical Problems in Engineering 2016. +MARBACH, D., PRILL, R. J., SCHAFFTER, T., MATTIUSSI, C., FLOREANO, D. and STOLOVITZKY, G. (2010). +Revealing strengths and weaknesses of methods for gene network inference. Proceedings of the National +Academy of Sciences 107 6286–6291. +MARCHETTI-BOWICK, M., YU, Y., WU, W. and XING, E. P. (2019). A penalized regression model for the joint +estimation of eQTL associations and gene network structure. The Annals of Applied Statistics 13 248–270. +MEINSHAUSEN, N. and BÜHLMANN, P. (2006). High-dimensional graphs and variable selection with the lasso. +The Annals of Statistics 34 1436–1462. +MICHAELSON, J. J., ALBERTS, R., SCHUGHART, K. and BEYER, A. (2010). Data-driven assessment of eQTL +mapping methods. BMC Genomics 11 1–16. +PENG, J., WANG, P., ZHOU, N. and ZHU, J. (2009). Partial correlation estimation by joint sparse regression +models. Journal of the American Statistical Association 104 735–746. +ROCKMAN, M. V. and KRUGLYAK, L. (2006). Genetics of global gene expression. Nature Reviews Genetics 7 +862–872. +ROTHMAN, A. J., LEVINA, E. and ZHU, J. (2010). Sparse multivariate regression with covariance estimation. +Journal of Computational and Graphical Statistics 19 947–962. +SCHÄFER, J. and STRIMMER, K. (2005). A shrinkage approach to large-scale covariance matrix estimation and +implications for functional genomics. Statistical Applications in Genetics and Molecular Biology 4. +SEGAL, E., FRIEDMAN, N., KAMINSKI, N., REGEV, A. and KOLLER, D. (2005). From signatures to models: +understanding cancer using microarrays. Nature Genetics 37 S38–S45. +TIBSHIRANI, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: +Series B (Methodological) 58 267–288. +VINGA, S. (2021). Structured sparsity regularization for analyzing high-dimensional omics data. Briefings in +Bioinformatics 22 77–87. +WANG, H., NIE, F., HUANG, H., KIM, S., NHO, K., RISACHER, S. L., SAYKIN, A. J., SHEN, L. and INITIA- +TIVE, A. D. N. (2012). Identifying quantitative trait loci via group-sparse multitask regression and feature +selection: an imaging genetics study of the ADNI cohort. Bioinformatics 28 229–237. +WANG, L., ZHOU, H., WANG, Y., YU, B., ZHANG, Y., LIU, W. and CHEN, Y. (2019). Three-parameter prestack +seismic inversion based on L1−2 minimization. Geophysics 84 R753–R766. +YIN, P., ESSER, E. and XIN, J. (2014). Ratio and difference of l1 and l2 norms and sparse representation with +coherent dictionaries. Communications in Information and Systems 14 87–109. +YIN, J. and LI, H. (2011). A sparse conditional Gaussian graphical model for analysis of genetical genomics +data. The Annals of Applied Statistics 5 2630. +YIN, P., LOU, Y., HE, Q. and XIN, J. (2015). Minimization of l1−2 for compressed sensing. SIAM Journal on +Scientific Computing 37 A536–A563. +YU, Y.-L. (2013). Better approximation and faster algorithm using the proximal average. Advances in Neural +Information Processing Systems 26. +YUAN, M. and LIN, Y. (2006). Model selection and estimation in regression with grouped variables. Journal of +the Royal Statistical Society: Series B (Statistical Methodology) 68 49–67. + +GFLasso +MRCE +ICLasso +l +1 +− +l +2 +-ICFL +4 +5 +6 +7 +8 +9 +Prediction errors in the ADNI-1 dataset + +0 +3 +6 +10 +20 +50 +100 +The ratio λ +1 +/τ +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Regression error +The prediction error as a function of the ratio +Our model + +0 +3 +6 +10 +20 +50 +100 +The ratio λ +1 +/τ +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Regression error +The prediction error as a function of the ratio +Our model + +0 +3 +6 +10 +20 +50 +100 +The ratio λ +1 +/τ +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Regression error +The prediction error as a function of the ratio +Our model + + diff --git a/LtE0T4oBgHgl3EQfSgD7/content/tmp_files/load_file.txt b/LtE0T4oBgHgl3EQfSgD7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5b2b54fff84227255f7881ab1387de28c59d584 --- /dev/null +++ b/LtE0T4oBgHgl3EQfSgD7/content/tmp_files/load_file.txt @@ -0,0 +1,954 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf,len=953 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='02225v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='ML] 4 Jan 2023 Submitted to the Annals of Applied Statistics L1−2 GLASSO: L1−2 REGULARIZED MULTI-TASK GRAPHICAL LASSO FOR JOINT ESTIMATION OF EQTL MAPPING AND GENE NETWORK BY WEI MIAO1,a, LAN YAO2,b 1College of Mathematics, Hunan University, amiaow@hnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='cn 2College of Mathematics, Hunan University, byao@hnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='cn A critical problem in genetics is to discover how gene expression is reg- ulated within cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Two major tasks of regulatory association learning are : (i) identifying SNP-gene relationships, known as eQTL mapping, and (ii) determining gene-gene relationships, known as gene network estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To share information between these two tasks, we focus on the unified model for joint estimation of eQTL mapping and gene network, and propose a L1−2 regularized multi-task graphical lasso, named L1−2 GLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Numerical ex- periments on artificial datasets demonstrate the competitive performance of L1−2 GLasso on capturing the true sparse structure of eQTL mapping and gene network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' L1−2 GLasso is further applied to real dataset of ADNI-1 and experimental results show that L1−2 GLasso can obtain sparser and more accurate solutions than other commonly-used methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Developments in sequencing technology allow us to obtain more and more genomic data since the publication of the first human genome sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Computational techniques can help us to mine meaningful information from raw data and understand how gene expression is regulated in cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In general, these problems include identifying cancer gene co-expression (co-expression: simultaneous expression of two or more genes) mod- ules, determining SNP-gene relationships through eQTL (expression quantitative trait locus) mapping and determining gene-gene relationships by estimating gene network structure, etc (Rockman and Kruglyak, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Gardner and Faith, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Given a dataset containing single nucleotide polymorphisms (SNPs) and mRNA expression, the problem is to understand the SNP-gene and gene-gene relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For example, assuming SNPs x = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=',xp) and genes y = (y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=',yq), the SNP-gene relationships in eQTL mapping are determined by a regression coefficient matrix and the gene-gene relationships in gene network estimation are captured by output structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' There have been many types of research on eQTL mapping and gene network estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The traditional method of eQTL mapping is to determine whether there is an association between a gene and an SNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Later, multivariate models have been developed to determine relationships between multiple SNPs and a gene (Michaelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' More recently, several models have been proposed to determine relationships between multiple SNPs and multiple genes (Kim and Xing, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' As for gene network estimation, the traditional method is to construct a graph and connect two related genes with an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To be specific, many previous studies inferred gene-gene re- lationships from gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For example, in Gaussian Graphical Models (GGM) framework, graphical models use graphs to represent dependencies between random vari- ables (Schäfer and Strimmer, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Segal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Li and Gui, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In GGM, multivariate vectors follow a multivariate normal distribution and have a specific structure of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The inverse of the covariance matrix is called the con- centration matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' GGM assumes that the expression variation pattern of a given gene can Keywords and phrases: Multivariate regression, structured sparsity, difference of l1 and l2 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 1 2 be predicted by a small subset of other genes (Meinshausen and Bühlmann, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The as- sumption leads to the sparsity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', multiple zeros) in the concentration matrix and reduces the problem to a well-known neighborhood selection or covariance selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In the concentration map modeling framework, the key idea is to use a partial correlation as a measure of the independence of any two genes, thereby directly distinguishing between direct and indirect interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In other approaches, Bayesian Networks are also utilized to establish the structure between genes (Marbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The Multi-task regression model can be used to jointly estimate the regression coefficient matrix and the output structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' One challenge to be faced is the high-dimensional data disas- ter which is very common in genetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In previous studies, sparse learning is a good way to deal with this problem and has attracted wide attention due to its advantages of sparse solu- tions, strong interpretability, and convenient computation (Bertsimas and Van Parys, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Furthermore, to enhance the expression ability, researchers have proposed various structured sparse models which combine sparse learning with structured regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In various fields of computing and engineering, it is an important research topic to construct a structured sparse model based on the prior assumption of sparsity and the specific structural character- istics of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Many models which are based on structured sparsity regularization have been reviewed (Vinga, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Group Lasso, which encourages related exit groups to have nonzero coef- ficients for the same subset of inputs, has been studied extensively (Yuan and Lin, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A computationally efficient way was provided to perform Lasso-regularized estimation of sparse concentration matrices (Friedman, Hastie and Tibshirani, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Graph-Guide Fused Lasso encourages pairs of outputs linked in a graph to have similar coefficient values (Kim and Xing, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Conditional Gaussian Graphical Models (CGGM) have been devel- oped to estimate both the output structure and the regression coefficients with structured sparsity at the same time (Yin and Li, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Li, Chun and Zhao, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Chun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' However, these models all require us to have prior knowledge of relationships between the output y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Another class of models, which focuses on estimating the conditional covariance of y|x rather than the covariance structure of the output y, has been developed to learn both the regression coefficient matrix and the output structure (Rothman, Levina and Zhu, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Un- der the influence of noisy data, these models may not end up with the true structure between outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Recently, a novel approach called Inverse-Covariance-Fused Lasso (ICLasso) which focuses on the covariance structure of the output y, can also jointly estimate the regression coefficient matrix and the output structure (Marchetti-Bowick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The structured sparsity regularization penalty is formed by the l1 norm in ICLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In addition, many other regularization penalties have also been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The projection operators that can enforce both l1 and l2 norms have been developed for encouraging sparsity in structured sparse models (Hoyer, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' One of the regularization penalties that has been studied a lot is the difference of l1 and l2 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The penalty is considered robust and can help select sparse solutions (Yin, Esser and Xin, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It has been used in nonnegative least squares (NNLS) and orthogonal matching pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The comparisons with l1 minimization for imaging data can be found in (Esser, Lou and Xin, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Some researchers have also applied the difference of l1 and l2 norms in sparse signal reconstruction problems to approximate the original l0-norm-based sparseness (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It can be seen in other areas such as compressed sensing, seismic inversions, etc (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Motivated by these studies, we propose a new model based on difference of l1 and l2 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Our model makes some important new contributions: (i) We introduce a new regular- ization penalty into the model inducing a better approximation and use a faster algorithm to solve the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (ii) Under the same parameter setting, the solved regression coefficient matrix is sparser compared with existing methods such as MRCE, ICLasso, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (iii) Our model outperforms other baseline methods in the recovery of the output structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 3 In Section 2, we give an introduction to several baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In Section 3, we describe our new model with a different penalty and the optimization algorithm in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In sections 4 and 5, we evaluate the effectiveness of our method on simulated and real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Finally, we summarize the article in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We assume that X ∈ Rn×p is a matrix of SNP genotypes and Y ∈ Rn×q is a matrix of gene expression values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Here, n represents the number of samples, q represents the number of genes, and p represents the number of SNPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We show the exact matrix form as: (1) \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 y11 y12 ··· y1q y21 y22 ··· y2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' yn1 yn2 ··· ynq \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb � �� � Y = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 x11 x12 ··· x1p x21 x22 ··· x2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' xn1 xn2 ··· xnp \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb � �� � X × \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 β11 β12 ··· β1q β21 β22 ··· β2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' βp1 βp2 ··· βpq \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb � �� � B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Multi-Task Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Multi-Task Lasso can be used for statistical tests to detect SNPs that are associated with genes (Tibshirani, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Given X and Y , the multivariate linear regression model is given by (2) yk = Xβk + ǫk,k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=',q, where βk = [β1k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=',βpk]T represents regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It can be used to detect SNPs that are significantly associated with genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We assume ǫk ∼ N(0,σ2) and the mathematical expression in matrix form is: (3) min B 1 n∥Y − XB∥2 F + λ∥B∥1, where B ∈ Rp×q represents the regression coefficient matrix, λ is the regularization parame- ter, which is used to control the degree of sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Multivariate regression with covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Multivariate regression with covariance estimation (MRCE) is a method that can jointly estimate the regression coeffi- cient matrix and the output structure (Rothman, Levina and Zhu, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It assumes that X has the linear relationship with Y : Y = XB + E, in which E ∼ N � 0,Ω−1� is a Gaussian noise matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We can calculate that Y | X ∼ N � XB,Ω−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MRCE can be expressed as follows: (4) min B,Ω 1 n tr � (Y − XB)T (Y − XB)Ω � − log det(Ω) + λ1∥B∥1 + λ2∥Ω∥1, where Ω represents the conditional inverse covariance of Y |X rather than the inverse covari- ance of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Based on previous assumptions, Ω is related to the Gaussian noise matrix E and it can not capture the exact relationship between the regression coefficient matrix B and the output structure in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Inverse-Covariance-Fused Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Marchetti-Bowick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2019) introduced a new model called Inverse-Covariance-Fused Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The model can also jointly estimate regres- sion coefficients and the output structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Compared with previous studies, the method cap- tures the marginal inverse covariance of Y rather than the conditional inverse covariance of Y |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' ICLasso (Inverse-Covariance-Fused Lasso) begins with two core modeling assumptions: x ∼ N(0,T) y|x ∼ N(xT B,E), where T represents the covariance of x and E represents the conditional covariance of y|x, Θ represents the marginal inverse covariance of Y , which is different from Ω in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' With these assumptions, we can derive the marginal distribution of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Based on the fact that the marginal distribution p(y) follows the Gaussian distribution, then use the law of total expectation and the law of total variance to derive the mean and covariance of y, as follows: (5) Ey(y) = Ex � Ey|x(y | x) � = Ex � xT B) � = 0 Covy(y) = Ex � Covy|x(y | x) � + Covx � Ey|x(y | x) � = Ex (E) + Covx � xT B � = E + BT TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We can calculate the distribution of y : (6) y ∼ N � 0,Θ−1� , where Θ−1 = E + BTTB is the marginal covariance of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' This is a connection between the inverse covariance of y and the regression coefficient matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For simplicity, we assume T = τ 2Ip×p and E = ε2Iq×q and then Θ−1 ∝ BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Given i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' observations of SNPs x ∈ Rp and genes y ∈ Rq, in order to jointly estimate the regression coefficient matrix B ∈ Rp×q and the inverse covariance matrix Θ ∈ Rq×q, the inverse-covariance-fused lasso optimization problem can be written as: (7) min B,Θ 1 n∥Y − XB∥2 F + 1 n tr � Y T Y Θ � − log det(Θ) + λ1∥B∥1 + λ2∥Θ∥1 + γ � (k,m) |θkm| · ∥β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='k + sgn(θkm)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='m∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' This objective effectively boils down to a combination of problems: Multi-Task Lasso and Sparse Inverse Covariance Estimation including a graph-guided fusion penalty form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The role of the penalty � (k,m) |θkm| · ∥β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='k + sgn(θkm)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='m∥1 is to encourage structural information sharing between B and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Estimating Model Parameters with a Fusion Penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Previous studies provide us with an idea to estimate these parameters and also encourage information sharing between B and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To do this, the model is formulated as a convex optimization problem, whose objective function is: (8) lossy|x(B) + lossy(Θ) + penalty(B,Θ), THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 5 where we can see: lossy|x(B) can be derived from the negative log-likelihood of y|x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' lossy(Θ) can be derived from the negative marginal log-likelihood of y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' penalty(B,Θ) is a penalty term that encourages shared information between the regres- sion coefficient matrix B and the output structure Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The l1 norm penalty ∥B∥1 and ∥Θ∥1 induce sparsity in the estimates of B and Θ, which make the model feasible even on high dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To obtain a sparser solution, we can naturally generalize the l1 norm penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Based on this framework, we propose our model in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The l1−2 Graphical Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Since the sparsity of B is reflected by the number of its nonzero terms, it is equivalent to the so-called l0 norm penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' ICLasso replaces l0 norm penalty with l1 norm penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A reconstruction framework based on the difference l1 and l2 norms was proposed (Esser, Lou and Xin, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It can be reformulated as follows: (9) minx λ∥x∥1 − τ∥x∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' y = Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It was proved that as long as the selection of appropriate λ and τ, the solution can be close to the solution of the problem with l0 norm penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To get sparser solutions, we propose the l1−2 Graphical Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Our model can be described as: (10) lossy|x(B) + lossy(Θ) + penalty(B,Θ) + penalty(B) + penalty(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Specifically, the expression is as followed: (11) min B,Θ 1 n∥Y − XB∥2 F + 1 n tr � Y T Y Θ � − log det(Θ) + λ1||B||1 − τ||B||2,1 + λ2∥Θ∥1 + γ � (k,m) |θkm| · ∥β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='k + sgn(θkm)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='m∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We define: (12) g12(B) = 1 n||Y − XB||2 F + λ1||B||1 − τ||B||2,1 (13) h(Θ) = 1 ntr(Y T Y Θ) − log det(Θ) + λ2||Θ||1 (14) GFL(B,−Θ) = q � k=1 q � m=1 |θkm| · ∥β·k + sgn(θkm)β·m∥1 The term 1 n∥Y −XB∥2 F , 1 n tr � Y T Y Θ � −log det(Θ) and γ � (k,m) |θkm|·∥βk + sgn(θkm)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='m∥1 are derived from Equation (8) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We describe the role of each item in detail: lossy|x(B): 1 n∥Y − XB∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' According to y | x ∼ N � xT B,ε2I � , we can derive its ex- pression by Maximum Likelihood Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The role of this term is to encourage the re- gression coefficient matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' lossy(Θ): 1 n tr � Y T Y Θ � − log det(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' According to y ∼ N � 0,Θ−1� , we can derive its expression by Maximum Likelihood Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The role of this term is to encourage the inverse covariance Θ to reflect the correlations among the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 6 penalty(B,Θ): γ � (k,m) |θkm| · ∥β·k + sgn(θkm)β·m∥1 is a graph-guided fusion penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It encourages the regression coefficients of closely related outputs to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' When yk is partially positively correlated with ym and βjk ̸= βjm for any j, it imposes a penalty proportional to θkm, and when yk and ym have a negative partial correlation for any j, it imposes a penalty proportional to −θkm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' penalty(B): λ1||B||1 − τ||B||2,1 = λ1 � j,k |βjk| + τ2 �p j=1 ��q k=1 β2 jk is an l1−2 norm penalty over the regression coefficient matrix that induces sparsity in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Compared to l1 norm, the difference of l1 and l2 norms is closer to the l0 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' penalty(Θ): λ2∥Θ∥1 = λ2 � k,m |θkm| is an l1 norm penalty that induces sparsity in Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Relationship to ICLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' l1-2-GLasso also implicitly assumes two underlying mod- eling assumptions: x ∼ N(0,T) and y|x ∼ N(xT B,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The difference between l1-2-GLasso and ICLasso is that we use difference of l1 and l2 norms to get sparse solutions that are closer to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The experimental results are shown in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We find that l1-2- GLasso can obtain sparser solutions than other models while maintaining the regression error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Previous work on the problem (11) propose some off-the-shelf algo- rithms to solve the above optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Based on these studies, we use the alternating minimization strategy to solve the l1-2-GLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Pay attention to the GFL(B,−Θ) term, it is clear that this term is a bi-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Thus, upon defining (15) g12(B) = 1 n∥Y − XB∥2 F + λ1∥B∥1 − τ||B||2,1 h(Θ) = 1 n tr � Y T Y Θ � − log det(Θ) + λ2∥Θ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The original objective can be rewritten as (16) min B,Θ g12(B) + h(Θ) + GFL(B,−Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Compared with ICLasso, l1-2-GLasso can obtain sparser regression coefficient matrix by difference of l1 and l2 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The difficulty of solving the problem also lies in this penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It can be seen that the term GFL(B,−Θ) is bi-convex, so we can use an alternating minimiza- tion strategy developed by Marchetti-Bowick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2019) to solve the problem (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' First, we fix Θ, so the problem becomes: (17) fΘ(B) = g12(B) + GFL(B,−Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Although our model introduces a new regularization term λ1||B||1 − τ||B||2,1, it can be decomposed into the form of a differentiable function and a non-differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For the matrix B = [β1,β2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=',βp]T , βiT is the ith row in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' According to the definition: (18) ||B||2,1 = p � i=1 (βiT βi) 1 2 , we define the Σ and the derivative of ||B||2,1 as: (19) Σ := \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 ||β1||2 0 ··· 0 0 1 ||β2||2 ··· 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 0 0 ··· 1 ||βp||2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 7 (20) ∂||B||2,1 ∂B = (∂ �p i=1(βiT βi) 1 2 ∂βi )p×1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 ||β1||2 0 ··· 0 0 1 ||β2||2 ··· 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 0 0 ··· 1 ||βp||2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb B = ΣB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' When we use a small τ, the term ∥Y −XB∥2 F −τ||B||2,1 is considered convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' This prob- lem can be solved by the proximal-average proximal gradient descent (PA-PG) algorithm Yu (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Compared to the proximal gradient descent, PA-PG converges consistently faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It was proved that with a suitable stepsize, the subgradient method converges in at most O(1/ǫ) steps for any accuracy ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' First, the derivation of ∥Y − XB∥2 F − τ||B||2,1 is XT (XB − Y ) − τΣ, then we take a gradient step of the form B − ν(XT (XB − Y ) − τΣ) and some small step size ν are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Other optimization procedures can follow Marchetti-Bowick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Finally, this sub-problem can be written as (21) �βjk, �βjm = arg min βjk,βjm 1 2ν (βjk − zjk)2 + (βjm − zjm)2 + |βjk + sgn(θkm)βjm|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We can find a closed-form solution for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The solution to this sub-problem is relatively simple, and the convergence speed of the algorithm can also be found in Yu (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Then fix B, and the problem becomes: (22) min Θ h(Θ) + GFL(B,−Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' This problem can be solved by adapting the block coordinate descent (BCD) algorithm (Friedman, Hastie and Tibshirani, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Finally, this sub-problem can be written as: (23) min α 1 2αT �Hjα + uT α+ − lT α−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We solve each coordinate using the coordinate descent method and applying a variant of the soft threshold operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Simulation Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In this section, we compare different models on synthetic data with known values of B and Θ, so that we can directly measure how well the true parameter values are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Models include Graph-Guided Fused Lasso (GFLasso), Sparse Mul- tivariate Regression with Covariance Estimation (MRCE), Inverse-Covariance-Fused Lasso (ICLasso), and l1-2-GLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For each model, we select hyperparameter values by minimizing the error on a held-out validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We evaluate each model from two dimensions: (i) Recovery of sparse structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (ii) Re- gression error of B and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (i) Recovery of sparse structures: To evaluate how well each model can estimate the sparsity structure of B and Θ, we calculate the F1 score for recovering the elements of B and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To do this, we choose a threshold at which each element value is considered “zero” or “nonzero” and then we calculate the Precision (P) and Recall (R) for different synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (ii) Regression error of B and Θ: We examine the prediction error of each model when using ˆB to predict Y from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For this analysis, we use a block-structured network over the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The outputs are divided into non-overlapping groups, and each group forms a fully connected subgraph in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 8 Here we describe our procedure for generating synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' At a high level, we first fix the sparse structure of the underlying components of the model, then generate coefficient values, and lastly sample X and Y according to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Given the number of samples n, the number of genes q, and the number of SNPs p, firstly we determine the module size in the gene network and fix the number of SNPs s associated with each gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Next, we randomly assign each gene to a module and select the set of s SNPs associated with each module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For each module, we randomly assign a major gene in this module and for each SNP xj associated with the primary gene yk, we generate its association strengths according to βjk ∼ Uniform(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For the other genes ym in this module, we generate the association strengths according to βjm ∼ Uniform(βjk,ρ2), where ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='1 (we can change this parameter for different synthetic data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Then, we consider four settings of the covariance matrices E and T to generate the simu- lated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' These are case 1: T = Ipp and E = Iqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' case 2: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = Iqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' case 3: T = Ipp and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' case 4: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Finally, based on (5), we generate Θ = (E + BT TB)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For each case, we average our results over 15 synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The comparison of results on a single synthetic dataset with n = 120, p = 60, q = 60 can be seen in Figure 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The detailed description is as follows: we identify 20 groups, and each group has 3 genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In each group, y1 is related to x1,x2,x3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' y4 is related to x4,x5,x6 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Let E = Iqq and T ̸= Ipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The real B and Θ are both given in the upper left corner of Figure 1, 2, the right side of Figure 1, 2 show the results of GFLasso and MRCE and the bottom row shows the results of ICLasso and l1-2-GLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' All models can find the structure of B, but GFLasso’s estimates are subject to significant error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' By comparing with other models, it can be found that the regression coefficient matrix calculated by l1-2-GLasso is closer to the real B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (a) True (b) GFLasso (c) MRCE (d) ICLasso (e) l1-2-GLasso FIG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The comparison of B with different models on single synthetic dataset (p = 60 and q = 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The real B is given in the upper left corner of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 9 (a) True (b) GFLasso (c) MRCE (d) ICLasso (e) l1-2-GLasso FIG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The comparison of Θ with different models on single synthetic dataset (p = 60 and q = 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The real Θ is given in the upper left corner of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' As can be seen from the Figure 2, only l1-2-GLasso and ICLasso can recover Θ more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The main results of our synthetic experiments are shown in Figure 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We evaluate our approach according to three metrics: (1) F1 score on B (2) F1 score on Θ (3) Regression error on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It can be seen that l1-2-GLasso is closer to the real data in terms of sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Comparing the size of different input dimensions, l1-2-GLasso is also superior to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score on B S mulat on case 1 l 1 − 2 GLasso ICLasso MRCE GFLasso (a) case 1: T = Ipp and E = Iqq 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score on B S mulat on case 2 l 1 − 2 GLasso ICLasso MRCE GFLasso (b) case 2: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = Iqq 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score on B Simulation case 3 l 1 − 2 GLasso ICLasso MRCE GFLasso (c) case 3: T = Ipp and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score on B S mulat on case 4 l 1 − 2 GLasso ICLasso MRCE GFLasso (d) case 4: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| FIG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The comparison of B on 15 synthetic datasets generated with different types of covariance structures (T , E) and with different numbers of SNPs p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 10 60 120 480 720 The umbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score o Θ Simulatio case 1 l 1 − 2 GLasso ICLasso (a) case 1: T = Ipp and E = Iqq 60 120 480 720 The umbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score o Θ Simulatio case 2 l 1 − 2 GLasso ICLasso (b) case 2: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = Iqq 60 120 480 720 The nu bers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score on Θ Si ulation case 3 l 1 − 2 GLasso ICLasso (c) case 3: T = Ipp and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| 60 120 480 720 The umbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 F1 score o Θ Simulatio case 4 l 1 − 2 GLasso ICLasso (d) case 4: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| FIG 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The comparison of Θ on 15 synthetic datasets generated with different types of covariance structures (T , E) and with different numbers of SNPs p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 Regress on error S mulat on case 1 l 1 − 2 GLasso ICLasso MRCE GFLasso (a) case 1: T = Ipp and E = Iqq 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 Regress on error S mulat on case 2 l 1 − 2 GLasso ICLasso MRCE GFLasso (b) case 2: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = Iqq 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 Regress on error S mulat on case 3 l 1 − 2 GLasso ICLasso MRCE GFLasso (c) case 3: T = Ipp and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| 60 120 480 720 The numbers of SNPs p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='0 Regress on error S mulat on case 4 l 1 − 2 GLasso ICLasso MRCE GFLasso (d) case 4: T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6|j−k| FIG 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The comparison of Regression Error on 15 synthetic datasets generated with different types of covariance structures (T , E) and with different numbers of SNPs p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In Figure 3, 4, we present the F1 score in the recovery of B and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To some extent, the F1 score reflects the ability of each model to learn the regression coefficient matrix and the THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 11 output structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In Figure 5, we show the regression error of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It can be seen that in terms of regression error, l1-2-GLasso can be a little better than ICLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' As shown in these Figures, our results clearly show that l1-2-GLasso outperforms other baselines in the four cases we considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 0 3 6 10 20 50 100 The ratio λ 1 /τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='07 Regression error The regression error with different ratio l 1 − 2 GLasso FIG 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The regression error with different ratio λ1 τ Finally, because l1-2-GLasso introduces a new hyperparameter τ, we also explore the in- fluence of different hyperparameters on the effect of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We use the dataset shown in Figure 1, and the regression error with different ratios λ1 τ is presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' As we can see, when λ1 τ < 3, the regression error is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A larger ratio leads to a more satisfying solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' When λ1 τ ≥ 3, the regression error becomes more and more stable, so we finally select λ1 τ = 10 to conduct the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Real dataset From eQTL Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Greenlaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2017) analyze a dataset ob- tained from the ADNI-1 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We compare l1-2-GLasso with other models on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The genes used in our analysis are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='TABLE 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='The Gene ID in the ADNI-1 database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Gene ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Measurement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Region of interest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_AmygVol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Amygdala ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_CerebCtx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Cerebral cortex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_CerebWM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Cerebral white matter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_HippVol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Hippocampus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_InfLatVent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Inferior lateral ventricle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_LatVent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Lateral ventricle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_EntCtx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Entorhinal cortex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_Fusiform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Fusiform gyrus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_InfParietal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Inferior parietal gyrus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_InfTemporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Inferior temporal gyrus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_MidTemporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Middle temporal gyrus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_Parahipp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Parahippocampal gyrus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_PostCing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Posterior cingulate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_Postcentral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Postcentral gyrus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Left_Precentral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Precentral gyurs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='The dataset is available for n = 632 subjects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and among all possible SNPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' we include only those SNPs belonging to the top 15 candidate genes listed on the AlzGene database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The dataset presented here is queried from the most recent genome build as of December 2014, from the ADNI-1 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' After quality control and imputation steps, the genetic dataset used for this study includes p = 486 SNPs from q = 15 genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We apply l1-2-GLasso and other models to discover SNPs that influence the expression levels of genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' This type of study is widely known as an expression quantitative trait locus (eQTL) mapping in the genetics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It is generally believed that when a genetic vari- ation in the genome such as an SNP perturbs the expression of a gene, the effect propagates through the gene network to influence the expressions of genes in downstream of the pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' First, we estimate the regression coefficient matrix using 480 samples and then compute the regression error on the remaining 152 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' As shown in Figure 7, l1-2-GLasso produces the smallest regression error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It should be noted that we randomly select 480 samples each time and repeat the sample 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We show the median (red) and the discrete distribution of the regression error on these 10 different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 13 GFLasso MRCE ICLasso l 1 − 2 GLasso 4 5 6 7 8 9 Regression error Regression errors on the ADNI-1 database FIG 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Each box shows the discrete distribution of the regression error on these 10 different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The median is shown by the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In the original methodology of Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2012), based on biological experiments, 24 SNPs that are highly correlated with 15 genes have been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' According to the results of biological experiments, we also apply the proposed model to find related SNPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' l1-2-GLasso finds 22 SNPs among the determined 24 SNPs and these 22 SNPs are highlighted in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In addition, we also compared the GFLasso, MRCE, and ICLasso on the dataset with our model in Table 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The SNPs in bold are selected by each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' It can be seen in Table 4, 5 that l1-2-GLasso and ICLasso can both estimate the structure of two larger subnetworks in the gene networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' This has been demonstrated by biological experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' They show that two SNPs: rs405509 and rs10787010 stand out as being potentially associated with the largest number of ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In Table 6, we show the SNPs associated with genes identified by different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' l1-2-GLasso is better than the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' To show the superiority of l1-2-GLasso, we can see from Table 2, 3, 4, GFLasso picks out 14 of the determined 24 SNPs , MRCE picks out 16 of the determined 24 SNPs and ICLasso picks out 17 of the determined SNPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The experimental results are in line with our assumptions about l1-2-GLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Compared with other models, l1-2-GLasso can more accurately determine the SNPs related to genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Back to the sparsity, we compare the regression coefficient matrix B calculated by l1-2- GLasso with ICLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' As we can see in Figure 8, the proposed model can get sparser solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For example, we set our sights on SNP rs4311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In the real dataset, the SNP rs4311 was detected to have an association with the gene InfParietal (L) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' ICLasso not only finds an association between the SNP rs4311 and the gene InfParietal (L) but also detects that the SNP rs4311 is associated with gene AmygVol (L), CerebCtx (L), LatVent (L), EntCtx (L), Fusiform (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' But in our model, l1-2-GLasso calculates the coefficients between SNP rs4311 and gene AmygVol (L), CerebCtx (L), LatVent (L), EntCtx (L), Fusiform (L) that are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='019947, 0, 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' For genes that are not correlated with an SNP, most regression coefficients calculated by l1-2-GLasso are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The result shows that our model achieves sparser solutions compared with ICLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' 14 (a) B calculated by ICLasso (b) B calculated by l1-2-GLasso FIG 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' We show the regression coefficient matrix calculated by ICLasso and l1-2-GLasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' l1-2-GLasso can get a sparser and more accurate regression coefficient matrix B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' TABLE 2 The 24 SNPs identified in biological experiments contain 14 SNPs chosen using GFLasso SNP Number of Phenotype Phenotype ID(Hemisphere) rs4311 1 InfParietal (L) rs405509 8 AmygVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfParietal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MidTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Postcentral (L) rs666004 1 InfTemporal (L) rs1433099 1 CerebCtx (L) rs1473180 4 CerebCtx (L) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' PostCing (L) rs1475345 1 Parahipp (L) rs1568400 1 Precentral (L) rs2149196 1 Postcentral (L) rs2418811 1 CerebWM (L) rs4935774 1 CerebWM (L) rs6107516 1 MidTemporal (L) rs6584307 1 Parahipp (L) rs11191692 1 EntCtx (L) rs12209631 2 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L) rs16924159 1 PostCing (L) rs1023024 1 Precentral (L) rs1269918 3 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfLatVent (L) rs2327389 1 AmygVol (L) rs2418811 1 CerebWM (L) rs2756271 4 EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Parahipp (L) rs7219773 1 Precentral (L) rs9314349 1 Parahipp (L) rs10787010 7 AmygVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfLatVent (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Precentral (L) rs10787011 1 EntCtx (L) rs11601726 2 CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' LatVent (L) THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 15 TABLE 3 The 24 SNPs identified in biological experiments contain 16 SNPs chosen using MRCE SNP Number of Phenotype Phenotype ID(Hemisphere) rs4311 1 InfParietal (L) rs405509 3 AmygVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L) rs666004 1 InfTemporal (L) rs1433099 1 CerebCtx (L) rs1473180 4 CerebCtx (L) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' PostCing (L) rs1475345 1 Parahipp (L) rs1568400 1 Precentral (L) rs2149196 1 Postcentral (L) rs2327389 1 AmygVol (L) rs4935774 1 CerebWM (L) rs6107516 1 MidTemporal (L) rs6584307 1 Parahipp (L) rs9314349 1 Parahipp (L) rs10787010 2 InfLatVent (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Precentral (L) rs11191692 1 EntCtx (L) rs12209631 2 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L) rs1023024 1 Precentral (L) rs1269918 3 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfLatVent (L) rs2418811 1 CerebWM (L) rs2756271 4 EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Parahipp (L) rs7219773 1 Precentral (L) rs10787011 1 EntCtx (L) rs11601726 2 CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' LatVent (L) rs16924159 1 PostCing (L) 16 TABLE 4 The 24 SNPs identified in biological experiments contain 17 SNPs chosen using ICLasso SNP Number of Phenotype Phenotype ID(Hemisphere) rs4311 1 InfParietal (L) rs405509 8 AmygVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfParietal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MidTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Postcentral (L) rs1433099 1 CerebCtx (L) rs1473180 4 CerebCtx (L) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' PostCing (L) rs1475345 1 Parahipp (L) rs1568400 1 Precentral (L) rs2149196 1 Postcentral (L) rs2418811 1 CerebWM (L) rs4935774 1 CerebWM (L) rs6107516 1 MidTemporal (L) rs6584307 1 Parahipp (L) rs9314349 1 Parahipp (L) rs10787010 7 AmygVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfLatVent (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Precentral (L) rs10787011 1 EntCtx (L) rs11191692 1 EntCtx (L) rs12209631 2 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L) rs16924159 1 PostCing (L) rs666004 1 InfTemporal (L) rs1023024 1 Precentral (L) rs1269918 3 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfLatVent (L) rs2327389 1 AmygVol (L) rs2756271 4 EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Parahipp (L) rs7219773 1 Precentral (L) rs11601726 2 CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' LatVent (L) THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 17 TABLE 5 The 24 SNPs identified in biological experiments contain 22 SNPs chosen using l1-2-GLasso SNP Number of phenotype Phenotype ID(Hemisphere) rs4311 1 InfParietal (L) rs405509 8 AmygVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfParietal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MidTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Postcentral (L) rs666004 1 InfTemporal (L) rs1023024 1 Precentral (L) rs1269918 3 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfLatVent (L) rs1433099 1 CerebCtx (L) rs1473180 4 CerebCtx (L) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' PostCing (L) rs1475345 1 Parahipp (L) rs1568400 1 Precentral (L) rs2149196 1 Postcentral (L) rs2418811 1 CerebWM (L) rs2756271 4 EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Parahipp (L) rs4935774 1 CerebWM (L) rs6107516 1 MidTemporal (L) rs6584307 1 Parahipp (L) rs9314349 1 Parahipp (L) rs10787010 7 AmygVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' EntCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Fusiform (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfLatVent (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' InfTemporal (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Precentral (L) rs10787011 1 EntCtx (L) rs11191692 1 EntCtx (L) rs11601726 2 CerebWM (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' LatVent (L) rs12209631 2 CerebCtx (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HippVol (L) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='rs16924159 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='PostCing (L) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='rs2327389 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='AmygVol (L) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='rs7219773 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='Precentral (L) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='TABLE 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='SNPs identified by different models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='SNP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='GFLasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='MRCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='ICLasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='l1-2-GLasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='rs4311 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='✘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='✘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='✘ ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' In this paper, we propose the Graphical Lasso based on difference of l1 and l2 norms, which introduce a new penalty for the sparsity of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Based on two important assumptions, we jointly estimate the regression coefficient matrix B and the output structure Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Similar to ICLasso, the optimization problem can be solved based on existing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Through the synthetic dataset, we demonstrate that l1-2-GLasso outperforms other models in the recovery of the eQTL associations and the gene network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The results on real datasets also show the superiority of l1-2-GLasso and confirm the assumptions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Also, we can use other penalty norm for B and Θ, like l 1 2 penalty which is closer to the l0 penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Future work will seek higher efficiency of the solution algorithm and decomposi- tion of large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The application domain of this model can also be extended to financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Data analyzed in this article are available in the R-package "bgsmtr".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' With the R console: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='data(bgsmtr-example-data) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='str(bgsmtr-example-data) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='SNP <- t(bgsmtr-example-data-SNP-data) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='BM <- t(bgsmtr-example-data-Brain-Measures) REFERENCES BERTSIMAS, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and VAN PARYS, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Sparse high-dimensional regression: Exact scalable algorithms and phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The Annals of Statistics 48 300–323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' CHUN, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', CHEN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', LI, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and ZHAO, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Joint conditional Gaussian graphical models with multiple sources of genomic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Frontiers in Genetics 4 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' ESSER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', LOU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and XIN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A method for finding structured sparse solutions to nonnegative least squares problems with applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' SIAM Journal on Imaging Sciences 6 2010–2046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' FRIEDMAN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', HASTIE, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and TIBSHIRANI, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Sparse inverse covariance estimation with the graphical lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Biostatistics 9 432–441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' THE GRAPHICAL LASSO BASED ON DIFFERENCE OF L1 AND L2 NORMS 19 GARDNER, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and FAITH, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Reverse-engineering transcription control networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Physics of Life Reviews 2 65–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' GREENLAW, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', SZEFER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', GRAHAM, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', LESPERANCE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', NATHOO, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and INITIATIVE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A Bayesian group sparse multi-task regression model for imaging genetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Bioinformatics 33 2513– 2522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' HOYER, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Non-negative matrix factorization with sparseness constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Journal of Machine Learn- ing Research 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' KIM, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and XING, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Statistical estimation of correlated genome associations to a quantitative trait network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' PLoS Genetics 5 e1000587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' KIM, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and XING, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Tree-guided group lasso for multi-response regression with structured sparsity, with an application to eQTL mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The Annals of Applied Statistics 6 1095–1117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' LI, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', CHUN, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and ZHAO, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Sparse estimation of conditional graphical models with application to gene networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Journal of the American Statistical Association 107 152–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' LI, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and GUI, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Gradient directed regularization for sparse Gaussian concentration graphs, with appli- cations to inference of genetic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Biostatistics 7 302–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' LIU, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', CHEN, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', ZHOU, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and LI, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' L1- and L2-Norm Joint Regularization Based Sparse Signal Reconstruction Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Mathematical Problems in Engineering 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MARBACH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', PRILL, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', SCHAFFTER, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', MATTIUSSI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', FLOREANO, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and STOLOVITZKY, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Revealing strengths and weaknesses of methods for gene network inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 107 6286–6291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MARCHETTI-BOWICK, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', YU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', WU, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and XING, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A penalized regression model for the joint estimation of eQTL associations and gene network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The Annals of Applied Statistics 13 248–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MEINSHAUSEN, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and BÜHLMANN, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' High-dimensional graphs and variable selection with the lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The Annals of Statistics 34 1436–1462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' MICHAELSON, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', ALBERTS, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', SCHUGHART, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and BEYER, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Data-driven assessment of eQTL mapping methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' BMC Genomics 11 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' PENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', WANG, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', ZHOU, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and ZHU, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Partial correlation estimation by joint sparse regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Journal of the American Statistical Association 104 735–746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' ROCKMAN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and KRUGLYAK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Genetics of global gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Nature Reviews Genetics 7 862–872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' ROTHMAN, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', LEVINA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and ZHU, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Sparse multivariate regression with covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Journal of Computational and Graphical Statistics 19 947–962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' SCHÄFER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and STRIMMER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Statistical Applications in Genetics and Molecular Biology 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' SEGAL, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', FRIEDMAN, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', KAMINSKI, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', REGEV, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and KOLLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' From signatures to models: understanding cancer using microarrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Nature Genetics 37 S38–S45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' TIBSHIRANI, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Regression shrinkage and selection via the lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Methodological) 58 267–288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' VINGA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Structured sparsity regularization for analyzing high-dimensional omics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Briefings in Bioinformatics 22 77–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' WANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', NIE, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', HUANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', KIM, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', NHO, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', RISACHER, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', SAYKIN, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', SHEN, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and INITIA- TIVE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Identifying quantitative trait loci via group-sparse multitask regression and feature selection: an imaging genetics study of the ADNI cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Bioinformatics 28 229–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' WANG, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', ZHOU, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', WANG, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', YU, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', ZHANG, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', LIU, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and CHEN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Three-parameter prestack seismic inversion based on L1−2 minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Geophysics 84 R753–R766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' YIN, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', ESSER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and XIN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Ratio and difference of l1 and l2 norms and sparse representation with coherent dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Communications in Information and Systems 14 87–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' YIN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and LI, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' A sparse conditional Gaussian graphical model for analysis of genetical genomics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' The Annals of Applied Statistics 5 2630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' YIN, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', LOU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=', HE, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and XIN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Minimization of l1−2 for compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' SIAM Journal on Scientific Computing 37 A536–A563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' YU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Better approximation and faster algorithm using the proximal average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' YUAN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' and LIN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Model selection and estimation in regression with grouped variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology) 68 49–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content=' GFLasso MRCE ICLasso l 1 − l 2 ICFL 4 5 6 7 8 9 Prediction errors in the ADNI-1 dataset 0 3 6 10 20 50 100 The ratio λ 1 /τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE0T4oBgHgl3EQfSgD7/content/2301.02225v1.pdf'} +page_content='03 0.' metadata={'source': 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Calabrese +Center for Advanced Systems Understanding (CASUS) +Helmholtz-Zentrum Dresden Rossendorf (HZDR) +Görlitz, Germany +Department of Ecological Modelling +Helmholtz Centre for Environmental Research – UFZ, +Leipzig, Germany +Department of Biology, University of Maryland +College Park, MD, USA +j.calabrese@hzdr.de +January 10, 2023 +ABSTRACT +Efficient personnel scheduling plays a significant role in matching workload demand in organizations. +However, staff scheduling is sometimes affected by unexpected events, such as the COVID-19 +pandemic, that disrupt regular operations. Since infectious diseases like COVID-19 transmit mainly +through close contact with individuals, an efficient way to prevent the spread is by limiting the number +of on-site employees in the workplace along with regular testing. Thus, determining an optimal +scheduling and testing strategy that meets the organization’s goals and prevents the spread of the virus +is crucial during disease outbreaks. In this paper, we formulate these challenges in the framework +of two Mixed Integer Non-linear Programming (MINLP) models. The first model aims to derive +optimal staff occupancy and testing strategies to minimize the risk of infection among employees, +while the second model aims at only optimal staff occupancy under a random testing strategy. To +solve the problems expressed in the models, we propose a canonical genetic algorithm as well as +two commercial solvers. Using both real and synthetic contact networks of employees, our results +show that following the recommended occupancy and testing strategy reduces the risk of infection +25%–60% under different scenarios. +Keywords personnel scheduling · presence strategy · testing strategy · pandemic · COVID-19 +1 +Introduction +Personnel scheduling decisions are crucial in many organizations since labor cost constitutes one of the major expenses +in operations management. Thus, any improvement in staffing and scheduling decisions would result in overall +organizational benefits. Staffing and scheduling decisions can be subject to unexpected events that should be managed +∗Corresponding Author +arXiv:2301.03382v1 [math.OC] 6 Jan 2023 + +arXiv Template +A PREPRINT +proactively to ensure that performance measures are met. A recent global-scale phenomenon that considerably impacts +scheduling decisions is the COVID-19 pandemic. During a pandemic, it is necessary to consider a hybrid work strategy +to limit the number of employees present in the workplace to ensure employee safety. Another efficient strategy to +mitigate the impact of a pandemic is implementing testing, which organizations may offer to their employees. Due +to the limitations in the testing capacity and sensitivity, efficient applications of tests is necessary to prevent virus +outbreaks in the workplace. Therefore, it is crucial to derive efficient staff scheduling and testing strategies to guarantee +safety in the workplace while achieving the organization’s goals. +The personnel scheduling problem in pandemic situations is an emerging topic that has not been extensively addressed. +The question of defining a scheduling plan that accounts for testing strategies to reduce the risk of infection while +ensuring low levels of understaffing remains unanswered. In this paper, we aim to fill this gap by developing two Mixed +Integer Non-linear Programming (MINLP) models considering a probabilistic graph-based approach to determine the +optimal workplace occupancy that minimizes the risk of infection. The graph-based approach assumes that employees +are in close contact with each other, which contributes to the virus’s spread. The main objective is to minimize the +expected risk of infection while constraining workplace occupancy to comply with COVID-19 regulations. +Our models deal with two different realistic scenarios. The first model, considering the situation where employees +frequently underestimate the adherence to testing protocols, provides both optimal personnel scheduling at the workplace +and their testing strategies. On the other hand, the second model assumes a random testing strategy for the employees +and derives only the optimal presence scheduling. We propose two approaches to solve the non-linear models. The first +approach applies commercial optimization solvers; APOPT for the first model and Gurobi for the second model. To this +end, we linearize an equation (the equation for computing and updating the probability of infection) in the models. The +second approach is a canonical genetic algorithm that utilizes penalization to satisfy the constraints of the models. We +consider both real contact network data of employees and randomly generated sparse and dense graphs while assessing +the models’ performance under several scenarios. The results show significant impacts of both presence rate and testing +schedule in minimizing the risk of infection. +This paper is organized into six sections. After reviewing related and recent studies in Section 2, the problem of +finding optimal presence and testing strategies are formulated in two different models in Section 3. Section 4 presents a +heuristic algorithm for solving the models compared with the use of commercial non-linear solvers. Numerical results +for different scenarios are presented in Section 5. Finally, a conclusion is drawn in Section 6. +2 +Related work +Personnel scheduling is one of the critical decisions in organizations, however, it is impacted by both expected events +like demand or capacity uncertainty and by unexpected events like the COVID-19 pandemic. While the first kind +of events usually can be handled by considering labor flexibility strategies (e.g., multiskilled staff, flexible contracts, +collaborative teams) to minimize the mismatch between supply and demand (Porto et al., 2019), the second kind is +difficult to handle. These events certainly affect the performance of some organizations (e.g., service sector, retail, +healthcare, manufacturing), which must continue with regular operations despite the global health crisis. +An extensive literature on personnel scheduling problems exists (Bechtold et al., 1991; Brucker et al., 2011; Ernst et al., +2004a,b). According to the classification defined in Ernst et al. (2004a), we can categorize this study as disruption +management, in which the aim is to derive robust schedules by reducing the impact of the effects caused by a health +emergency, such as a pandemic. +The personnel scheduling problem in a pandemic situation is an emerging topic that largely started with the COVID-19 +pandemic (Jordan et al., 2020; Choi, 2021). In contrast to existing studies on disruption management (Mac-Vicar et al., +2017; Abdelghany et al., 2004, 2008; Shebalov and Klabjan, 2006; Moz and Pato, 2003) that focus mainly on developing +strategies to cope with staffing operational disruptions (e.g., demand variations, airline crew delays, nurse absenteeism), +this problem concerns employees’ health and safety, requiring additional considerations, such as the control of the +virus spread among the staff while satisfying staffing levels. The existing studies in the literature are focused on +developing scheduling policies to prevent the spread of the virus in organizations and closed spaces. For residential care +facilities, Moosavi et al. (2022) developed a task scheduling model to minimize the number of employees assigned +to residents to control the spread of the virus. To solve the model, the authors proposed a population-based heuristic +algorithm that guarantees solution quality against benchmark solution approaches. Güler and Geçici (2020), studied the +problem of scheduling physicians during the COVID-19 pandemic in a hospital in Turkey. The authors proposed a +Mixed Integer Programming (MIP) model to solve a shift scheduling problem to guarantee the safety of the physicians +while keeping a balanced workload in the hospital. +2 + +arXiv Template +A PREPRINT +During the COVID-19 pandemic, demand for hospital care often exceeds supply. Gao et al. (2022) studied a Medical +Staff Rebalancing (MSR) problem to allocate medical staff to different areas, considering the demand as the number of +infected patients in the allocation regions. To address the MSR problem, the authors proposed two robust optimization +models that account for uncertainty in data availability while ensuring allocation fairness during emergencies. Similarly, +Abadi et al. (2021) proposed a scheduling model to minimize the workload unbalance of the nurses in charge of +COVID-19 patients. To solve this problem, they developed a Hybrid Salp Swarm Algorithm and Genetic Algorithm +(HSSAGA) and showed their algorithm outperformed the state-of-the-art solution approaches. +For a pharmaceutical distribution warehouse in Italy, Zucchi et al. (2021) developed a Mixed Integer Linear Programming +(MILP) model to solve a shift scheduling problem. The aim was to minimize the deviation in allocated contractual +hours of employees during the COVID-19 pandemic to keep operations ongoing while guaranteeing the safety of the +employees. Guerriero and Guido (2022), proposed a flexible staff scheduling approach for a University administrative +department during the pandemic. By allowing a hybrid work system, they developed a days-off optimization model +considering employee preferences and availability. Alwadood et al. (2021) considered the personnel scheduling problem +for a hotel housekeeping department used as a quarantine center for foreign travelers. This study proposed a weekly +schedule for the staff using a Binary Integer Programming model that minimizes the workforce on duty to decrease the +risk of infection. +3 +Modeling a graph-based personnel scheduling problem during pandemic situations +Personnel scheduling during a pandemic requires special handling to ensure employee safety while continuing regular +operations. Since the virus that spreads COVID-19, the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS- +CoV-2), is transmitted through individual contacts, the World Health Organization (WHO COVID-19, 2020) suggested +several measures to control the spread of the virus, including social distancing, testing, and vaccination (Abdin et al., +2023). Thus, to limit the spread of the virus, the number of employees present in the workplace must be reduced, +creating a conflict between the size of the workforce and the risk of infection. On the one hand, having full workforce +occupancy leads to accomplishing staffing metrics (e.g., workload demand). However, the chance of a disease outbreak +will be high, compromising the safety of the employees. +This section proposes two MINLP models to solve the personnel scheduling problem during disease outbreaks, +particularly the COVID-19 pandemic. The models aim to minimize the risk of infection in organizations by considering +flexible allocation policies (i.e., teleworking and tests for the employees) and capacity constraints to impose social +distancing in the workplace. We aim to derive a presence strategy to find the optimal schedule of employees (i.e., +working remotely or at the workplace) and a testing strategy to determine the testing days of the employees. We consider +an organization with n employees who are in contact with each other and have to be allocated in a discrete-time horizon, +d = 1, 2, ..., D (i.e., a week, D = 5) such that the risk of infection in the workplace is minimized. The proposed models +compute the probability of infection for each employee under two different cases. The first model assumes that the +employees comply with the testing protocols following the suggested testing days. The second model does not impose +strict regulations on testing, so the employees perform tests arbitrarily during the evaluated time horizon. The notation +and assumptions considered in the proposed models are listed below. +Nomenclature +Parameters +n +Number of employees. +ei +i − th employee, indexed by i = 1, 2, ..., n. +pij +Probability of employee i and employee j have contact if both come to the workplace. +βi +probability of infection of employee i per contact, depending if vaccinated or not. +FN +Probability of false negativity of COVID-19 tests. +PId +i +Probability of infection of employee i in day d. +D +Scheduling Interval, i.e., d = 1, 2, ..., D days. +TCi +Test capacity for employee i (i.e., the maximum number of available tests for employee i in the scheduling +interval D days). +br +Background Risk; probability of infection in the neighborhood of the organization based on the 7-days incidence +reports in the county. +m +Number of task flow constraints in the organization. +m′ +Number of room capacity constraints in the organization. +Decision Variables +xd +i +Binary variable that indicates the allocation of employees; xd +i = 1 if employee i works in the workplace at day +d, otherwise xd +i = 0. +td +i +Binary variable that indicates the testing status; td +i = 1 if employee i is tested on day d, otherwise td +i = 0. +Assumptions: +3 + +arXiv Template +A PREPRINT +• The tests are performed in the morning before employees come to the workplace, and if the result is positive, +they stay at home. +• We initialize the probability of infections to background risk, i.e., PI0 +i = br, for i = 1, 2, ..., n. Indeed, we +assume for the starting day of the scheduling interval (i.e., Monday), the employees’ risk is the same as the +background risk in the organization’s neighborhood. +3.1 +Computing the probability of infection in a graph-based approach +In this subsection, we propose a graph-based approach to compute the probability of infection in the workplace. Let +PId +i be the probability of infection for ei at the end of the working day d, for i = 1, 2, ..., n and d = 0, 1, ..., D. +The objective function is to minimize the expected risk of infection inside the facilities, which is directly related to +minimizing the probability of infection of employees. Thus, +Minimize Z = +1 +D × n +D +� +d=1 +n +� +i=1 +PId +i . +(1) +To compute the probability of infection, PId +i , we present a recursive formula based on the presence and testing strategies +for the employees. As mentioned in the assumptions, PI0 +i as the initial probability of infection is set to br, which +means a risk of infection based on the number of incidences reported in the organization’s neighborhood. Now, we +iteratively compute PId +i by having PId−1 +i +. Considering the graph of connectivity among the employees, PId +i can be +computed based on PId−1 +i +and PId−1 +j +for all employees which may be in contact with employee i at day d − 1. If an +employee with a probability of infection p performs a test and the result is negative, the infection probability will reduce +to p × FN, where FN is the probability of false negativity for the tests. Now, let the binary variable td +i indicate ei is +tested on day d before coming to the workplace (i.e., td +i = 1) or not (i.e., td +i = 0). So, the updating recursive formula +can be written as +PI +′d +i += PId−1 +i +× (1 − td +i ) + PId−1 +i +× td +i × FN. +(2) +Equation (2) works well when the employees follow the testing strategy. If they do not follow this strategy, we can +apply a random testing strategy, that is, assuming a testing probability pr(test) for each employee. For example, if the +employees take two tests in five days, the Probability of testing per day is pr(test) = 2 +5. If ei performs a test at day d +with probability pr(test), then we can use the following equation for applying the effect of tests and computing PI +′d +i , +PI +′d +i += (1 − pr(test)) × PId−1 +i ++ pr(test) × PId−1 +i +× FN. +(3) +Therefore, regarding the fact that the employees follow a recommended testing strategy, Eq. (2), or a random test +strategy, Eq. (3), we apply the effect of performing the tests and update the probability of infection for all employees. +Then, we update the probabilities of infection for the employees based on their contacts. If ei comes to the workplace +on day d (i.e. xd +i = 1), and with the probability of pij contacts with ej (suppose xd +j = 1 as well), then the probability of +infection for him/her can be updated as +PId +i←j = 1 − [(1 − PI +′d +i ) × (1 − pij × βi × PI +′d +j )], +(4) +where βi is the probability of infection for ei in case that ej is infected. For the sake of simplicity, we only assume two +possible values for βi whether the employee is vaccinated or not. PId +i←j denotes the effect of contact with ej. Thus, by +applying all possible contacts ei may have during day d, the infection probability at the end of the day can be computed +as follows +PId +i = 1 − [(1 − PI +′d +i ) × +n +� +j=1&j̸=i +(1 − pij × βi × xd +j × PI +′d +j )]. +(5) +Thus, a two-step procedure is performed to update the probability of infection of the employees. First, we apply the +effect of testing (Equation (2) or (3)), and then apply the effect of contacts among the employees in the workplace, (5). +Note that, without loss of generality, we assume that working from home is free of risk, i.e., if the employees ei do not +4 + +arXiv Template +A PREPRINT +come to the workplace on the day d, PId +i = PId−1 +i +. For the sake of simplicity, let’s denote the two-step procedure by a +function based on the related parameters if the employees follow the recommended testing strategy. +PId = Update(PId−1, xd, td), +(6) +and, if they follow a random testing strategy +PId = Update(PId−1, xd, pr(test)), +(7) +where PId−1 = {PId−1 +1 +, PId−1 +2 +, . . . , PId−1 +n +}, xd = {xd−1 +1 +, xd−1 +2 +, . . . , xd−1 +n +}, and td = {td−1 +1 +, td−1 +2 +, . . . , td−1 +n +} are +the infection probability, presence indicator and testing indicators for all the employees in the day d, respectively. +3.2 +Personnel scheduling and testing strategy models +In this section, we define two MINLP models considering the probability of infection equations defined in Subsection +3.1. Model 1 assumes that the employees follow a recommended testing strategy, and Model 2 is based on the fact that +employees do not follow the testing protocols. Thus, they test themselves randomly with a probability of pr(test) for +each day. We also consider a set of constraints. We keep the model simple and easy to understand. In practice, different +organizations may have their specific limits and constraints, which can be added to the model. Here, we consider two +families of upper and lower bound constraints: the first family of constraints is related to satisfying the on-site tasks in +the organization. They are defined as, +n +� +i∈Ck +xd +i ≥ bk, for k = 1, 2, ..., m, +(8) +where Ck is the k − th subset of employees such that at least bk number of them have to present at the workplace on +day d. The second family of constraints refers to capacity limitations in the workplace. In fact, during the COVID-19 +pandemic, there have been regulations on the maximum number of employees who can be simultaneously (in a day) be +present in the workplace. So, we can model them as follows, +n +� +i∈C′ +k +xd +i ≤ b′ +k, for k = 1, 2, ..., m′, +(9) +Model 1 would result in a lower risk of infection compared to Model 2, but it requires the employees to follow the +recommended testing strategy. In contrast, Model 2 depicts a more flexible testing scheme, in which the employees +apply the offered tests randomly during the scheduling period. The models are defined as follows, +Model 1: Personnel scheduling with testing strategy +Minimize Z = +1 +D × n +D +� +d=1 +n +� +i=1 +PId +i , +Subject to : +PI0 = {β1br, β2br, . . . , βnbr} +PId = Update(PId−1, xd, td), +n +� +i∈Ck +xd +i ≥ bk, for k = 1, 2, ..., m, +n +� +i∈C′ +k +xd +i ≤ b′ +k, for k = 1, 2, ..., m′, +D +� +d=1 +td +i ≤ TCi, for i = 1, 2, ..., n, +xd +i , td +i ∈ {0, 1}, for i = 1, 2, ..., n, and d = 1, 2, ..., D +(10) +5 + +arXiv Template +A PREPRINT +TCi refers to the test capacity for ei; the maximum number of available test kits for ei in a period of D days. This +capacity may be the same for all employees or distributed among the employees regarding the number of connections +each employee has. +Model 1 has two decision variables, presence scheduling, xd +i , and testing schedule, td +i , for i = 1, 2, . . . , n and +d = 1, 2, . . . , D. So, the model will (optimally) derive which employees to allocate in the workplace and when, and on +which days to perform the tests. If in an organization, the employees do not follow the suggested testing strategy, and +they use the tests arbitrarily during the scheduling period, Model 1 will not fit with that organization. In this case, The +following model, which assumes the tests can be used by the employees with a probability, is a better match for that +organization. +Model 2: Personnel scheduling without testing strategy +Minimize Z = +1 +D × n +D +� +d=1 +n +� +i=1 +PId +i , +Subject to : +PI0 = {β1br, β2br, . . . , βnbr} +PId = Update(PId−1, xd, pr(test)), +n +� +i∈Ck +xd +i ≥ bk, for k = 1, 2, ..., m, +n +� +i∈C′ +k +xd +i ≤ b′ +k, for k = 1, 2, ..., m′, +xd +i ∈ {0, 1}, for i = 1, 2, ..., n, and d = 1, 2, ..., D +(11) +Both Model 1 and Model 2 are MINLP and, like the general scheduling problem (Hoong, 1996) with hard constraints, +are NP-hard. Furthermore, considering the upper bound and lower bound constraints of the problem, finding even a +feasible solution that satisfies the constraints is an intractable problem. The main difficulty of the models is updating +the risk of infection for the employees after daily contacts, Eq. 5. It is an exponential equation and impossible for most +algorithms to cope with. Therefore, in the following, we present an efficient simplification to handle this issue. +Relaxation +The term �n +j=1&j̸=i(1 − pij × βi × xd +j × PI +′d +j ) in Eq. 5 is the only exponential equation of the proposed models. As +explained, this term is used for updating the infection risk of an employee after his/her daily contacts. In practice, the +value of this term is so small (based on the data and experiments, that it is in order of 10−5. So, to relax the models +and remove this exponential term, we use the linear Taylor expansion of the formula, (1 − x)n ≈ 1 − nx. Thus, the +simplified approximation of Eq. 5 can be written as below, +PId +i = 1 − [(1 − PI +′d +i ) × (1 − Σn +j=1&j̸=i(pij × βi × xd +j × PI +′d +j ))], +(12) +To evaluate the accuracy of this simplification, we compared the above linear equation with Equation (5) using the +parameters reported in the example presented in subsection 5.1. The comparison showed that the equations result in +almost the same values with a precision on the order of 10−9 on average. Thus, it is a suitable linear approximation in +practice. +4 +Solution approach +To solve the proposed Model 1 and Model 2, we developed different solution approaches. For each model, we apply a +nonlinear commercial solver; GEKKO and APOPT (v1.0) (Beal et al., 2018) solver for the first model, and Gurobi +5.6.3 optimization solver (Gurobi Optimization, LLC, 2022) for the second model. To apply these solvers, we replace +the Equation (5) with linearization explained in Equation (12). In addition to applying these solvers, we propose a +Genetic Algorithm (GA) and tailored it for both Model 1 and Model 2. In the following, we briefly explain the GA and +its operators. +6 + +arXiv Template +A PREPRINT +Genetic algorithms are random search algorithms which work based on heuristic exploration and exploitation operations +Coello et al. (2007). It starts with a random set of solutions (chromosomes), called the population. Then, the GA +evolves the population generation by generation using some exploration and exploitation operators. To this end, the +selection operator chooses some high-fitness solutions as the parent chromosomes and put them in the mating pool. +Then, the Crossover operator takes a pair of such parents and produces (usually) two new chromosomes, i.e., the +children. Indeed, it combines part of (some genes) from the first parent with the other part of the second parent and vice +versa. The mutation operator mimics the natural mutation and changes some genes of a child solution at random to +explore a new search space and prevent the (premature) convergence of the population. Both of the operators play the +role of exploration. Finally, at the end of each iteration, from the combined parent and children populations, a set of +high-fitness chromosomes are picked for the next generations. +There are several kinds of crossover, mutation and selection operators Coello et al. (2007); Deb (2011). We present a +canonical GA with standard tournament selection operators, single-point crossover, and swap-mutation operators. Since +the decision variables are binary, we directly use them to represent a chromosome. Also, to satisfy the constraints of the +models, we penalize the infeasible chromosomes by adding a penalty value, that is, the number of violations of the +constraints that occurs. Finally, we define the fitness function as the sum of the objective function, the expected risk of +infection defined in Equation (1), and the penalty value. Note that the objective function is always a value between zero +and one. So, such a definition of the fitness function causes emphasizes feasible solutions in the search space at first and +then improving the solutions in terms of the risk value. So, the GA can also be applied to finding a feasible solution, +e.g., the first solution with a penalty value of zero found in the population. We utilize the proposed GA in two folds, +finding a random feasible solution that satisfies all the constraints, and a (feasible) suboptimal solution that minimizes +the expected risk of infection among the employees. We use such random solution(s) in comparing and showing the +impact of the presence and testing strategies. +We applied GA to solve the proposed Model 1 and Model 2. Its complexity depends on the size of its population and +the number of generations, and there is a trade-off between the complexity and the optimality of the obtained solutions. +On the other hand, it is straightforward to apply the GA on either Equation (5) or its relaxed linear equation, Equation +(12). For n employees and a period of D days, any chromosome can be evaluated in O(n × D × M) time, where M is +the size of all constraints in the problem. Therefore, the time complexity of a GA with population size p in g iterations +is O(p × g × n × D × M) time. Thus, all the parameters have a linear impact on the algorithm’s complexity. Finally, +GA is a random search, meaning multiple runs of it on the same instance of a problem may result in different solutions. +Despite GA, the time complexity of solvers Gurobi and APOPT in GEKKO depends on the number of employees and +the size of constraints, which have an exponential impact on the complexity of such solvers in the worst case. Therefore, +these solvers use different approaches to avoid the long-running time, such as solving the problem in the dual space and +applying predetermined optimality gaps, pre-solving, and branch and bound techniques. We applied APOPT to Model 1 +and Gurobi to Model 2 with the linear Equation (12). Although the running time of these solvers depends on the whole +of the parameters in the models, with a logical optimality gap size like 10−5, they are faster than GA. But the objective +value of obtained solutions by GA is better than the ones obtained by the solvers. In Subsection 5.4, we evaluate the +performance of the different solution approaches in terms of running time and solution optimality by comparing the +APOPT solver and GA for Model 1 and the Gurobi solver and GA for Model 2. +5 +Numerical results +In this section, we evaluate the proposed models and algorithms on several test problems. The aim is to find the +optimal presence and testing strategies that result in the minimum expected risk of infection of the employees. We +compare the models and the algorithms separately. Further, we analyze the impact and sensitivity of the models and +parameters. The experiments are composed of four parts. In the first part, we assume a small-size organization and +show the optimal presence and testing strategies. In the following part, we consider real data on employee contact +networks in organizations and random connectivity graphs. In the third part, the results for random connectivity graphs +are represented, and finally, the fourth part compares effectiveness of the algorithms. Since in the first three parts, the +aim is to compare the models and impact of the introduced parameters, we run the presented genetic algorithms 30 +times and report the average objective’s values. We consider the following values for the model’s parameters. +• We consider a time horizon of a week, e.g., five working days, D = 5 for running the experiments. +• We define the probability of disease transmission as β = 0.1. We choose this value as a probable pessimistic +case from a possible range of values reported in previous studies Lelieveld et al. (2020); Yang and Shaman +(2021); World Health Organization (2022); Lyngse et al. (2021) regarding the first variants of COVID-19 +(Alpha, Delta, and Omicron, which is more transmissible than the previous ones). +7 + +arXiv Template +A PREPRINT +• We calculate the background risk based on 7-day incidences of COVID-19 infections in Saxony, Germany, +in the period of June-August 2022. We consider 300 incidences on average for this period and set the daily +background risk to br = 1 +7 × +300 +100,000. +• The initial risk of infection, PI0, e.g., the risk of infection on Mondays, is determined by applying the +background risk and two days weekend. That is, PI0 = 1 − (1 − br)2. +• We apply the impact of vaccine (fully vaccinated) on the initial risk of infections and transmission probability +during the employee’s interaction. Based on the related recent researches (e.g., see Polack et al. (2020); Baden +et al. (2020); Emary et al. (2021); Fiolet et al. (2021)), we set the transmission rate of vaccinated employees to +(1 − 0.85)β, which implies 85% immunity for them. +5.1 +Base case study +In this subsection, we present the base case study in which we consider a small-size organization with 20 employees. +The organization is divided into two cross-functional sections: the first section includes 12 employees, i.e., e1, e2, ..., e12, +and the second one includes e13, e14, ..., e12. The connectivity graph that represents the employee’s contact network is +shown in Figure 1. We consider the following allocation rules on the presence of the employees: +(i) Each employee has to be present at the workplace at least 2 days a week, +(ii) The whole workplace occupation should remain between Min Occupation = 50% and Max Occupation = +75% of all the employees. That means at least 10 and at most 15 employees can be present at the workplace +daily. +(iii) At least 30% of employees in each section should be present at the workplace. That means at least 4 employees +from the first section and 3 from the second section should be present daily at the workplace. +In addition, we assume that the probability of false negativity of the tests is FN = 0.2, and two tests are available per +employee per week. That means, in the first model, the probability of testing per (working) day is pr(test) = 2 +5 = 0.4, +and in the second model, TC = 2. Table 1 and Table 2 show the obtained results for Model 10 and Model 11. As can +be extracted from the results, the presence strategy aims to satisfy the occupancy constraints by selecting employees +who have a weak connection rate with each other for onsite work. Further, the strategy satisfies the minimum 50% +occupancy with exactly 10 employees every day, except with a strategy of 11 employees on Thursdays in the second +model. Also, in the obtained results for the second model, the testing strategy suggests employees apply the tests (2 +tests are available per employee) on the first days of the week. This is because we initialize the risk of infection on +Mondays with a higher value after two days of weekend. So, the strategy tries to reduce this value by suggesting tests +before the employees are in contact with each other. +The expected risk of infection for the suggested strategies is 4.61e-5 for the first model and 1.99e-5 for the second +model. That means if the employees follow the suggested testing strategy, they can reduce the risk to 43% of the risk +when they follow a random testing strategy. +It may also be interesting to see what happens if the employees perform one test or three tests per week. To answer this +question, we run the models for TC = 1 and 3, and pr(test) = 0.2 and 0.6. We skip reporting the strategies and only +focus on the objective value. The risk of infection for TC = 1, and TC = 3, are 2.44e-5 and 1.85e-5, respectively, +and the risk of infection for the second model where the employees follow a random testing strategy with probabilities +pr(test) = 0.2 and pr(test) = 0.6 are 6.05e-5 and 3.85e-5, respectively. Having some high-level information and the +price of each test, a manager can optimally decide how many tests are better to offer to the employees based on the +testing strategy and the model that may fit the organization. +5.2 +Results on real data +We consider publicly available face-to-face interaction data collected by the SocioPatterns collaboration (Socio Patterns +Collaboration). This dataset contains the interactions among 92 employees recorded in 20-second intervals in an office +building in France from 24th June to 3rd July 2013 (Génois et al., 2015). Since the probability of contact for the +employees is not explicitly reported in the available data, we pre-process the data to adapt the dataset to our proposed +models. For employee i and employee j, we first aggregated the contacts between them over the above-mentioned period +and calculated the average number of contacts per day (cij). Next, for each employee i, we divided the total number of +contacts made by that employee per day by the number of colleagues he/she has and calculate the degree-normalized +average contacts (di) per day. Here, we defined employee i and employee j as colleagues if they have at least one +contact over this period. Finally, we calculated the probability of contacts between i and j as: +8 + +arXiv Template +A PREPRINT +Figure 1: Base case study contact network: An organization with 20 employees in two cross-functional sections. All the +employees except number 15 are fully vaccinated. The edge’s label shows the probability of occurring contact between +a pair of employees. +Table 1: Obtained presence strategy for the example illustrated in Figure 1. 1 indicates present at the workplace, and +0 means working from home. The results are shown for five working days for 20 employees. In this scenario, each +employee performs a test with a probability of 0.4 per day. This presence strategy results in the expected risk of +infection 4.61e-5. +Employee +Monday +Tuesday +Wednesday +Thursday +Friday +1 +1 +0 +1 +1 +0 +2 +1 +0 +1 +1 +0 +3 +0 +1 +0 +0 +1 +4 +1 +0 +0 +0 +1 +5 +0 +1 +0 +0 +1 +6 +0 +1 +0 +0 +1 +7 +0 +1 +1 +1 +0 +8 +1 +0 +0 +1 +1 +9 +1 +0 +1 +1 +0 +10 +0 +1 +1 +1 +0 +11 +1 +0 +1 +1 +0 +12 +0 +1 +0 +0 +1 +13 +1 +0 +0 +0 +1 +14 +0 +1 +1 +1 +0 +15 +0 +0 +1 +1 +0 +16 +1 +1 +1 +0 +0 +17 +1 +0 +1 +1 +0 +18 +0 +1 +0 +0 +1 +19 +0 +1 +0 +0 +1 +20 +1 +0 +0 +0 +1 +pij = +� +1, +if +cij +di ≥ 1 or +cij +dj ≥ 1 +max{ cij +di , cij +dj } +if +cij +di < 1 and +cij +dj < 1. +We assume 95% of employees are fully vaccinated, and the probability of false negativity for the tests is FN = 0.2. +Under such conditions, we run the models for different scenarios as follows: +(1) Each employee has to be present at the workplace at least (i) 2 days in a week, and (ii) 3 days in a week, +(2) The feasible minimum and maximum occupancy are set to (i) [30%, 70%] and (ii) [40%, 80%] +(3) The number of available tests per employee per week is TC = 1, 2, 3 for the first model and correspondingly +pr(test) = 0.2, 0.4, 0.6 for the second model. +Therefore, we will have 2 × 2 × 3 = 12 different scenarios. We run both the models under these scenarios and report +the results in Table 3. We scale the risk’s values by 105. Columns M1 and M2 show the expected risk of infection for +the first and the second model, respectively. Also, to compare the impact of the models, we compute random feasible +solutions (the solutions that satisfy all the constraints of the problem) for the presence and testing of the employees. +The column R in the table shows the expected risk of infection for such solutions. This value is the average risk of +infection for 30 different random solutions. As expected, the risk of infection improves from the random strategy to the +strategy in the second model, and also from the second model to the first model. On average, following the suggested +presence strategy can reduce the risk of infection 26% compared to the random strategy. And if the employees follow +9 + +0.6 +0.arXiv Template +A PREPRINT +Table 2: Obtained presence and testing strategies for the example illustrated in Figure 1. The left binary digit is a +presence indicator, present at the workplace (1), or works from home (0). The right binary digit shows he/she performs +a test (1) or not (0). Note that it is possible an employee stays at home but does a test. This presence and testing strategy +results in the expected risk of infection 1.99e-5. +Employee +Monday +Tuesday +Wednesday +Thursday +Friday +1 +0 ; 1 +1 ; 0 +1 ; 0 +1 ; 1 +0 ; 0 +2 +0 ; 1 +1 ; 0 +1 ; 0 +1 ; 1 +0 ; 0 +3 +0 ; 1 +1 ; 0 +1 ; 0 +1 ; 1 +0 ; 0 +4 +1 ; 1 +0 ; 1 +0 ; 0 +1 ; 0 +1 ; 0 +5 +1 ; 0 +0 ; 1 +0 ; 1 +0 ; 0 +1 ; 0 +6 +1 ; 0 +1 ; 1 +0 ; 0 +0 ; 0 +0 ; 1 +7 +1 ; 0 +0 ; 1 +0 ; 1 +1 ; 0 +1 ; 0 +8 +1 ; 1 +0 ; 1 +0 ; 0 +0 ; 0 +1 ; 0 +9 +0 ; 1 +1 ; 0 +0 ; 1 +1 ; 0 +1 ; 0 +10 +0 ; 1 +1 ; 0 +0 ; 1 +0 ; 0 +1 ; 0 +11 +0 ; 1 +1 ; 0 +0 ; 1 +1 ; 0 +1 ; 0 +12 +1 ; 1 +0 ; 1 +1 ; 0 +1 ; 0 +0 ; 0 +13 +0 ; 1 +1 ; 0 +1 ; 1 +0 ; 0 +0 ; 0 +14 +1 ; 0 +0 ; 1 +0 ; 1 +1 ; 0 +1 ; 0 +15 +0 ; 0 +0 ; 1 +1 ; 1 +1 ; 0 +0 ; 0 +16 +0 ; 1 +1 ; 0 +1 ; 1 +1 ; 0 +1 ; 0 +17 +1 ; 1 +0 ; 1 +1 ; 0 +0 ; 0 +0 ; 0 +18 +1 ; 0 +0 ; 1 +1 ; 1 +0 ; 0 +0 ; 0 +19 +1 ; 0 +0 ; 0 +0 ; 1 +0 ; 1 +1 ; 0 +20 +0 ; 1 +1 ; 1 +1 ; 0 +0 ; 0 +0 ; 0 +Figure 2: Graph representation of employees contact network with 92 employees. +both the suggested presence and testing strategies, the risk of infection can be reduced to 60% of the random strategy, +and 45% of the risk compared to the suggested strategy in the second model. +Thus, efficiently using the tests together with the presence strategy can significantly reduce the expected average risk +among the employees. Also, when the tests are used randomly by the employees, in columns R and M2, increasing the +test capacity, TC, has an almost linear impact on the risk of infection, while if the employees follow the suggested +testing strategy, M1, the infection risk considerably reduces when TC increases from one to two compared to when it +increases from two to three. So, the managers can offer an optimal number of free tests per week to the employees. +This can be optimally chosen regarding the price of test kits and the acceptable level of risk of infection. Note that +the background risk can be used as a reference point to determine such a level, and the minimum and the maximum +occupation are determined by considering workplace capacities and COVID-19 regulations, workfellow, and the type of +services the employees provide. +We assumed two different cases where the employees have to be present at least two days a week and where they +have to be present at least three days a week. As expected, the risk of infection for the first case is less than in the +second case. This constraint forces all employees to be present at the workplace, even those who are at higher risk +of infection (e.g., the unvaccinated employees and employees with a higher degree of connection). The other family +of constraints, which we applied to this real data, is two general constraints addressing the whole number of on-site +employees per day. We considered two scenarios [30%, 70%] and [40%, 80%], and the results are reported separately. +Despite the first family of constraints, this type of constraint allows flexibility in choosing low-risk employees to satisfy +10 + +arXiv Template +A PREPRINT +Table 3: Obtained results for the real data under 12 different scenarios. The reported risk values are scaled in 105.For +each scenario, three strategies are computed: (i) a feasible random strategy, denoted by column R. This solution satisfies +all the constraints and is random in terms of both presence and testing strategies. (ii) The solution for the second model, +denoted by M2, is for the case that the employees follow the suggested presence strategy with a random testing strategy. +(iii) The solution for the first model, denoted by M1, for the case that the employees follow suggested strategies for +both presence and testing. +min presence: 2 days +min presence: 3 days +Occupancy +TC +R +M2 +M1 +R +M2 +M1 +1 +5.81 +4.31 +2.38 +9.51 +7.97 +4.40 +[30% , 70%] +2 +4.47 +3.02 +1.47 +7.54 +6.00 +2.99 +3 +3.31 +1.96 +1.10 +5.77 +4.11 +2.27 +1 +6.68 +4.90 +2.81 +9.92 +7.88 +4.66 +[40% , 80%] +2 +5.12 +3.51 +1.93 +7.60 +5.97 +2.90 +3 +4.04 +2.57 +1.53 +6.20 +4.01 +2.32 +Table 4: Obtained results for sparse random graphs. The reported risk values are scaled in 105, and R, M2, M1 and +TC are the same as explained in the previous results. Here n is the number of nodes and FN is the probability of false +negativity for each test. +TC = 1 +TC = 2 +TC = 3 +n +FN +R +M2 +M1 +R +M2 +M1 +R +M2 +M1 +40 +0.3 +9.69 +7.36 +4.31 +7.22 +6.02 +3.17 +5.85 +4.62 +2.71 +40 +0.1 +8.19 +6.98 +2.85 +6.49 +5.22 +1.91 +5.19 +3.59 +1.6 +100 +0.3 +10.91 +9.13 +5.89 +8.85 +7.36 +4.43 +7.21 +5.58 +3.6 +100 +0.1 +10.08 +8.59 +4.4 +8.08 +6.35 +2.78 +6.27 +4.2 +2.25 +250 +0.3 +13.25 +11.53 +8.54 +10.49 +8.89 +6.22 +8.5 +6.56 +4.53 +250 +0.1 +12.59 +10.62 +6.8 +9.27 +7.68 +4.13 +7.05 +4.96 +2.81 +the minimum necessary number of on-site employees per day. The optimal solution (the solution which minimizes the +risk of infection) usually belongs to the boundary of these constraints. Therefore, the obtained strategy for [30%, 70%] +are better than for [40%, 80%]. +5.3 +Results on random graphs +In this part, we evaluate the models on randomly generated connectivity graphs. To this end, we first choose a set of +nodes (each node represent an employee), and then for any pair of nodes i and j assume a weight in pij ∈ [0, 1] as the +probability of a contact between them. We generated three small (with n = 40 nodes), medium (with n = 100 nodes) +and large (with n = 250 nodes) size graphs with two sparse and dense connections. In the sparse graphs, we set pij +to 1 with probability 0.05, and set it to 0.5 with probability 0.1, otherwise (with probability 0.85) we set it to 0. In +the dense graphs, pij = 1 with probability of 0.1, pij = 0.5 with probability of 0.2, and pij = 0 with probability of +0.7. We also assume three possible cases TC = 1, 2 and 3 for the number of available tests per employee per week, +as well as two scenarios FN = 0.1 and 0.3 as the probability of false negativity for each test. So, in general, there +are 3 × 3 × 2 = 18 scenarios for the size of graphs and the test’s parameters. We considered the constraints that each +employee should present at the workplace at least 3 days a week and the daily occupation in the workplace is allowed to +be in [50%, 75%]. Table 4 and Table 5 show the obtained risk of infection for 18 sparse graphs and 18 dense graphs, +respectively. The reported risk values are scaled in 105 and they are the average of 30 independent runs. +Table 5: Obtained results for dense random graphs. The reported risk values are scaled in 105, and R, M2, M1 and TC +are the same as explained in the previous results. Here n is the number of nodes and FN is the probability of false +negativity for each test. +TC = 1 +TC = 2 +TC = 3 +n +FN +R +M2 +M1 +R +M2 +M1 +R +M2 +M1 +40 +0.3 +13.42 +11.50 +6.38 +11.52 +9.27 +4.68 +9.28 +6.99 +3.89 +40 +0.1 +12.77 +10.86 +4.19 +9.76 +7.97 +2.74 +8.28 +5.26 +2.10 +100 +0.3 +16.39 +11.78 +7.30 +11.53 +9.27 +5.41 +9.41 +6.93 +4.33 +100 +0.1 +14.65 +11.00 +5.47 +10.08 +7.96 +3.49 +7.81 +5.24 +2.44 +250 +0.3 +24.17 +20.28 +14.24 +18.67 +14.93 +9.54 +13.77 +10.6 +6.78 +250 +0.1 +23.08 +18.55 +11.02 +17.16 +12.07 +5.38 +11.14 +8.31 +5.35 +11 + +arXiv Template +A PREPRINT +Table 6: Comparison results between APOPT solver and GA. The results are an average of 30 runs. The running times +are in second, and the objective values are scaled in factor 105. +running time +APOPT Solver +Genetic Algorithm +n +APOPT +GA +min +mean +max +min +mean +max +10 +1.95 +18.86 +2.15 +2.70 +3.31 +2.15 +2.69 +3.30 +20 +8.97 +60.93 +2.19 +2.48 +2.66 +2.23 +2.46 +2.75 +40 +28.66 +394 +2.31 +2.59 +2.52 +2.36 +2.51 +2.87 +100 +924 +2291 +2.53 +2.82 +3.87 +2.53 +2.72 +3.18 +The results in each table show the impact of the number of tests and their sensitivity, as well as following a random or +suggested presence testing strategy by the employees. For example, in sparse and small-size graphs (n = 40), when +the employees use only one test per week (TC = 1) and present at the workplace by a random strategy and apply the +tests randomly on a day of the week (column R) if the tests have a false negativity rate FN = 0.3, the results expected +average risk of infection is 9.69e-5, while for the case FN = 0.1, it results in 8.19e-5. These risk values can be reduced +to 7.36e-5 and 6.98e-5 when the employees follow the suggested presence strategy with a random test strategy (column +M2). Finally, following both suggested presence and testing strategies (column M1) results in 4.31e-5 and 2.85e-5, +respectively. The table shows the results for two (TC = 2) and three (TC = 3) tests per week. For example, two tests +with accepting suggested presence and testing strategies result in a risk of infection 3.17e-5 and 1.91e-5 for FN = 0.3 +and FN = 0.1, respectively. That means 56% and 70% improvements (risk reduction) comparing the case when they +do not follow the strategies. Thus, by comparing such results and the model which is more fit for the organization, an +optimal case can be chosen. The same results are reported for the dense graphs, and the managers may be interested in +observing the impact of the rate of contact among the employees on the expected risk of infection. In our experiments, +the probability of contact between any pair of employees in the dense graphs is two times more than in the sparse +graphs. However, the average risk of infection in columns R, M2 and M1 increased by factors 1.54, 1.49 and 1.41, +respectively. This kind of information can be efficiently used in establishing contact regulations in the workplace space. +In fact, in addition to the presence strategy and testing strategy, the rate of contact is an influential factor to control the +risk of infection. +5.4 +Comparison of solution algorithms +As explained, we presented a GA and applied it to both models. Further, we solve the problem defined in the first model, +Model 10, using GEKKO and APOPT solver, and the problem defined in the second model, Model 11, using Gurobi +solver. In this part, we compare these algorithms in terms of running time and efficiency in finding the optimal solution. +Note that, since the problem is an NP-hard problem, none of the algorithms can guarantee to find the optimal solution +in polynomial time. Thus, we compare the obtained solutions with the algorithms. +The models and algorithms were implemented in Python 3.7 on a standard PC (Intel (R) Core(TM) i7 and 32G RAM). +For a graph with a specific number of nodes, we run the algorithms for 30 different sets of edges and weights and report +the average running time (in seconds) of the algorithms as well as the best, mean, and worst of the objective value in the +obtained solutions. We consider graphs with the number of nodes n = 10, 20, 40, 100 with different sparse and dense +topologies and run the algorithms for a scheduling period D = 5. We run the GA with population size = 100 + 2 × n, +maximum generation = 200 + 2 × n, and mutation probability 1 +n. Table 6 shows the result for GA and GEKKO +with APOPT solver for Model 1, and Table 7 shows the result for GA and Gurobi for Model 2. +From Table 6, it is clear the APOPT solver reaches the solutions significantly quicker than GA. However, there is +always a trade-off between the running time of GA and the efficiency of the obtained solutions, APOPT dominates GA +in terms of running time. On the other hand, the GA slightly outperforms (on average about 2%) APOPT in terms of +optimality. In general, both approaches can be used for Model 1. From Table 7, it is obvious Gurobi solver is quite +faster than GA. On the other hand, the GA outperforms (on average about 4%) Gurobi in terms of optimality. We ran +Gurobi for very large graphs, e.g., with 1000 nodes, and observed it still is efficient and can compute the solution in less +than 10 seconds. In general, since the running time of GA is acceptable for real-size instances of the problem, it seems +that both approaches can be applied to Model 2. +6 +Conclusion and future work +During the COVID-19 pandemic, the most efficient strategy to prevent the spread of the virus was (and is) the +implementation of teleworking and regular testing of the employees. However, to date, there is not a clear understanding +12 + +arXiv Template +A PREPRINT +Table 7: Comparison results between Gurobi solver and GA. The results are an average of 30 runs. The running times +are in second, and the objective values are scaled in factor 105. +running time +Gurobi Solver +Genetic Algorithm +n +Gurobi +GA +min +mean +max +min +mean +max +10 +3.06 +14.92 +5.04 +5.83 +6.28 +5.04 +5.76 +5.92 +20 +12.98 +52.35 +5.08 +6.16 +6.54 +5.08 +5.68 +6.44 +40 +0.55 +322 +5.28 +5.62 +6.94 +5.17 +5.52 +6.82 +100 +0.34 +1882 +5.47 +7.37 +8.16 +5.42 +6.74 +7.95 +of how to define efficient personnel scheduling plans while considering testing capacities so as to guarantee the safety +of the employees that should keep working despite a pandemic. +This paper tackled such a situation by developing two MINLP models for deriving efficient scheduling plans during +a pandemic, taking into account teleworking strategies and testing capacities. The main objective is to minimize the +expected average risk of infection among the employees, considering work flow constraints and occupancy limitations +in the workplace to comply with the COVID-19 regulations. The first model focuses on scheduling the presence of +employees in the workplaces as well as scheduling their tests. However, since in practice the employee may not follow +a testing schedule, we presented a second model, which aims at optimizing the presence scheduling under a random +testing strategy. We compared the models under various scenarios and discovered that by implementing these strategies, +an organization can reduce the risk of infection by 25% to 60%. Further, we performed a sensitivity analysis by +tuning several influential parameters and showed several scenarios that can significantly help managers in establishing +regulations in the workplace. For instance, they can see the impact of connection weights, when it changes from dense +graphs to sparse graphs, or when different occupation constraints are considered. +The models we proposed assume that the number of available tests for all the employees is the same, while the tests +should be distributed according to the degree of connection for the employees and the number of days on which they +should be present at the workplace. The models can be easily extended to cover this case. However, by modeling a +variable test availability, the search space of the problem may increase, which may require more efficient solution +approaches such as branch and bound and hybrid methods. +Acknowledgements +This work was partially funded by the Where2Test project, which is financed by SMWK with tax funds on the basis of +the budget approved by the Saxon State Parliament. This work was also partially funded by the Center of Advanced +Systems Understanding (CASUS) which is financed by Germany’s Federal Ministry of Education and Research (BMBF) +and by the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis of the budget approved +by the Saxon State Parliament. +References +Abadi, M.Q.H., Rahmati, S., Sharifi, A., Ahmadi, M., 2021. Hssaga: designation and scheduling of nurses for taking +care of COVID-19 patients using novel method of hybrid salp swarm algorithm and genetic algorithm. Applied Soft +Computing 108, 107449. +Abdelghany, A., Ekollu, G., Narasimhan, R., Abdelghany, K., 2004. A proactive crew recovery decision support tool +for commercial airlines during irregular operations. Annals of Operations Research 127, 309–331. +Abdelghany, K.F., Abdelghany, A.F., Ekollu, G., 2008. An integrated decision support tool for airlines schedule +recovery during irregular operations. European Journal of Operational Research 185, 825–848. +Abdin, A.F., Fang, Y.P., Caunhye, A., Alem, D., Barros, A., Zio, E., 2023. An optimization model for planning testing +and control strategies to limit the spread of a pandemic–the case of covid-19. European journal of operational +research 304, 308–324. +Alwadood, Z., Noor, N.M., Mainor, N.A., 2021. An optimization model for hotel housekeeping personnel scheduling +in pandemic outbreak. Menemui Matematik (Discovering Mathematics) 43, 83–92. +Baden, L.R., El Sahly, H.M., Essink, B., Kotloff, K., Frey, S., Novak, R., Diemert, D., Spector, S.A., Rouphael, N., +Creech, C.B., et al., 2020. Efficacy and safety of the mrna-1273 sars-cov-2 vaccine. New England Journal of +Medicine . +13 + +arXiv Template +A PREPRINT +Beal, L., Hill, D., Martin, R., Hedengren, J., 2018. +Gekko optimization suite. +Processes 6, 106. +doi:doi:10.3390/pr6080106. +Bechtold, S.E., Brusco, M.J., Showalter, M.J., 1991. A comparative evaluation of labor tour scheduling methods. +Decision Sciences 22, 683–699. +Brucker, P., Qu, R., Burke, E., 2011. Personnel scheduling: Models and complexity. European Journal of Operational +Research 210, 467–473. +Choi, T.M., 2021. Fighting against COVID-19: what operations research can help and the sense-and-respond framework. +Annals of Operations Research , 1–17. +Coello, C.A.C., Lamont, G.B., Van Veldhuizen, D.A., et al., 2007. Evolutionary algorithms for solving multi-objective +problems. volume 5. Springer. +Deb, K., 2011. Multi-objective optimisation using evolutionary algorithms: an introduction, in: Multi-objective +evolutionary optimisation for product design and manufacturing. Springer, pp. 3–34. +Emary, K.R., Golubchik, T., Aley, P.K., Ariani, C.V., Angus, B., Bibi, S., Blane, B., Bonsall, D., Cicconi, P., Charlton, +S., et al., 2021. Efficacy of chadox1 ncov-19 (azd1222) vaccine against sars-cov-2 variant of concern 202012/01 (b. +1.1. 7): an exploratory analysis of a randomised controlled trial. The Lancet 397, 1351–1362. +Ernst, A.T., Jiang, H., Krishnamoorthy, M., Owens, B., Sier, D., 2004a. An annotated bibliography of personnel +scheduling and rostering. Annals of Operations Research 127, 21–144. +Ernst, A.T., Jiang, H., Krishnamoorthy, M., Sier, D., 2004b. Staff scheduling and rostering: A review of applications, +methods and models. European journal of operational research 153, 3–27. +Fiolet, T., Kherabi, Y., MacDonald, C.J., Ghosn, J., Peiffer-Smadja, N., 2021. Comparing COVID-19 vaccines for their +characteristics, efficacy and effectiveness against SARS-CoV-2 and variants of concern: A narrative review. Clinical +Microbiology and Infection . +Gao, X., Huang, G., Zhao, Q., Cao, C., Jiang, H., 2022. Robust optimization model for medical staff rebalancing +problem with data contamination during COVID-19 pandemic. International Journal of Production Research 60, +1737–1766. +Génois, M., Vestergaard, C.L., Fournet, J., Panisson, A., Bonmarin, I., Barrat, A., 2015. Data on face-to-face contacts in +an office building suggest a low-cost vaccination strategy based on community linkers. Network Science 3, 326–347. +Guerriero, F., Guido, R., 2022. Modeling a flexible staff scheduling problem in the era of COVID-19. Optimization +Letters 16, 1259–1279. +Güler, M.G., Geçici, E., 2020. A decision support system for scheduling the shifts of physicians during COVID-19 +pandemic. Computers & Industrial Engineering 150, 106874. +Gurobi Optimization, LLC, 2022. Gurobi Optimizer Reference Manual. URL: https://www.gurobi.com. +Hoong, C.L., 1996. On the complexity of manpower shift scheduling. Computers & Operations Research 23, 93–102. +Jordan, R.E., Adab, P., Cheng, K., 2020. COVID-19: risk factors for severe disease and death. +Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., Pöschl, +U., 2020. Model calculations of aerosol transmission and infection risk of COVID-19 in indoor environments. +International Journal of Environmental Research and Public Health 17, 8114. +Lyngse, F.P., Mortensen, L.H., Denwood, M.J., Christiansen, L.E., Møller, C.H., Skov, R.L., Spiess, K., Fomsgaard, +A., Lassauniere, R., Rasmussen, M., et al., 2021. SARS-CoV-2 Omicron VOC transmission in Danish households. +medRxiv . +Mac-Vicar, M., Ferrer, J.C., Muñoz, J.C., Henao, C.A., 2017. Real-time recovering strategies on personnel scheduling +in the retail industry. Computers & Industrial Engineering 113, 589–601. +Moosavi, A., Ozturk, O., Patrick, J., 2022. Staff scheduling for residential care under pandemic conditions: The case of +COVID-19. Omega 112, 102671. +Moz, M., Pato, M.V., 2003. An integer multicommodity flow model applied to the rerostering of nurse schedules. +Annals of Operations Research 119, 285–301. +World Health Organization, 2022. Enhancing response to Omicron SARS-CoV-2 variant: Technical brief and priority +actions for member states. Available from: https://www.who.int/publications/m/item/enhancing-readiness-for- +omicron-(b.1.1.529)-technical-brief-and-priority-actions-for-member-states. +Polack, F.P., Thomas, S.J., Kitchin, N., Absalon, J., Gurtman, A., Lockhart, S., Perez, J.L., Marc, G.P., Moreira, E.D., +Zerbini, C., et al., 2020. Safety and efficacy of the bnt162b2 mrna covid-19 vaccine. New England Journal of +Medicine . +14 + +arXiv Template +A PREPRINT +Porto, A.F., Henao, C.A., López-Ospina, H., González, E.R., 2019. Hybrid flexibility strategy on personnel scheduling: +Retail case study. Computers & Industrial Engineering 133, 220–230. +Shebalov, S., Klabjan, D., 2006. Robust airline crew pairing: Move-up crews. Transportation science 40, 300–312. +Socio Patterns Collaboration, . https://www.sociopatterns.org. [Retrieved: June, 2022]. +WHO COVID-19, 2020. Dashboard. Geneva: World Health Organization. Available from: https://covid19.who.int/. +Yang, W., Shaman, J., 2021. SARS-CoV-2 transmission dynamics in South Africa and epidemiological characteristics +of the Omicron variant. medRxiv . +Zucchi, G., Iori, M., Subramanian, A., 2021. Personnel scheduling during COVID-19 pandemic. Optimization Letters +15, 1385–1396. +15 + diff --git a/OdE1T4oBgHgl3EQfuAUH/content/tmp_files/load_file.txt b/OdE1T4oBgHgl3EQfuAUH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..04eb7aef943d693436922d819fdc4e7ffde010b0 --- /dev/null +++ b/OdE1T4oBgHgl3EQfuAUH/content/tmp_files/load_file.txt @@ -0,0 +1,1202 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf,len=1201 +page_content='PERSONNEL SCHEDULING AND TESTING STRATEGY DURING PANDEMICS: THE CASE OF COVID-19 A PREPRINT Mansoor Davoodi∗ Center for Advanced Systems Understanding (CASUS) Helmholtz-Zentrum Dresden Rossendorf (HZDR) Görlitz, Germany m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='davoodi-monfared@iasbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='ir Ana Batista Center for Advanced Systems Understanding (CASUS) Helmholtz-Zentrum Dresden Rossendorf (HZDR) Görlitz, Germany a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='batista-german@hzdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='de Abhishek Senapati Center for Advanced Systems Understanding (CASUS) Helmholtz-Zentrum Dresden Rossendorf (HZDR) Görlitz, Germany a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='senapati@hzdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='de Justin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Calabrese Center for Advanced Systems Understanding (CASUS) Helmholtz-Zentrum Dresden Rossendorf (HZDR) Görlitz, Germany Department of Ecological Modelling Helmholtz Centre for Environmental Research – UFZ, Leipzig, Germany Department of Biology, University of Maryland College Park, MD, USA j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='calabrese@hzdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='de January 10, 2023 ABSTRACT Efficient personnel scheduling plays a significant role in matching workload demand in organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' However, staff scheduling is sometimes affected by unexpected events, such as the COVID-19 pandemic, that disrupt regular operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Since infectious diseases like COVID-19 transmit mainly through close contact with individuals, an efficient way to prevent the spread is by limiting the number of on-site employees in the workplace along with regular testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, determining an optimal scheduling and testing strategy that meets the organization’s goals and prevents the spread of the virus is crucial during disease outbreaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In this paper, we formulate these challenges in the framework of two Mixed Integer Non-linear Programming (MINLP) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The first model aims to derive optimal staff occupancy and testing strategies to minimize the risk of infection among employees, while the second model aims at only optimal staff occupancy under a random testing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To solve the problems expressed in the models, we propose a canonical genetic algorithm as well as two commercial solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Using both real and synthetic contact networks of employees, our results show that following the recommended occupancy and testing strategy reduces the risk of infection 25%–60% under different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Keywords personnel scheduling · presence strategy · testing strategy · pandemic · COVID-19 1 Introduction Personnel scheduling decisions are crucial in many organizations since labor cost constitutes one of the major expenses in operations management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, any improvement in staffing and scheduling decisions would result in overall organizational benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Staffing and scheduling decisions can be subject to unexpected events that should be managed ∗Corresponding Author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='03382v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='OC] 6 Jan 2023 arXiv Template A PREPRINT proactively to ensure that performance measures are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' A recent global-scale phenomenon that considerably impacts scheduling decisions is the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' During a pandemic, it is necessary to consider a hybrid work strategy to limit the number of employees present in the workplace to ensure employee safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Another efficient strategy to mitigate the impact of a pandemic is implementing testing, which organizations may offer to their employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Due to the limitations in the testing capacity and sensitivity, efficient applications of tests is necessary to prevent virus outbreaks in the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Therefore, it is crucial to derive efficient staff scheduling and testing strategies to guarantee safety in the workplace while achieving the organization’s goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The personnel scheduling problem in pandemic situations is an emerging topic that has not been extensively addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The question of defining a scheduling plan that accounts for testing strategies to reduce the risk of infection while ensuring low levels of understaffing remains unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In this paper, we aim to fill this gap by developing two Mixed Integer Non-linear Programming (MINLP) models considering a probabilistic graph-based approach to determine the optimal workplace occupancy that minimizes the risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The graph-based approach assumes that employees are in close contact with each other, which contributes to the virus’s spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The main objective is to minimize the expected risk of infection while constraining workplace occupancy to comply with COVID-19 regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Our models deal with two different realistic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The first model, considering the situation where employees frequently underestimate the adherence to testing protocols, provides both optimal personnel scheduling at the workplace and their testing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' On the other hand, the second model assumes a random testing strategy for the employees and derives only the optimal presence scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We propose two approaches to solve the non-linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The first approach applies commercial optimization solvers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' APOPT for the first model and Gurobi for the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To this end, we linearize an equation (the equation for computing and updating the probability of infection) in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The second approach is a canonical genetic algorithm that utilizes penalization to satisfy the constraints of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We consider both real contact network data of employees and randomly generated sparse and dense graphs while assessing the models’ performance under several scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The results show significant impacts of both presence rate and testing schedule in minimizing the risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This paper is organized into six sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' After reviewing related and recent studies in Section 2, the problem of finding optimal presence and testing strategies are formulated in two different models in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Section 4 presents a heuristic algorithm for solving the models compared with the use of commercial non-linear solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Numerical results for different scenarios are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Finally, a conclusion is drawn in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 2 Related work Personnel scheduling is one of the critical decisions in organizations, however, it is impacted by both expected events like demand or capacity uncertainty and by unexpected events like the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' While the first kind of events usually can be handled by considering labor flexibility strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', multiskilled staff, flexible contracts, collaborative teams) to minimize the mismatch between supply and demand (Porto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2019), the second kind is difficult to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' These events certainly affect the performance of some organizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', service sector, retail, healthcare, manufacturing), which must continue with regular operations despite the global health crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' An extensive literature on personnel scheduling problems exists (Bechtold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Brucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Ernst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2004a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' According to the classification defined in Ernst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2004a), we can categorize this study as disruption management, in which the aim is to derive robust schedules by reducing the impact of the effects caused by a health emergency, such as a pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The personnel scheduling problem in a pandemic situation is an emerging topic that largely started with the COVID-19 pandemic (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Choi, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In contrast to existing studies on disruption management (Mac-Vicar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Abdelghany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2004, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Shebalov and Klabjan, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Moz and Pato, 2003) that focus mainly on developing strategies to cope with staffing operational disruptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', demand variations, airline crew delays, nurse absenteeism), this problem concerns employees’ health and safety, requiring additional considerations, such as the control of the virus spread among the staff while satisfying staffing levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The existing studies in the literature are focused on developing scheduling policies to prevent the spread of the virus in organizations and closed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For residential care facilities, Moosavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2022) developed a task scheduling model to minimize the number of employees assigned to residents to control the spread of the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To solve the model, the authors proposed a population-based heuristic algorithm that guarantees solution quality against benchmark solution approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Güler and Geçici (2020), studied the problem of scheduling physicians during the COVID-19 pandemic in a hospital in Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The authors proposed a Mixed Integer Programming (MIP) model to solve a shift scheduling problem to guarantee the safety of the physicians while keeping a balanced workload in the hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 2 arXiv Template A PREPRINT During the COVID-19 pandemic, demand for hospital care often exceeds supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2022) studied a Medical Staff Rebalancing (MSR) problem to allocate medical staff to different areas, considering the demand as the number of infected patients in the allocation regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To address the MSR problem, the authors proposed two robust optimization models that account for uncertainty in data availability while ensuring allocation fairness during emergencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Similarly, Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2021) proposed a scheduling model to minimize the workload unbalance of the nurses in charge of COVID-19 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To solve this problem, they developed a Hybrid Salp Swarm Algorithm and Genetic Algorithm (HSSAGA) and showed their algorithm outperformed the state-of-the-art solution approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For a pharmaceutical distribution warehouse in Italy, Zucchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2021) developed a Mixed Integer Linear Programming (MILP) model to solve a shift scheduling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The aim was to minimize the deviation in allocated contractual hours of employees during the COVID-19 pandemic to keep operations ongoing while guaranteeing the safety of the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Guerriero and Guido (2022), proposed a flexible staff scheduling approach for a University administrative department during the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' By allowing a hybrid work system, they developed a days-off optimization model considering employee preferences and availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Alwadood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2021) considered the personnel scheduling problem for a hotel housekeeping department used as a quarantine center for foreign travelers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This study proposed a weekly schedule for the staff using a Binary Integer Programming model that minimizes the workforce on duty to decrease the risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 3 Modeling a graph-based personnel scheduling problem during pandemic situations Personnel scheduling during a pandemic requires special handling to ensure employee safety while continuing regular operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Since the virus that spreads COVID-19, the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS- CoV-2), is transmitted through individual contacts, the World Health Organization (WHO COVID-19, 2020) suggested several measures to control the spread of the virus, including social distancing, testing, and vaccination (Abdin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, to limit the spread of the virus, the number of employees present in the workplace must be reduced, creating a conflict between the size of the workforce and the risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' On the one hand, having full workforce occupancy leads to accomplishing staffing metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', workload demand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' However, the chance of a disease outbreak will be high, compromising the safety of the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This section proposes two MINLP models to solve the personnel scheduling problem during disease outbreaks, particularly the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The models aim to minimize the risk of infection in organizations by considering flexible allocation policies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', teleworking and tests for the employees) and capacity constraints to impose social distancing in the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We aim to derive a presence strategy to find the optimal schedule of employees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', working remotely or at the workplace) and a testing strategy to determine the testing days of the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We consider an organization with n employees who are in contact with each other and have to be allocated in a discrete-time horizon, d = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', D (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', a week, D = 5) such that the risk of infection in the workplace is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The proposed models compute the probability of infection for each employee under two different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The first model assumes that the employees comply with the testing protocols following the suggested testing days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The second model does not impose strict regulations on testing, so the employees perform tests arbitrarily during the evaluated time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The notation and assumptions considered in the proposed models are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Nomenclature Parameters n Number of employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' ei i − th employee, indexed by i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' pij Probability of employee i and employee j have contact if both come to the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' βi probability of infection of employee i per contact, depending if vaccinated or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' FN Probability of false negativity of COVID-19 tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' PId i Probability of infection of employee i in day d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' D Scheduling Interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', d = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', D days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' TCi Test capacity for employee i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', the maximum number of available tests for employee i in the scheduling interval D days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' br Background Risk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' probability of infection in the neighborhood of the organization based on the 7-days incidence reports in the county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' m Number of task flow constraints in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' m′ Number of room capacity constraints in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Decision Variables xd i Binary variable that indicates the allocation of employees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' xd i = 1 if employee i works in the workplace at day d, otherwise xd i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' td i Binary variable that indicates the testing status;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' td i = 1 if employee i is tested on day d, otherwise td i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Assumptions: 3 arXiv Template A PREPRINT The tests are performed in the morning before employees come to the workplace, and if the result is positive, they stay at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We initialize the probability of infections to background risk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', PI0 i = br, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Indeed, we assume for the starting day of the scheduling interval (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Monday), the employees’ risk is the same as the background risk in the organization’s neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1 Computing the probability of infection in a graph-based approach In this subsection, we propose a graph-based approach to compute the probability of infection in the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Let PId i be the probability of infection for ei at the end of the working day d, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', n and d = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The objective function is to minimize the expected risk of infection inside the facilities, which is directly related to minimizing the probability of infection of employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, Minimize Z = 1 D × n D � d=1 n � i=1 PId i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (1) To compute the probability of infection, PId i , we present a recursive formula based on the presence and testing strategies for the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' As mentioned in the assumptions, PI0 i as the initial probability of infection is set to br, which means a risk of infection based on the number of incidences reported in the organization’s neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Now, we iteratively compute PId i by having PId−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Considering the graph of connectivity among the employees, PId i can be computed based on PId−1 i and PId−1 j for all employees which may be in contact with employee i at day d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' If an employee with a probability of infection p performs a test and the result is negative, the infection probability will reduce to p × FN, where FN is the probability of false negativity for the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Now, let the binary variable td i indicate ei is tested on day d before coming to the workplace (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', td i = 1) or not (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', td i = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, the updating recursive formula can be written as PI ′d i = PId−1 i × (1 − td i ) + PId−1 i × td i × FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2) Equation (2) works well when the employees follow the testing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' If they do not follow this strategy, we can apply a random testing strategy, that is, assuming a testing probability pr(test) for each employee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For example, if the employees take two tests in five days, the Probability of testing per day is pr(test) = 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' If ei performs a test at day d with probability pr(test), then we can use the following equation for applying the effect of tests and computing PI ′d i , PI ′d i = (1 − pr(test)) × PId−1 i + pr(test) × PId−1 i × FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (3) Therefore, regarding the fact that the employees follow a recommended testing strategy, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2), or a random test strategy, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (3), we apply the effect of performing the tests and update the probability of infection for all employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Then, we update the probabilities of infection for the employees based on their contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' If ei comes to the workplace on day d (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' xd i = 1), and with the probability of pij contacts with ej (suppose xd j = 1 as well), then the probability of infection for him/her can be updated as PId i←j = 1 − [(1 − PI ′d i ) × (1 − pij × βi × PI ′d j )], (4) where βi is the probability of infection for ei in case that ej is infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For the sake of simplicity, we only assume two possible values for βi whether the employee is vaccinated or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' PId i←j denotes the effect of contact with ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, by applying all possible contacts ei may have during day d, the infection probability at the end of the day can be computed as follows PId i = 1 − [(1 − PI ′d i ) × n � j=1&j̸=i (1 − pij × βi × xd j × PI ′d j )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (5) Thus, a two-step procedure is performed to update the probability of infection of the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' First, we apply the effect of testing (Equation (2) or (3)), and then apply the effect of contacts among the employees in the workplace, (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Note that, without loss of generality, we assume that working from home is free of risk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', if the employees ei do not 4 arXiv Template A PREPRINT come to the workplace on the day d, PId i = PId−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For the sake of simplicity, let’s denote the two-step procedure by a function based on the related parameters if the employees follow the recommended testing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' PId = Update(PId−1, xd, td), (6) and, if they follow a random testing strategy PId = Update(PId−1, xd, pr(test)), (7) where PId−1 = {PId−1 1 , PId−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' , PId−1 n }, xd = {xd−1 1 , xd−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' , xd−1 n }, and td = {td−1 1 , td−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' , td−1 n } are the infection probability, presence indicator and testing indicators for all the employees in the day d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2 Personnel scheduling and testing strategy models In this section, we define two MINLP models considering the probability of infection equations defined in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Model 1 assumes that the employees follow a recommended testing strategy, and Model 2 is based on the fact that employees do not follow the testing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, they test themselves randomly with a probability of pr(test) for each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We also consider a set of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We keep the model simple and easy to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In practice, different organizations may have their specific limits and constraints, which can be added to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Here, we consider two families of upper and lower bound constraints: the first family of constraints is related to satisfying the on-site tasks in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' They are defined as, n � i∈Ck xd i ≥ bk, for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', m, (8) where Ck is the k − th subset of employees such that at least bk number of them have to present at the workplace on day d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The second family of constraints refers to capacity limitations in the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In fact, during the COVID-19 pandemic, there have been regulations on the maximum number of employees who can be simultaneously (in a day) be present in the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, we can model them as follows, n � i∈C′ k xd i ≤ b′ k, for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', m′, (9) Model 1 would result in a lower risk of infection compared to Model 2, but it requires the employees to follow the recommended testing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In contrast, Model 2 depicts a more flexible testing scheme, in which the employees apply the offered tests randomly during the scheduling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The models are defined as follows, Model 1: Personnel scheduling with testing strategy Minimize Z = 1 D × n D � d=1 n � i=1 PId i , Subject to : PI0 = {β1br, β2br, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' , βnbr} PId = Update(PId−1, xd, td), n � i∈Ck xd i ≥ bk, for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', m, n � i∈C′ k xd i ≤ b′ k, for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', m′, D � d=1 td i ≤ TCi, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', n, xd i , td i ∈ {0, 1}, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', n, and d = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', D (10) 5 arXiv Template A PREPRINT TCi refers to the test capacity for ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' the maximum number of available test kits for ei in a period of D days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This capacity may be the same for all employees or distributed among the employees regarding the number of connections each employee has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Model 1 has two decision variables, presence scheduling, xd i , and testing schedule, td i , for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' , n and d = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' , D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, the model will (optimally) derive which employees to allocate in the workplace and when, and on which days to perform the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' If in an organization, the employees do not follow the suggested testing strategy, and they use the tests arbitrarily during the scheduling period, Model 1 will not fit with that organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In this case, The following model, which assumes the tests can be used by the employees with a probability, is a better match for that organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Model 2: Personnel scheduling without testing strategy Minimize Z = 1 D × n D � d=1 n � i=1 PId i , Subject to : PI0 = {β1br, β2br, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' , βnbr} PId = Update(PId−1, xd, pr(test)), n � i∈Ck xd i ≥ bk, for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', m, n � i∈C′ k xd i ≤ b′ k, for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', m′, xd i ∈ {0, 1}, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', n, and d = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', D (11) Both Model 1 and Model 2 are MINLP and, like the general scheduling problem (Hoong, 1996) with hard constraints, are NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Furthermore, considering the upper bound and lower bound constraints of the problem, finding even a feasible solution that satisfies the constraints is an intractable problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The main difficulty of the models is updating the risk of infection for the employees after daily contacts, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' It is an exponential equation and impossible for most algorithms to cope with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Therefore, in the following, we present an efficient simplification to handle this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Relaxation The term �n j=1&j̸=i(1 − pij × βi × xd j × PI ′d j ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5 is the only exponential equation of the proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' As explained, this term is used for updating the infection risk of an employee after his/her daily contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In practice, the value of this term is so small (based on the data and experiments, that it is in order of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, to relax the models and remove this exponential term, we use the linear Taylor expansion of the formula, (1 − x)n ≈ 1 − nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, the simplified approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5 can be written as below, PId i = 1 − [(1 − PI ′d i ) × (1 − Σn j=1&j̸=i(pij × βi × xd j × PI ′d j ))], (12) To evaluate the accuracy of this simplification, we compared the above linear equation with Equation (5) using the parameters reported in the example presented in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The comparison showed that the equations result in almost the same values with a precision on the order of 10−9 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, it is a suitable linear approximation in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 4 Solution approach To solve the proposed Model 1 and Model 2, we developed different solution approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For each model, we apply a nonlinear commercial solver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' GEKKO and APOPT (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='0) (Beal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2018) solver for the first model, and Gurobi 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3 optimization solver (Gurobi Optimization, LLC, 2022) for the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To apply these solvers, we replace the Equation (5) with linearization explained in Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In addition to applying these solvers, we propose a Genetic Algorithm (GA) and tailored it for both Model 1 and Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In the following, we briefly explain the GA and its operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 6 arXiv Template A PREPRINT Genetic algorithms are random search algorithms which work based on heuristic exploration and exploitation operations Coello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' It starts with a random set of solutions (chromosomes), called the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Then, the GA evolves the population generation by generation using some exploration and exploitation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To this end, the selection operator chooses some high-fitness solutions as the parent chromosomes and put them in the mating pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Then, the Crossover operator takes a pair of such parents and produces (usually) two new chromosomes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', the children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Indeed, it combines part of (some genes) from the first parent with the other part of the second parent and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The mutation operator mimics the natural mutation and changes some genes of a child solution at random to explore a new search space and prevent the (premature) convergence of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Both of the operators play the role of exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Finally, at the end of each iteration, from the combined parent and children populations, a set of high-fitness chromosomes are picked for the next generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' There are several kinds of crossover, mutation and selection operators Coello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Deb (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We present a canonical GA with standard tournament selection operators, single-point crossover, and swap-mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Since the decision variables are binary, we directly use them to represent a chromosome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Also, to satisfy the constraints of the models, we penalize the infeasible chromosomes by adding a penalty value, that is, the number of violations of the constraints that occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Finally, we define the fitness function as the sum of the objective function, the expected risk of infection defined in Equation (1), and the penalty value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Note that the objective function is always a value between zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, such a definition of the fitness function causes emphasizes feasible solutions in the search space at first and then improving the solutions in terms of the risk value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, the GA can also be applied to finding a feasible solution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', the first solution with a penalty value of zero found in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We utilize the proposed GA in two folds, finding a random feasible solution that satisfies all the constraints, and a (feasible) suboptimal solution that minimizes the expected risk of infection among the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We use such random solution(s) in comparing and showing the impact of the presence and testing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We applied GA to solve the proposed Model 1 and Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Its complexity depends on the size of its population and the number of generations, and there is a trade-off between the complexity and the optimality of the obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' On the other hand, it is straightforward to apply the GA on either Equation (5) or its relaxed linear equation, Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For n employees and a period of D days, any chromosome can be evaluated in O(n × D × M) time, where M is the size of all constraints in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Therefore, the time complexity of a GA with population size p in g iterations is O(p × g × n × D × M) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, all the parameters have a linear impact on the algorithm’s complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Finally, GA is a random search, meaning multiple runs of it on the same instance of a problem may result in different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Despite GA, the time complexity of solvers Gurobi and APOPT in GEKKO depends on the number of employees and the size of constraints, which have an exponential impact on the complexity of such solvers in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Therefore, these solvers use different approaches to avoid the long-running time, such as solving the problem in the dual space and applying predetermined optimality gaps, pre-solving, and branch and bound techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We applied APOPT to Model 1 and Gurobi to Model 2 with the linear Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Although the running time of these solvers depends on the whole of the parameters in the models, with a logical optimality gap size like 10−5, they are faster than GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' But the objective value of obtained solutions by GA is better than the ones obtained by the solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='4, we evaluate the performance of the different solution approaches in terms of running time and solution optimality by comparing the APOPT solver and GA for Model 1 and the Gurobi solver and GA for Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5 Numerical results In this section, we evaluate the proposed models and algorithms on several test problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The aim is to find the optimal presence and testing strategies that result in the minimum expected risk of infection of the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We compare the models and the algorithms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Further, we analyze the impact and sensitivity of the models and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The experiments are composed of four parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In the first part, we assume a small-size organization and show the optimal presence and testing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In the following part, we consider real data on employee contact networks in organizations and random connectivity graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In the third part, the results for random connectivity graphs are represented, and finally, the fourth part compares effectiveness of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Since in the first three parts, the aim is to compare the models and impact of the introduced parameters, we run the presented genetic algorithms 30 times and report the average objective’s values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We consider the following values for the model’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We consider a time horizon of a week, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', five working days, D = 5 for running the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We define the probability of disease transmission as β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We choose this value as a probable pessimistic case from a possible range of values reported in previous studies Lelieveld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Yang and Shaman (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' World Health Organization (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Lyngse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2021) regarding the first variants of COVID-19 (Alpha, Delta, and Omicron, which is more transmissible than the previous ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 7 arXiv Template A PREPRINT We calculate the background risk based on 7-day incidences of COVID-19 infections in Saxony, Germany, in the period of June-August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We consider 300 incidences on average for this period and set the daily background risk to br = 1 7 × 300 100,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The initial risk of infection, PI0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', the risk of infection on Mondays, is determined by applying the background risk and two days weekend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' That is, PI0 = 1 − (1 − br)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We apply the impact of vaccine (fully vaccinated) on the initial risk of infections and transmission probability during the employee’s interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Based on the related recent researches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', see Polack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Baden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Emary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Fiolet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (2021)), we set the transmission rate of vaccinated employees to (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='85)β, which implies 85% immunity for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1 Base case study In this subsection, we present the base case study in which we consider a small-size organization with 20 employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The organization is divided into two cross-functional sections: the first section includes 12 employees, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', e12, and the second one includes e13, e14, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', e12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The connectivity graph that represents the employee’s contact network is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We consider the following allocation rules on the presence of the employees: (i) Each employee has to be present at the workplace at least 2 days a week, (ii) The whole workplace occupation should remain between Min Occupation = 50% and Max Occupation = 75% of all the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' That means at least 10 and at most 15 employees can be present at the workplace daily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (iii) At least 30% of employees in each section should be present at the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' That means at least 4 employees from the first section and 3 from the second section should be present daily at the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In addition, we assume that the probability of false negativity of the tests is FN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2, and two tests are available per employee per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' That means, in the first model, the probability of testing per (working) day is pr(test) = 2 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='4, and in the second model, TC = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Table 1 and Table 2 show the obtained results for Model 10 and Model 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' As can be extracted from the results, the presence strategy aims to satisfy the occupancy constraints by selecting employees who have a weak connection rate with each other for onsite work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Further, the strategy satisfies the minimum 50% occupancy with exactly 10 employees every day, except with a strategy of 11 employees on Thursdays in the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Also, in the obtained results for the second model, the testing strategy suggests employees apply the tests (2 tests are available per employee) on the first days of the week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This is because we initialize the risk of infection on Mondays with a higher value after two days of weekend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, the strategy tries to reduce this value by suggesting tests before the employees are in contact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The expected risk of infection for the suggested strategies is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='61e-5 for the first model and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='99e-5 for the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' That means if the employees follow the suggested testing strategy, they can reduce the risk to 43% of the risk when they follow a random testing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' It may also be interesting to see what happens if the employees perform one test or three tests per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To answer this question, we run the models for TC = 1 and 3, and pr(test) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We skip reporting the strategies and only focus on the objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The risk of infection for TC = 1, and TC = 3, are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='44e-5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='85e-5, respectively, and the risk of infection for the second model where the employees follow a random testing strategy with probabilities pr(test) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2 and pr(test) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='6 are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='05e-5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='85e-5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Having some high-level information and the price of each test, a manager can optimally decide how many tests are better to offer to the employees based on the testing strategy and the model that may fit the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2 Results on real data We consider publicly available face-to-face interaction data collected by the SocioPatterns collaboration (Socio Patterns Collaboration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This dataset contains the interactions among 92 employees recorded in 20-second intervals in an office building in France from 24th June to 3rd July 2013 (Génois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Since the probability of contact for the employees is not explicitly reported in the available data, we pre-process the data to adapt the dataset to our proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For employee i and employee j, we first aggregated the contacts between them over the above-mentioned period and calculated the average number of contacts per day (cij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Next, for each employee i, we divided the total number of contacts made by that employee per day by the number of colleagues he/she has and calculate the degree-normalized average contacts (di) per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Here, we defined employee i and employee j as colleagues if they have at least one contact over this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Finally, we calculated the probability of contacts between i and j as: 8 arXiv Template A PREPRINT Figure 1: Base case study contact network: An organization with 20 employees in two cross-functional sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' All the employees except number 15 are fully vaccinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The edge’s label shows the probability of occurring contact between a pair of employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Table 1: Obtained presence strategy for the example illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 1 indicates present at the workplace, and 0 means working from home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The results are shown for five working days for 20 employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In this scenario, each employee performs a test with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='4 per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This presence strategy results in the expected risk of infection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='61e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Employee Monday Tuesday Wednesday Thursday Friday 1 1 0 1 1 0 2 1 0 1 1 0 3 0 1 0 0 1 4 1 0 0 0 1 5 0 1 0 0 1 6 0 1 0 0 1 7 0 1 1 1 0 8 1 0 0 1 1 9 1 0 1 1 0 10 0 1 1 1 0 11 1 0 1 1 0 12 0 1 0 0 1 13 1 0 0 0 1 14 0 1 1 1 0 15 0 0 1 1 0 16 1 1 1 0 0 17 1 0 1 1 0 18 0 1 0 0 1 19 0 1 0 0 1 20 1 0 0 0 1 pij = � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' if cij di ≥ 1 or cij dj ≥ 1 max{ cij di ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' cij dj } if cij di < 1 and cij dj < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We assume 95% of employees are fully vaccinated, and the probability of false negativity for the tests is FN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Under such conditions, we run the models for different scenarios as follows: (1) Each employee has to be present at the workplace at least (i) 2 days in a week, and (ii) 3 days in a week, (2) The feasible minimum and maximum occupancy are set to (i) [30%, 70%] and (ii) [40%, 80%] (3) The number of available tests per employee per week is TC = 1, 2, 3 for the first model and correspondingly pr(test) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='6 for the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Therefore, we will have 2 × 2 × 3 = 12 different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We run both the models under these scenarios and report the results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We scale the risk’s values by 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Columns M1 and M2 show the expected risk of infection for the first and the second model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Also, to compare the impact of the models, we compute random feasible solutions (the solutions that satisfy all the constraints of the problem) for the presence and testing of the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The column R in the table shows the expected risk of infection for such solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This value is the average risk of infection for 30 different random solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' As expected, the risk of infection improves from the random strategy to the strategy in the second model, and also from the second model to the first model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' On average, following the suggested presence strategy can reduce the risk of infection 26% compared to the random strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' And if the employees follow 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='arXiv Template A PREPRINT Table 2: Obtained presence and testing strategies for the example illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The left binary digit is a presence indicator, present at the workplace (1), or works from home (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The right binary digit shows he/she performs a test (1) or not (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Note that it is possible an employee stays at home but does a test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This presence and testing strategy results in the expected risk of infection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='99e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Employee Monday Tuesday Wednesday Thursday Friday 1 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 1 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 0 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 0 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 1 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 0 2 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 1 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 0 1 ;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 0 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 0 Figure 2: Graph representation of employees contact network with 92 employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' both the suggested presence and testing strategies, the risk of infection can be reduced to 60% of the random strategy, and 45% of the risk compared to the suggested strategy in the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, efficiently using the tests together with the presence strategy can significantly reduce the expected average risk among the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Also, when the tests are used randomly by the employees, in columns R and M2, increasing the test capacity, TC, has an almost linear impact on the risk of infection, while if the employees follow the suggested testing strategy, M1, the infection risk considerably reduces when TC increases from one to two compared to when it increases from two to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, the managers can offer an optimal number of free tests per week to the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This can be optimally chosen regarding the price of test kits and the acceptable level of risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Note that the background risk can be used as a reference point to determine such a level, and the minimum and the maximum occupation are determined by considering workplace capacities and COVID-19 regulations, workfellow, and the type of services the employees provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We assumed two different cases where the employees have to be present at least two days a week and where they have to be present at least three days a week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' As expected, the risk of infection for the first case is less than in the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This constraint forces all employees to be present at the workplace, even those who are at higher risk of infection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', the unvaccinated employees and employees with a higher degree of connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The other family of constraints, which we applied to this real data, is two general constraints addressing the whole number of on-site employees per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We considered two scenarios [30%, 70%] and [40%, 80%], and the results are reported separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Despite the first family of constraints, this type of constraint allows flexibility in choosing low-risk employees to satisfy 10 arXiv Template A PREPRINT Table 3: Obtained results for the real data under 12 different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The reported risk values are scaled in 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='For each scenario, three strategies are computed: (i) a feasible random strategy, denoted by column R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This solution satisfies all the constraints and is random in terms of both presence and testing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (ii) The solution for the second model, denoted by M2, is for the case that the employees follow the suggested presence strategy with a random testing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' (iii) The solution for the first model, denoted by M1, for the case that the employees follow suggested strategies for both presence and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' min presence: 2 days min presence: 3 days Occupancy TC R M2 M1 R M2 M1 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='32 Table 4: Obtained results for sparse random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The reported risk values are scaled in 105, and R, M2, M1 and TC are the same as explained in the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Here n is the number of nodes and FN is the probability of false negativity for each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' TC = 1 TC = 2 TC = 3 n FN R M2 M1 R M2 M1 R M2 M1 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='81 the minimum necessary number of on-site employees per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The optimal solution (the solution which minimizes the risk of infection) usually belongs to the boundary of these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Therefore, the obtained strategy for [30%, 70%] are better than for [40%, 80%].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3 Results on random graphs In this part, we evaluate the models on randomly generated connectivity graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' To this end, we first choose a set of nodes (each node represent an employee), and then for any pair of nodes i and j assume a weight in pij ∈ [0, 1] as the probability of a contact between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We generated three small (with n = 40 nodes), medium (with n = 100 nodes) and large (with n = 250 nodes) size graphs with two sparse and dense connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In the sparse graphs, we set pij to 1 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='05, and set it to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='5 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1, otherwise (with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='85) we set it to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In the dense graphs, pij = 1 with probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1, pij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='5 with probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='2, and pij = 0 with probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We also assume three possible cases TC = 1, 2 and 3 for the number of available tests per employee per week, as well as two scenarios FN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3 as the probability of false negativity for each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' So, in general, there are 3 × 3 × 2 = 18 scenarios for the size of graphs and the test’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We considered the constraints that each employee should present at the workplace at least 3 days a week and the daily occupation in the workplace is allowed to be in [50%, 75%].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Table 4 and Table 5 show the obtained risk of infection for 18 sparse graphs and 18 dense graphs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The reported risk values are scaled in 105 and they are the average of 30 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Table 5: Obtained results for dense random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The reported risk values are scaled in 105, and R, M2, M1 and TC are the same as explained in the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Here n is the number of nodes and FN is the probability of false negativity for each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' TC = 1 TC = 2 TC = 3 n FN R M2 M1 R M2 M1 R M2 M1 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='42 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='38 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='52 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='16 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='38 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='35 11 arXiv Template A PREPRINT Table 6: Comparison results between APOPT solver and GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The results are an average of 30 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The running times are in second, and the objective values are scaled in factor 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' running time APOPT Solver Genetic Algorithm n APOPT GA min mean max min mean max 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='95 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='75 40 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='66 394 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='87 100 924 2291 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='18 The results in each table show the impact of the number of tests and their sensitivity, as well as following a random or suggested presence testing strategy by the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For example, in sparse and small-size graphs (n = 40), when the employees use only one test per week (TC = 1) and present at the workplace by a random strategy and apply the tests randomly on a day of the week (column R) if the tests have a false negativity rate FN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3, the results expected average risk of infection is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='69e-5, while for the case FN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1, it results in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='19e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' These risk values can be reduced to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='36e-5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='98e-5 when the employees follow the suggested presence strategy with a random test strategy (column M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Finally, following both suggested presence and testing strategies (column M1) results in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='31e-5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='85e-5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The table shows the results for two (TC = 2) and three (TC = 3) tests per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For example, two tests with accepting suggested presence and testing strategies result in a risk of infection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='17e-5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='91e-5 for FN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3 and FN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' That means 56% and 70% improvements (risk reduction) comparing the case when they do not follow the strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, by comparing such results and the model which is more fit for the organization, an optimal case can be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The same results are reported for the dense graphs, and the managers may be interested in observing the impact of the rate of contact among the employees on the expected risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In our experiments, the probability of contact between any pair of employees in the dense graphs is two times more than in the sparse graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' However, the average risk of infection in columns R, M2 and M1 increased by factors 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='54, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='49 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='41, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This kind of information can be efficiently used in establishing contact regulations in the workplace space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In fact, in addition to the presence strategy and testing strategy, the rate of contact is an influential factor to control the risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='4 Comparison of solution algorithms As explained, we presented a GA and applied it to both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Further, we solve the problem defined in the first model, Model 10, using GEKKO and APOPT solver, and the problem defined in the second model, Model 11, using Gurobi solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In this part, we compare these algorithms in terms of running time and efficiency in finding the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Note that, since the problem is an NP-hard problem, none of the algorithms can guarantee to find the optimal solution in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Thus, we compare the obtained solutions with the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The models and algorithms were implemented in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='7 on a standard PC (Intel (R) Core(TM) i7 and 32G RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For a graph with a specific number of nodes, we run the algorithms for 30 different sets of edges and weights and report the average running time (in seconds) of the algorithms as well as the best, mean, and worst of the objective value in the obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We consider graphs with the number of nodes n = 10, 20, 40, 100 with different sparse and dense topologies and run the algorithms for a scheduling period D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We run the GA with population size = 100 + 2 × n, maximum generation = 200 + 2 × n, and mutation probability 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Table 6 shows the result for GA and GEKKO with APOPT solver for Model 1, and Table 7 shows the result for GA and Gurobi for Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' From Table 6, it is clear the APOPT solver reaches the solutions significantly quicker than GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' However, there is always a trade-off between the running time of GA and the efficiency of the obtained solutions, APOPT dominates GA in terms of running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' On the other hand, the GA slightly outperforms (on average about 2%) APOPT in terms of optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In general, both approaches can be used for Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' From Table 7, it is obvious Gurobi solver is quite faster than GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' On the other hand, the GA outperforms (on average about 4%) Gurobi in terms of optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We ran Gurobi for very large graphs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', with 1000 nodes, and observed it still is efficient and can compute the solution in less than 10 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' In general, since the running time of GA is acceptable for real-size instances of the problem, it seems that both approaches can be applied to Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 6 Conclusion and future work During the COVID-19 pandemic, the most efficient strategy to prevent the spread of the virus was (and is) the implementation of teleworking and regular testing of the employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' However, to date, there is not a clear understanding 12 arXiv Template A PREPRINT Table 7: Comparison results between Gurobi solver and GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The results are an average of 30 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The running times are in second, and the objective values are scaled in factor 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' running time Gurobi Solver Genetic Algorithm n Gurobi GA min mean max min mean max 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='06 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='83 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='92 20 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='98 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='68 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='44 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='55 322 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='82 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='34 1882 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='37 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='74 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='95 of how to define efficient personnel scheduling plans while considering testing capacities so as to guarantee the safety of the employees that should keep working despite a pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This paper tackled such a situation by developing two MINLP models for deriving efficient scheduling plans during a pandemic, taking into account teleworking strategies and testing capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The main objective is to minimize the expected average risk of infection among the employees, considering work flow constraints and occupancy limitations in the workplace to comply with the COVID-19 regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The first model focuses on scheduling the presence of employees in the workplaces as well as scheduling their tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' However, since in practice the employee may not follow a testing schedule, we presented a second model, which aims at optimizing the presence scheduling under a random testing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' We compared the models under various scenarios and discovered that by implementing these strategies, an organization can reduce the risk of infection by 25% to 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Further, we performed a sensitivity analysis by tuning several influential parameters and showed several scenarios that can significantly help managers in establishing regulations in the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' For instance, they can see the impact of connection weights, when it changes from dense graphs to sparse graphs, or when different occupation constraints are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The models we proposed assume that the number of available tests for all the employees is the same, while the tests should be distributed according to the degree of connection for the employees and the number of days on which they should be present at the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The models can be easily extended to cover this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' However, by modeling a variable test availability, the search space of the problem may increase, which may require more efficient solution approaches such as branch and bound and hybrid methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Acknowledgements This work was partially funded by the Where2Test project, which is financed by SMWK with tax funds on the basis of the budget approved by the Saxon State Parliament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' This work was also partially funded by the Center of Advanced Systems Understanding (CASUS) which is financed by Germany’s Federal Ministry of Education and Research (BMBF) and by the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis of the budget approved by the Saxon State Parliament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' References Abadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Rahmati, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Sharifi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Ahmadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Hssaga: designation and scheduling of nurses for taking care of COVID-19 patients using novel method of hybrid salp swarm algorithm and genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Applied Soft Computing 108, 107449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Abdelghany, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Ekollu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Narasimhan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Abdelghany, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' A proactive crew recovery decision support tool for commercial airlines during irregular operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Annals of Operations Research 127, 309–331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Abdelghany, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Abdelghany, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Ekollu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' An integrated decision support tool for airlines schedule recovery during irregular operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' European Journal of Operational Research 185, 825–848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Abdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Fang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Caunhye, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Alem, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Barros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Zio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' An optimization model for planning testing and control strategies to limit the spread of a pandemic–the case of covid-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' European journal of operational research 304, 308–324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Alwadood, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Noor, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Mainor, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' An optimization model for hotel housekeeping personnel scheduling in pandemic outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Menemui Matematik (Discovering Mathematics) 43, 83–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Baden, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', El Sahly, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Essink, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Kotloff, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Frey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Novak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Diemert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Spector, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Rouphael, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Creech, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Efficacy and safety of the mrna-1273 sars-cov-2 vaccine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' New England Journal of Medicine .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 13 arXiv Template A PREPRINT Beal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Hill, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Martin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Hedengren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Gekko optimization suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Processes 6, 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' doi:doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='3390/pr6080106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Bechtold, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Brusco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Showalter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' A comparative evaluation of labor tour scheduling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Decision Sciences 22, 683–699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Brucker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Qu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Burke, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Personnel scheduling: Models and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' European Journal of Operational Research 210, 467–473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Choi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Fighting against COVID-19: what operations research can help and the sense-and-respond framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Annals of Operations Research , 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Coello, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Lamont, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Van Veldhuizen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Evolutionary algorithms for solving multi-objective problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' volume 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Deb, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Multi-objective optimisation using evolutionary algorithms: an introduction, in: Multi-objective evolutionary optimisation for product design and manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 3–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Emary, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Golubchik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Aley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Ariani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Angus, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Bibi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Blane, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Bonsall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Cicconi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Charlton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Efficacy of chadox1 ncov-19 (azd1222) vaccine against sars-cov-2 variant of concern 202012/01 (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 7): an exploratory analysis of a randomised controlled trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' The Lancet 397, 1351–1362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Ernst, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Krishnamoorthy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Owens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Sier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2004a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' An annotated bibliography of personnel scheduling and rostering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Annals of Operations Research 127, 21–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Ernst, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Krishnamoorthy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Sier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2004b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Staff scheduling and rostering: A review of applications, methods and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' European journal of operational research 153, 3–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Fiolet, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Kherabi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', MacDonald, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Ghosn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Peiffer-Smadja, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Comparing COVID-19 vaccines for their characteristics, efficacy and effectiveness against SARS-CoV-2 and variants of concern: A narrative review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Clinical Microbiology and Infection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Cao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Robust optimization model for medical staff rebalancing problem with data contamination during COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' International Journal of Production Research 60, 1737–1766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Génois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Vestergaard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Fournet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Panisson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Bonmarin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Barrat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Data on face-to-face contacts in an office building suggest a low-cost vaccination strategy based on community linkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Network Science 3, 326–347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Guerriero, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Guido, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Modeling a flexible staff scheduling problem in the era of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Optimization Letters 16, 1259–1279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Güler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Geçici, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' A decision support system for scheduling the shifts of physicians during COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Computers & Industrial Engineering 150, 106874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Gurobi Optimization, LLC, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Gurobi Optimizer Reference Manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='gurobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Hoong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' On the complexity of manpower shift scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Computers & Operations Research 23, 93–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Jordan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Adab, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Cheng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' COVID-19: risk factors for severe disease and death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Lelieveld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Helleis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Borrmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Cheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Drewnick, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Haug, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Klimach, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Sciare, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Pöschl, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Model calculations of aerosol transmission and infection risk of COVID-19 in indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' International Journal of Environmental Research and Public Health 17, 8114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Lyngse, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Mortensen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Denwood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Christiansen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Møller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Skov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Spiess, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Fomsgaard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Lassauniere, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Rasmussen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' SARS-CoV-2 Omicron VOC transmission in Danish households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' medRxiv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Mac-Vicar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Ferrer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Muñoz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Henao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Real-time recovering strategies on personnel scheduling in the retail industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Computers & Industrial Engineering 113, 589–601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Moosavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Ozturk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Patrick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Staff scheduling for residential care under pandemic conditions: The case of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Omega 112, 102671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Moz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Pato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' An integer multicommodity flow model applied to the rerostering of nurse schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Annals of Operations Research 119, 285–301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' World Health Organization, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Enhancing response to Omicron SARS-CoV-2 variant: Technical brief and priority actions for member states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Available from: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='int/publications/m/item/enhancing-readiness-for- omicron-(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='529)-technical-brief-and-priority-actions-for-member-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Polack, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Thomas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Kitchin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Absalon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Gurtman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Lockhart, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Perez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Marc, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Moreira, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Zerbini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Safety and efficacy of the bnt162b2 mrna covid-19 vaccine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' New England Journal of Medicine .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 14 arXiv Template A PREPRINT Porto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Henao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', López-Ospina, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', González, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Hybrid flexibility strategy on personnel scheduling: Retail case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Computers & Industrial Engineering 133, 220–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Shebalov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Klabjan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Robust airline crew pairing: Move-up crews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Transportation science 40, 300–312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Socio Patterns Collaboration, .' metadata={'source': 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+page_content=' Geneva: World Health Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Available from: https://covid19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content='int/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Shaman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' SARS-CoV-2 transmission dynamics in South Africa and epidemiological characteristics of the Omicron variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' medRxiv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Zucchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Iori, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', Subramanian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Personnel scheduling during COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' Optimization Letters 15, 1385–1396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfuAUH/content/2301.03382v1.pdf'} diff --git a/Q9AzT4oBgHgl3EQfW_zX/content/tmp_files/2301.01312v1.pdf.txt b/Q9AzT4oBgHgl3EQfW_zX/content/tmp_files/2301.01312v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1dad9a117154ac6bf1466c877b2dbe89fbed0f7 --- /dev/null +++ b/Q9AzT4oBgHgl3EQfW_zX/content/tmp_files/2301.01312v1.pdf.txt @@ -0,0 +1,1963 @@ +MNRAS 000, 1–10 (2015) +Preprint 5 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Binary black hole spins: model selection with GWTC-3 +Carole Périgois1,2,★, , Michela Mapelli1,2,3,†, , Filippo Santoliquido1,2,3, , Yann Bouffanais1,2, , +and Roberta Rufolo1. +1Physics and Astronomy Department Galileo Galilei, University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy +2INFN - Padova, Via Marzolo 8, I–35131 Padova, Italy +3INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The origin of the spins of stellar-mass black holes is still controversial, and angular momentum transport inside massive stars +is one of the main sources of uncertainty. Here, we apply hierarchical Bayesian inference to derive constraints on spin models +from the 59 most confident binary black hole merger events in the third gravitational-wave transient catalogue (GWTC-3). We +consider up to five parameters: chirp mass, mass ratio, redshift, effective spin, and precessing spin. For model selection, we use +a set of binary population synthesis simulations spanning drastically different assumptions for black hole spins and natal kicks. +In particular, our spin models range from maximal to minimal efficiency of angular momentum transport in stars. We find that, +if we include the precessing spin parameter into our analysis, models predicting only vanishingly small spins are in tension with +GWTC-3 data. On the other hand, models in which most spins are vanishingly small, but that also include a sub-population of +tidally spun-up black holes are a good match to the data. Our results show that the precessing spin parameter has a crucial impact +on model selection. +Key words: black hole physics – gravitational waves – binaries: general – stars: black holes +1 INTRODUCTION +The third observing run (O3) of the Advanced LIGO (Aasi et al. 2015) +and Virgo (Acernese et al. 2015) detectors has brought the number +of compact binary merger observations up to 90 events with a prob- +ability of astrophysical origin > 0.5 (Abbott et al. 2019, 2021d,a,b). +In particular, the 63 confident detections of binary black hole (BBH) +mergers (with a false alarm rate FAR< 0.25 yr−1) lead to more ac- +curate constraints on the mass and spin distribution of these systems +(Abbott et al. 2021c). +The intrinsic distribution of primary black hole (BH) masses in- +ferred by the LIGO–Virgo–KAGRA collaboration (hereafter, LVK) +shows several sub-structures, including a main peak at ≈ 10 M⊙, +a secondary peak at ≈ 30 − 40 M⊙, and a long tail extending up +to ∼ 80 M⊙ (e.g., Abbott et al. 2021c). The inferred distribution +of mass ratios has a strong preference for equal-mass systems, but +several BBHs are confidently unequal-mass (e.g.,GW190517 Abbott +et al. 2020). Focusing on BH spins, we can safely exclude that all +BHs are maximally spinning (Farr et al. 2017, 2018; Abbott et al. +2019). Typical spin magnitudes in BBHs are small, with ∼ 50% of +BHs having 𝜒 ≲ 0.3 (e.g., Wysocki et al. 2019; Abbott et al. 2021d), +although not all BHs in the LVK sample have zero spin (Roulet & +Zaldarriaga 2019; Miller et al. 2020). For example, GW151226 (Ab- +bott et al. 2016a) and GW190517 (Abbott et al. 2021c) confidently +possess spin. LVK data also support some mild evidence for spin- +orbit misalignment (e.g., Tiwari et al. 2018; Abbott et al. 2021d,c; +★ E-mail: caroleperigois@outlook.com (CP) +† E-mail:michela.mapelli@unipd.it +Venumadhav et al. 2020; Olsen et al. 2022; Callister et al. 2021, +2022). +These results provide crucial insights to understand BBH forma- +tion and evolution (e.g., Gerosa et al. 2013; Stevenson et al. 2015; +Rodriguez et al. 2016; Stevenson et al. 2017; Talbot & Thrane 2017; +Fishbach & Holz 2017; Vitale et al. 2017; Zevin et al. 2017; Farr +et al. 2018; Barrett et al. 2018; Taylor & Gerosa 2018; Arca Sedda & +Benacquista 2019; Roulet & Zaldarriaga 2019; Wysocki et al. 2019; +Bouffanais et al. 2019, 2021a,b; Kimball et al. 2020; Baibhav et al. +2020; Arca Sedda et al. 2020; Zevin et al. 2021; Mapelli et al. 2021, +2022). Moreover, the mass and spin of BHs (BHs) carry the memory +of their progenitor stars and therefore are a key to unravel the details +of massive star evolution and collapse (e.g., Fryer & Kalogera 2001; +Heger et al. 2003; Belczynski et al. 2010; Mapelli et al. 2013; Fragos +& McClintock 2015; Marchant et al. 2016; Eldridge & Stanway 2016; +de Mink & Mandel 2016; Spera & Mapelli 2017; Bavera et al. 2020; +Belczynski et al. 2020; Mandel et al. 2021; Fryer et al. 2022; Olejak +et al. 2022; Chattopadhyay et al. 2022; van Son et al. 2022; Briel +et al. 2022; Stevenson & Clarke 2022; Broekgaarden et al. 2022a,b). +In particular, the spin magnitude of a stellar-origin BH should retain +the imprint of the spin of the core of its progenitor star (e.g., Qin +et al. 2018, 2019; Fuller & Ma 2019; Bavera et al. 2020; Belczynski +et al. 2020; Olejak & Belczynski 2021; Stevenson 2022). +Several models have been proposed to infer the spin magnitude +of the BH from that of the progenitor star. The main open question +concerns the efficiency of angular momentum transport within a star +(e.g., Maeder & Meynet 2000; Cantiello et al. 2014; Fuller et al. +2019). If angular momentum is efficiently transferred from the core +to the outer layers, mass loss by stellar winds can dissipate most of +© 2015 The Authors +arXiv:2301.01312v1 [astro-ph.HE] 3 Jan 2023 + +2 +C. Périgois et al. +it, leading to a low-spinning stellar core and then to a low-spinning +BH. If instead the core retains most of its initial angular momentum +until the final collapse, the BH will be fast spinning. +In the shellular model (Zahn 1992; Ekström et al. 2012; Limongi +& Chieffi 2018; Costa et al. 2019), angular momentum is mainly +transported by meridional currents and shear instabilities, leading to +relatively inefficient spin dissipation. In contrast, according to the +Tayler-Spruit dynamo mechanism (Spruit 2002), differential rotation +induces the formation of an unstable magnetic field configuration, +leading to an efficient transport of angular momentum via magnetic +torques. Building upon the Tayler-Spruit mechanism, Fuller & Ma +(2019) derived a new model with an even more efficient angular +momentum dissipation, predicting that the core of a single massive +star might end its life with almost no rotation. +Electromagnetic observations yield controversial results. Aster- +oseismology favours slowly rotating cores in the late evolutionary +stages, but the vast majority of stars with an asteroseismic esti- +mate of the spin are low-mass stars (Mosser et al. 2012; Gehan +et al. 2018; Aerts et al. 2019). Continuum-fitting derived spins of +BHs in high-mass X-ray binaries are extremely high (e.g., Reynolds +2021; Miller-Jones et al. 2021; Fishbach & Kalogera 2022), but such +measurements might be affected by substantial observational biases +(e.g., Reynolds 2021). Finally, BH spins inferred from quasi peri- +odic oscillations yield notably smaller values than continuum fitting. +For example, the estimate of the dimensionless spin of the BH in +GRO J1655–40 is 𝜒 = 0.7 ± 0.1 and 0.290 ± 0.003 from continuum +fitting (Shafee et al. 2006) and quasi-periodic oscillations (Motta +et al. 2014), respectively. +In a binary system, the evolution of the spin is further affected +by tidal forces and accretion, which tend to spin up a massive star, +whereas non-conservative mass transfer and common-envelope ejec- +tion enhance mass loss, leading to more efficient spin dissipation +(Kushnir et al. 2016; Hotokezaka & Piran 2017; Zaldarriaga et al. +2018; Qin et al. 2018). For example, the model by Bavera et al. +(2020) shows that the second-born BH can be highly spinning if its +progenitor was tidally spin up when it was a Wolf-Rayet star orbiting +about the first-born BH. +Furthermore, the orientation of the BH spin with respect to the +orbital angular momentum of the binary system encodes information +about binary evolution processes. In a tight binary system, tides and +mass transfer tend to align the stellar spins with the orbital angular +momentum (Gerosa et al. 2018, but see Stegmann & Antonini 2021 +for a possible spin flip process induced by mass transfer). If the binary +system is in the field, the supernova kick is the main mechanism that +can misalign the spin of a compact object with respect to the orbital +angular momentum, by tilting the orbital plane (e.g., Kalogera 2000). +Finally, the spins of BHs in dynamically formed binary systems are +expected to be isotropically distributed, because close encounters in +a dense stellar cluster reset any previous signature of alignment (e.g., +Rodriguez et al. 2016; Mapelli et al. 2021). +Here, we perform a model-selection hierarchical Bayesian analysis +on confident LVK BBHs (𝑝astro > 0.9 and 𝐹𝐴𝑅 < 0.25 yr−1). We +consider models of field BBHs for three of the most used angular- +momentum transport models: (i) the shellular model as implemented +in the Geneva stellar evolution code (Ekström et al. 2012), (ii) +the Tayler-Spruit dynamo model as implemented in the mesa code +(Cantiello et al. 2014), and (iii) the model by Fuller & Ma (2019). +Hereafter, we will refer to these three models simply as GENEVA +(G), MESA (M) and FULLER (F) models, following the description +in Belczynski et al. (2020). +For each of these models, we consider an additional variation +accounting for the Wolf-Rayet (WR) star tidal spin-up mechanism +described by Bavera et al. (2020). Also, we account for spin tilts +induced by core-collapse supernova explosions. +This paper is organized as follows. Section 2 presents our +population-synthesis models. Section 3 describes the hierarchical +Bayesian framework we used and discusses the LVK events used in +our study. We lay down the results in Section 4, and summarize our +conclusions in Section 5. +2 ASTROPHYSICAL MODELS +2.1 mobse and natal kicks +We simulated our binary systems with the code mobse (Mapelli et al. +2017; Giacobbo et al. 2018). mobse is a custom and upgraded version +of bse (Hurley et al. 2000, 2002), in which we introduced metallicity- +dependent stellar winds for OB (Vink et al. 2001), WR (Belczynski +et al. 2010), and luminous blue variable stars (Giacobbo & Mapelli +2018). mobse includes a formalism for electron-capture (Giacobbo +& Mapelli 2019), core-collapse (Fryer et al. 2012), and (pulsational) +pair-instability supernovae (Mapelli et al. 2020). Here, we adopt the +rapid core-collapse supernova prescription, which enforces a gap +between the maximum mass of neutron stars and the minimum mass +of BHs (2–5 M⊙, Özel et al. 2010; Farr et al. 2011). +We model natal kicks of neutron stars and BHs according to three +different models, as shown in Fig. 1: +• A unified kick model, in which both neutron stars and BHs +receive a kick 𝑣kick ∝ 𝑚ej/𝑚rem, where 𝑚ej is the mass of the ejecta +and 𝑚rem the mass of the compact remnant (Giacobbo & Mapelli +2020, hereafter GM20). This model naturally produces low-kicks for +electron-capture, stripped and ultra-stripped supernovae (Tauris et al. +2015, 2017). Hereafter, we call this model GM20. +• A model in which compact-object kicks are drawn from a +Maxwellian curve with one-dimensional root-mean-square 𝜎 = 265 +km s−1, consistent with observations of Galactic pulsars (Hobbs et al. +2005). This realistically represents the upper limit for BH natal kicks. +Hereafter, we name this model 𝜎265. +• A model in which compact-object kicks are drawn from a +Maxwellian curve with 𝜎 = 150 km s−1. This value of 𝜎 is more +similar to what suggested from indirect measurements of Galactic +BH kicks (e.g., Repetto et al. 2017; Atri et al. 2019). Hereafter, we +refer to this model as 𝜎150. +For more details about mobse, see Giacobbo & Mapelli (2018). +mobse is an open-source code and can be downloaded from this link. +2.2 Spin magnitude +We have implemented four models for the spin magnitude in mobse, +the first three from Belczynski et al. (2020), and the fourth from +Bouffanais et al. (2019). Given the large uncertainties on angular +momentum transport, we do not claim that these four models are a +complete description of the underlying physics: our models must be +regarded as toy models, which bracket the current uncertainties on +BH spins. +MNRAS 000, 1–10 (2015) + +BBH spins: model selection with GWTC-3 +3 +0 +50 +100 +150 +200 +250 +300 +350 +400 +VCM [km s +1] +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +PDF +150 +265 +GM20 +Figure 1. Probability distribution function (PDF) of the binary kick velocities +in the centre of mass (𝑉CM), for our sample of simulated BBH mergers. The +centre-of-mass kick velocity takes into account both the first and the second +supernova event in each binary system (Perna et al. 2022). Dashed dark-cyan +line: model GM20; solid black line: 𝜎150; dotted red line: 𝜎265. This figure +only shows the kick velocity of the stellar progenitors of BBHs that merge +within the lifetime of the Universe. +𝑏 +𝑚1 (M⊙) +𝑚2 (M⊙) +𝑎low +𝑍 +2.58 +16.0 +24.2 +0.13 +≥ 0.010 +3.578 +31.0 +37.8 +0.25 +[0.004, 0.010) +2.434 +18.0 +27.7 +0.0 +[0.0012, 0.004) +3.666 +32.0 +38.8 +0.25 +< 0.0012 +Table 1. Parameters adopted in model G. See Eq. 1 for details. +2.2.1 Geneva (G) model +In the Geneva (hereafer, G) model, the dimensionless natal spin +magnitude of a BH (𝜒) can be approximated as: +𝜒 = +���� +���� +0.85 +𝑀CO ≤ 𝑚1 +𝑎 𝑀CO + 𝑏 +𝑚1 < 𝑀CO < 𝑚2 +𝑎low +𝑀CO, ≥ 𝑚2 +(1) +where 𝑎 = −0.088 for all models, 𝑀CO is the final carbon-oxygen +mass of the progenitor star, while the values of 𝑏, 𝑚1, 𝑚2, and 𝑎low +depend on metallicity, as indicated in Table 1. This model springs +from a fit by Belczynski et al. (2020) to some evolutionary tracks by +the Geneva group (Ekström et al. 2012), in which angular momentum +transport is relatively inefficient. +2.2.2 MESA (M) model +In the M model, we use the fits done by Belczynski et al. (2020) to a +set of stellar tracks run with the mesa code. mesa models the trans- +port of angular momentum according to the Tayler-Spruit magnetic +dynamo (Spruit 2002, see also Cantiello et al. 2014). This yields a +dimensionless natal BH spin +𝜒 = +� +𝑎1 𝑀CO + 𝑏1 +if 𝑀CO ≤ 𝑚1 +𝑎2 𝑀CO + 𝑏2 +if 𝑀CO > 𝑚1, +(2) +𝑎1 +𝑏1 +𝑎2 +𝑏2 +𝑚1 (M⊙) +𝑍 +−0.0016 +0.115 +– +– +∞ +≥ 0.010 +−0.0006 +0.105 +– +– +∞ +[0.004, 0.010) +0.0076 +0.050 +−0.0019 +0.165 +12.09 +[0.0012, 0.004) +−0.0010 +0.125 +– +– +∞ +≤ 0.0012 +Table 2. Parameters adopted in model M. See Eq. 2 for details. +where 𝑎1, 𝑏1, and 𝑚1 are given in Table 2. +2.2.3 Fuller (F) model +Fuller & Ma (2019) predict that angular momentum transport can +be even more efficient than the one predicted by the Tayler-Spruit +dynamo. Belczynski et al. (2020) summarize the results of the model +by Fuller & Ma (2019) simply as 𝜒 = 0.01 for all single stars and +metallicities. +2.2.4 Maxwellian model (Max) +Finally, we also introduce a toy model in which we represent the +spin of a BH as a random number drawn from a Maxwellian curve +with one-dimensional root-means square 𝜎𝜒 = 0.1 and truncated +to 𝜒max = 1.0. This model has been first introduced by Bouffanais +et al. (2019), because it is a good match to the distribution arising +from LVK data (e.g., Abbott et al. 2019, 2021d,c). Hereafter, we will +indicate this Maxwellian toy model as Max, for brevity. +2.3 Tidal spin up +The progenitor star of the second-born BH can be substantially +spun-up by tidal interactions. In the scenario explored by Bavera +et al. (2020), a common-envelope or an efficient stable mass transfer +episode can lead to the formation of a BH–WR binary system, in +which the WR star is the result of mass stripping. The orbital period +of this BH–WR binary system can be sufficiently short to lead to +efficient tidal synchronisation and spin-orbit coupling. The WR star +is then efficiently spun-up. If the WR star then collapses to a BH +directly, the final spin of the BH will retain the imprint of the final +WR spin. +Based on the simulations by Bavera et al. (2020), Bavera et al. +(2021) derive a fitting formula to describe the spin-up of the WR star +and the final spin of the second-born BH: +𝜒 = +� +𝛼WR log2 +10 (𝑃/[day]) + 𝛽WR log10 (𝑃/day) +if𝑃 ≤ 1 d +0 +otherwise, +(3) +where 𝑃 is the orbital period of the BH–WR sytem, 𝛼WR = +𝑓 +� +𝑀WR, 𝑐𝛼 +1 , 𝑐𝛼 +2 , 𝑐𝛼 +3 +� +and 𝛽WR = 𝑓 +� +𝑀𝑊 𝑅, 𝑐𝛽 +1 , 𝑐𝛽 +2 , 𝑐𝛽 +3 +� +. In this def- +inition, +𝑓 (𝑀WR, 𝑐1, 𝑐2, 𝑐3) = +−𝑐1 +𝑐2 + exp (−𝑐3𝑀WR/[M⊙]) , +(4) +where 𝑀WR is the mass of the WR star, while the coefficients 𝑐1, +𝑐2 and 𝑐3 have been determined through non-linear least-square +minimization and can be found in Bavera et al. (2021). +In mobse, we can use these fits for the spin of the second-born +BH, while still adopting one of the models presented in the previous +subsections (G, M, F, and Max) for the first-born BH. +MNRAS 000, 1–10 (2015) + +4 +C. Périgois et al. +Model Name +Spin Magnitude𝑎 +B21𝑏 +Kick Model𝑐 +G +Geneva (G) +no +GM20, 𝜎265, 𝜎150 +G_B21 +Geneva (G) +yes +GM20, 𝜎265, 𝜎150 +M +MESA (M) +no +GM20, 𝜎265, 𝜎150 +M_B21 +MESA (M) +yes +GM20, 𝜎265, 𝜎150 +F +Fuller (F) +no +GM20, 𝜎265, 𝜎150 +F_B21 +Fuller (F) +yes +GM20, 𝜎265, 𝜎150 +Max +Maxwellian (Max) +no +GM20, 𝜎265, 𝜎150 +Max_B21 +Maxwellian (Max) +yes +GM20, 𝜎265, 𝜎150 +Table 3. Description of the runs performed for this work. 𝑎Model for the spin +magnitude (Section 2.2). 𝑏Correction of the spin magnitude accounting for +tidal spin up, as described in B21 (Section 2.3). 𝑐Model for the natal kick +(Section 2.1). +2.4 Spin orientation +We assume that natal kicks are the only source of misalignment +between the orbital angular momentum vector of the binary system +and the direction of BH spins (Rodriguez et al. 2016; Gerosa et al. +2018). Furthermore, we conservatively assume that accretion onto the +first-born BH cannot change the direction of its spin (Maccarone et al. +2007). For simplicity, we also neglect the spin-flip process recently +described by (Stegmann & Antonini 2021). Under such assumptions, +we can derive the angle between the direction of the spins of the two +compact objects and that of the orbital angular momentum of the +binary system as (Gerosa et al. 2013; Rodriguez et al. 2016) +cos 𝛿 = cos (𝜈1) cos (𝜈2) + sin (𝜈1) sin (𝜈2) cos (𝜙), +(5) +where 𝜈𝑖 is the angle between the new (�𝐿new) and the old (�𝐿old) +orbital angular momentum after a supernova (𝑖 = 1, 2 corre- +sponding to the first and second supernova), so that cos (𝜈) = +�𝐿new · �𝐿old/(𝐿new 𝐿old), while 𝜙 is the phase of the projection of the +orbital angular momentum into the orbital plane. +2.5 Setup of mobse runs +Hereafter, we consider eight possible models for the spins (see also +Table 3): +• the first four models (hereafter, G, M, F, and Max) adopt the +Geneva, Mesa, Fuller and Maxwellian models for both the first- and +second-born BHs, +• the other four models (hereafter, G_B21, M_B21, F_B21, and +Max_B21) adopt the fits by Bavera et al. (2021) for the second-born +BH and the Geneva, Mesa, Fuller and Maxwellian models for the +first-born BH. +For each of these eight spin models we consider three different kick +models: the GM20, 𝜎265, and 𝜎150 models discussed in Section 2.1. +Finally, for each of these 24 models, we considered 12 metallicities +(𝑍 = 0.0002, 0.0004, 0.0008, 0.0012, 0.0016, 0.002, 0.004, 0.006, +0.008, 0.012, 0.016, and 0.02). For each metallicity, we ran 107 +(2 × 107) binary systems if 𝑍 ≤ 0.002 (𝑍 ≥ 0.004). Hence, for each +model we ran 1.8 × 108 binary systems, for a total of 4.32 × 109 +binary systems encompassing the eight models. +We sampled the initial conditions for each binary system as fol- +lows. We have randomly drawn the zero-age main sequence mass of +the primary stars from a Kroupa (Kroupa 2001) initial mass func- +tion in the range 5 − 150 M⊙. The initial orbital parameters (semi- +major axis, orbital eccentricity and mass ratio) of binary stars have +been randomly drawn as already described in Santoliquido et al. +(2021). In particular, we derive the mass ratios 𝑞 ≡ 𝑚2/𝑚1 (with +𝑚2 ≤ 𝑚1) as F (𝑞) ∝ 𝑞−0.1 with 𝑞 ∈ [0.1, 1], the orbital period 𝑃 +from F (Π) ∝ − 0.55 with Π = log10 (𝑃/d) ∈ [0.15, 5.5] and the +eccentricity 𝑒 from F (𝑒) ∝ 𝑒−0.42 with 0 ≤ 𝑒 ≤ 0.9 (Sana et al. +2012). +As to the main binary evolution parameters, here we use 𝛼 = 1 +for common envelope, while the parameter 𝜆 depends on the stel- +lar structure as described in Claeys et al. (2014). The other binary +evolution parameters are set-up as described in Santoliquido et al. +(2021). +2.6 Merger rate density +We estimate the evolution of BBH mergers with redshift by using +our semi-analytic code CosmoRate (Santoliquido et al. 2020, 2021). +With CosmoRate, we convolve our mobse catalogues (Section 2.5) +with an observation-based metallicity-dependent star formation rate +(SFR) density evolution of the Universe, SFRD(𝑧, 𝑍), in order to +estimate the merger rate density of BBHs as +RBBH(𝑧) = +∫ 𝑧 +𝑧max +�∫ 𝑍max +𝑍min +SFRD(𝑧′, 𝑍) F (𝑧′, 𝑧, 𝑍) d𝑍 +� d𝑡(𝑧′) +d𝑧′ +d𝑧′, +(6) +where +d𝑡(𝑧′) +d𝑧′ += [𝐻0 (1 + 𝑧′)]−1 [(1 + 𝑧′)3Ω𝑀 + ΩΛ]−1/2. +(7) +In the above equation, 𝐻0 is the Hubble constant, Ω𝑀 and ΩΛ are +the matter and energy density, respectively. We adopt the values in +Aghanim et al. (2020). The term F (𝑧′, 𝑧, 𝑍) is given by: +F (𝑧′, 𝑧, 𝑍) = +1 +MTOT(𝑍) +dN (𝑧′, 𝑧, 𝑍) +d𝑡(𝑧) +, +(8) +where MTOT(𝑍) is the total simulated initial stellar mass, and +dN (𝑧′, 𝑧, 𝑍)/d𝑡(𝑧) is the rate of BBHs forming from stars with ini- +tial metallicity 𝑍 at redshift 𝑧′ and merging at 𝑧, extracted from our +mobse catalogues. In CosmoRate, SFRD(𝑧, 𝑍) is given by +SFRD(𝑧′, 𝑍) = 𝜓(𝑧′) 𝑝(𝑧′, 𝑍), +(9) +where 𝜓(𝑧′) is the cosmic SFR density at formation redshift 𝑧′, and +𝑝(𝑧′, 𝑍) is the log-normal distribution of metallicities 𝑍 at fixed +formation redshift 𝑧′, with average 𝜇(𝑧′) and spread 𝜎𝑍. Here, we +take both 𝜓(𝑧) and 𝜇(𝑧) from Madau & Fragos (2017). Finally, we +assume a metallicity spread 𝜎𝑍 = 0.3. +2.7 Hyper-parametric model description +For each of our models (Table 3), described by their hyper-parameters +𝜆, we predict the distributions of BBH mergers +d𝑁 +d𝜃 (𝜆) = 𝑁𝜆 𝑝(𝜃|𝜆), +(10) +where 𝜃 are the merger parameters, and 𝑁𝜆 is the total number of +mergers predicted by the model. Assuming an instrumental horizon +redshift 𝑧max = 1.5, 𝑁𝜆 can be calculated as +𝑁𝜆 = +∫ 𝑧max +0 +R(𝑧) d𝑉c +d𝑧 +𝑇obs +(1 + 𝑧) d𝑧, +(11) +where d𝑉c +d𝑧 is the comoving volume and𝑇obs the observation duration. +To model the population of merging BBHs, we have chosen +five observable parameters 𝜃 = {Mc, 𝑞, 𝑧, 𝜒eff, 𝜒p}, where Mc = +(𝑚1 𝑚2)3/5/(𝑚1 + 𝑚2)1/5 is the chirp mass in the source frame with +𝑚1 (𝑚2) the masses of the primary (secondary) BH of the binary, +MNRAS 000, 1–10 (2015) + +BBH spins: model selection with GWTC-3 +5 +Figure 2. Predicted detectable distribution of chirp mass, for each kick model: +GM20 (solid dark-cyan line), 𝜎150 (dotted black line) and 𝜎265 (dashed +red line). For detectable distribution we mean the distribution of simulated +BBHs with sufficiently high signal-to-noise ratio (Section 3). The shaded gray +area is the distribution we obtain by stacking the posterior samples of the 59 +confident detections from GWTC-3 (Appendix A). +𝑞 = 𝑚2/𝑚1. and 𝑧 is the redshift of the merger. In addition, we used +two spin parameters: the effective spin (𝜒eff) and the precessing spin +(𝜒p). The effective spin 𝜒eff is the mass-weighted projection of the +two individual BH spins on the binary orbital angular momentum �𝐿 +𝜒eff = ( �𝜒1 + 𝑞 �𝜒2) +1 + 𝑞 +· +�𝐿 +𝐿 , +(12) +where �𝜒1,2 = �𝑠1,2 𝑐/(𝐺 𝑚2 +1,2) is the dimensionless spin parameter +of the two BHs. The precessing spin 𝜒p is defined as +𝜒p = max �𝜒1,⊥, 𝐴 𝜒2,⊥ +� , +(13) +where 𝜒1,⊥ (𝜒2,⊥) is the spin component of the primary (secondary) +BH perpendicular to the orbital angular momentum vector �𝐿, and +𝐴 = (4 𝑞 + 3) 𝑞/(4 + 3 𝑞). +To compute the distributions 𝑝(𝜃|𝜆), we constructed a catalogue of +106 sources for all possible combinations of hyper-parameters 𝜆, us- +ing the merger rate density and the metallicity given by CosmoRate. +From these catalogues we derived continuous estimations of 𝑝(𝜃|𝜆) +by making use of a Gaussian kernel density estimation assuming a +bandwidth of 0.15. +3 HIERARCHICAL BAYESIAN INFERENCE +Given a set H = {ℎ𝑘}𝑁obs +𝑘=1 of 𝑁obs GW observations, the posterior +distribution of a set of hyper-parameters 𝜆 associated to an astrophys- +ical model can be described as an in-homogeneous Poisson distribu- +tion (e.g., Loredo 2004; Mandel et al. 2019; Thrane & Talbot 2019; +Bouffanais et al. 2019, 2021a,b): +𝑝(𝜆, 𝑁𝜆|H) = 𝑒−𝜇𝜆 𝜋(𝜆, 𝑁𝜆) +𝑁obs +� +𝑘=1 +𝑁𝜆 +∫ +L𝑘 (ℎ𝑘 |𝜃) 𝑝(𝜃|𝜆) d𝜃, +(14) +where 𝑁obs is the number of events observed by the LVK, with an +ensemble of parameters 𝜃, 𝑁𝜆 is the number of predicted mergers +by the model (as calculated in eq. 11), 𝜇𝜆 the number of predicted +observations given a model and a detector, 𝜋(𝜆, 𝑁𝜆) are the prior +distributions on 𝜆 and 𝑁𝜆, and L𝑘 ({ℎ}𝑘 |𝜃) is the likelihood of the +𝑘th observation. +The predicted number of events 𝜇𝜆 can be written in terms of +detection efficiency 𝛽(𝜆) for a given model: +𝜇𝜆 = 𝑁𝜆 𝛽(𝜆), +with +𝛽(𝜆) = +∫ +𝜃 +𝑝(𝜃|𝜆) 𝑝det(𝜃) d𝜃, +(15) +where 𝑝det(𝜃) is the detection probability for a set of parameters +𝜃. This probability can be inferred by computing the optimal signal +to noise ratio (SNR) of the sources and comparing it to a detection +threshold. In our case we chose as reference a threshold 𝜌thr = 8 in +the LIGO Livingston detector, for which we approximated the sensi- +tivity using the measurements for the three runs separately (Abadie +et al. 2010; Abbott et al. 2016b; Wysocki et al. 2018). The values +for the event’s log-likelihood were derived from the posterior and +prior samples released by the LVK. Hence, the integral in eq. 14 is +approximated with a Monte Carlo approach as +I𝑘 = +∫ +L𝑘 (ℎ𝑘 |𝜃) 𝑝(𝜃|𝜆) d𝜃 ≈ +1 +𝑁𝑘𝑠 +𝑁 𝑘 +𝑠 +∑︁ +𝑖=1 +𝑝(𝜃𝑘 +𝑖 |𝜆) +𝜋𝑘 (𝜃𝑘 +𝑖 ) +, +(16) +where 𝜃𝑘 +𝑖 is the 𝑖th posterior sample of the 𝑘th detection and 𝑁𝑘𝑠 +is the total number of posterior samples for the 𝑘th detection. To +compute the prior term in the denominator, we also used Gaussian +kernel density estimation. +Finally, we can also choose to neglect the information coming +from the number of sources predicted by the model when estimating +the posterior distribution. By doing so, we can have some insights on +the impact of the rate on the analysis. In practice, this can be done by +marginalising eq. 14 over 𝑁𝜆 using a prior 𝜋(𝑁𝜆) ∼ 1/𝑁𝜆 (Fishbach +et al. 2018), which yields to the following expression for a model +log-likelihood +L = 𝑝(𝜆|{ℎ}𝑘) ∼ 𝜋(𝜆) +𝑁obs +� +𝑘=1 +� I𝑘 +𝛽(𝜆) +� +. +(17) +We adopted the formalism described in eqs. 14–17 to perform a +hierarchical Bayesian inference to compare the astrophysical models +presented Sec. 2 with the third gravitational-wave transient catalogue +(GWTC-3), the most updated catalogue of gravitational-wave events +from the LVK (Abbott et al. 2021b,c). GWTC-3 contains 90 event +candidates with probability of astrophysical origin 𝑝astro > 0.5. From +GWTC-3, we extract 59 confident detections of BBHs with a false +alarm rate FAR < 0.25 yr−1. In this sub-sample, we do not include +binary neutron stars and neutron star – BH systems, and we also +exclude the other BBH candidates with an higher FAR. Our chosen +FAR threshold ensures a sufficiently pure sample for our analysis +(Abbott et al. 2021c). A list of the events used in this study is available +in Appendix A. For the observable parameters 𝜃, we use the choice +described in Section 2.7, namely 𝜃 = {Mc, 𝑞, 𝑧, 𝜒eff, 𝜒p}. +4 RESULTS +4.1 Chirp mass +The chirp mass distribution (Fig. 2) does not depend on the spin +model, by construction. Therefore, we only show different natal kicks. +Models 𝜎150 and 𝜎265 show a similar distribution of chirp masses +with two peaks of similar importance, one at Mc ≈ 8 M⊙ and the +MNRAS 000, 1–10 (2015) + +0.175 +GM20 +g150 +0.150 +0265 +0.125 +LVKevents +0.100 +PDF +0.075 +0.050 +0.025 +0.000 +5 +10 +15 +20 +25 +30 +35 +40 +45 +Mc [Mo]6 +C. Périgois et al. +Figure 3. Predicted detectable distribution of 𝜒p (left) and 𝜒eff (right) for all of our models. Different colours refer to the spin model: G, M, F and Max. Solid +(dashed) lines include (do not include) the tidal spin-up model by B21. From top to bottom: GM20, 𝜎150, and 𝜎265. The shaded gray areas are the distributions +we obtain by stacking the posterior samples of the 59 confident detections from GWTC-3 (Appendix A). +other (broader) peak at Mc ≈ 15 M⊙. In contrast, model GM20 +has a much stronger preference for low-mass BHs, with a dominant +peak at Mc ≈ 8 M⊙. The reason for this difference is that all BHs in +tight binary systems receive slow natal kicks in model GM20 (Fig. 1). +This happens because stars in tight binary systems lose their envelope +during mass transfer episodes; hence, the mass of supernova ejecta +(𝑚ej) is small, triggering low kicks in model GM20. +Figure 2 also compares the detectable distribution of our models +with the stacked posterior samples from the confident BBH detections +in GWTC-3. This figure highlights two main differences between the +population synthesis models and the posterior samples: the peak +at Mc ≈ 15 M⊙ is stronger in the models than it is in the data, +while the data present a more significant excess at Mc ≈ 25 − 30 +M⊙ than the models. Finally, the peak at Mc ≈ 9 M⊙ in the data +approximately matches the peak at Mc ≈ 8 M⊙ in the models. The +main features of our population synthesis models (in particular, the +peaks at Mc ≈ 8 − 10 M⊙ and Mc ≈ 15 − 20 M⊙) are also common +to other population-synthesis models (e.g., Belczynski et al. 2020; +van Son et al. 2022) and mostly spring from the core-collapse SN +prescriptions by Fryer et al. (2012). Alternative core-collapse SN +models (e.g., Mapelli et al. 2020; Mandel et al. 2021; Patton et al. +MNRAS 000, 1–10 (2015) + +G +M +F +Max +LVK events +G B21 +M B21 +F B21 +Max B21 +GM20 +101 +PDF +10-1 +10-3 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9-1.00-0.75-0.50-0.25 +50.00 +0.25 +0.50 +0.75 +1.00 +Xp +Xeff +0150 +101 +10~3 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9-1.00-0.75-0.50-0.250.00 +0.25 +0.50 +0.75 +1.00 +Xp +Xeff +0265 +101 +10-3 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9-1.00-0.75-0.50-0.250.00 +0.25 +0.50 +0.75 +1.00 +Xp +XeffBBH spins: model selection with GWTC-3 +7 +2022; Olejak et al. 2022) produce different features and deserve +further investigation (Iorio et al., in prep.). +4.2 Spin parameters +Figure 3 shows the detectable distribution of spin parameters 𝜒p and +𝜒eff for all of our models. By construction, large spins are much more +common in models G and G_B21, while models F and F_B21 have a +strong predominance of vanishingly small spins. Models M, M_B21, +Max and Max_B21 are intermediate between the other two extreme +models. +Including or not the correction by B21 has negligible impact on +the distribution of 𝜒p and 𝜒eff for models G, because of the predom- +inance of large spin magnitudes. In contrast, introducing the spin-up +correction by B21 has a key impact on models F, because it is the +only way to account for mild to large spins in these models. The +correction by B21 is important also for models M and Max, being +responsible for the large-spin wings. +Finally, our model with slow kicks (GM20) results in a distribution +of 𝜒p that is more peaked at zero (for models G, M and Max) with +respect to the other two kick models (𝜎150 and 𝜎265). In fact, the +supernova kicks in model GM20 are not large enough to appreciably +misalign BH spins (see Fig. 1). +A similar effect is visible in the distribution of 𝜒eff: model 𝜎265 +produces a distribution of 𝜒eff that is less asymmetric about the zero +with respect to models 𝜎150 and especially GM20. +4.3 Model Selection +Figure 4 and Table 4 report the values of the log-likelihood log L +defined in Eq. 17. We can quantify the difference between two models +A and B by computing the average absolute difference in percentage +ΔlogL(A, B) = +� +2 +��logLA +𝑖 − logLB +𝑖 +�� +logLA +𝑖 + logLB +𝑖 +� +𝑣𝑎𝑟 +, +(18) +on the non-A,B variation 𝑣𝑎𝑟 (𝑣𝑎𝑟 would be kick(spin) if A and +B are spin(kick) models). For example to compare the two mod- +els G and G_B21, A and B become G_B21 and G and 𝑣𝑎𝑟 = +{GM20, 𝜎150, 𝜎265}. +The tidal spin-up mechanism (B21) affects the spin of a small +part of the population of each model (Fig. 3). However, it im- +proves the likelihood of the F and M models significantly (e.g., +ΔlogL(M_B21, M) = 89%, Table 4). This improvement of the log- +likelihood can be explained by the presence of higher values of 𝜒p and +𝜒eff in the distribution of populations M_B21 and F_B21 compared +to M and F (Fig. 3). +The F model yields L(F) = −∞ if we do not include the tidal +spin-up correction, regardless of the kick model. This indicates that +the LVK data do not support vanishingly small BH spins for the +entire BBH population. However, it is sufficient to inject a tiny sub- +population of spinning BHs, by switching on the B21 correction, +and the F model becomes one of the best considered models. In fact, +the F_B21 models only includes 0.4% of BHs with 𝜒 > 0.01 and +achieves log L > 200 (for spin models 𝜎150 and 𝜎265). +The G and G_B21 spin models exhibit lower log-likelihood values +than the others for all kicks models: logL ⩽ 150 for 𝜎150 and 𝜎265, +and logL < 0 for GM20. This happens because the distribution of +𝜒eff has non-negligible support for extreme values 𝜒eff < −0.5 and +𝜒eff > 0.5 (Fig. 3). +The +kick +models +𝜎150 +and +𝜎265 +show +similar +results +(ΔlogL(𝜎150, 𝜎265) < 3%) for every spin assumptions. Also, +Table 4. Log-likelihood L (Eq. 17) estimated with five merger parameters +𝜃 = +� +Mc , 𝑧 , 𝜒eff , 𝑞 , 𝜒p +� +. +Model Name +GM20 +𝜎150 +𝜎265 +G +-1 +149 +145 +G_B21 +-12 +150 +141 +M +0 +162 +171 +M_B21 +36 +232 +232 +F +-∞ +-∞ +-∞ +F_B21 +88 +250 +242 +Max +92 +255 +254 +Max_B21 +106 +257 +250 +Table 5. Log-likelihood L (Eq. 17) estimated with four merger parameters +𝜃 = {Mc , 𝑧 , 𝜒eff , 𝑞}. Here, we ignore 𝜒p. +Model Name +GM20 +𝜎150 +𝜎265 +G +35 +146 +147 +G_B21 +47 +149 +154 +M +141 +192 +190 +M_B21 +130 +199 +180 +F +85 +146 +138 +F_B21 +185 +207 +180 +Max +161 +208 +155 +Max_B21 +160 +206 +200 +for all spin assumptions, the GM20 kick model scores a signifi- +cantly lower likelihood than the other models 𝜎150 and 𝜎265 with +ΔlogL(𝜎150, GM20) ∼ ΔlogL(𝜎265, GM20) ∼ 150%. This re- +sult can be explained by the high peak of model GM20 at low chirp +masses (Mc ∼ 8M⊙, see Sec.4.1 and Fig.2) and by the low value of +𝜒p compared to the other kick models (Fig. 3). +Models Max and Max_B21 are possibly the best match to the data, +but this is not surprising, because they were built as a toy model to vi- +sually match the data. Among the astrophysically-motivated models +(i.e., after excluding the Max model), M, M_B21 and F_B21 (with +kick models 𝜎150 and 𝜎265) are the most favoured by the data. This +might be interpreted as a support for the Tayler-Spruit instability +mechanism (adopted in models M) and for the tidal spin-up model +by B21. +4.4 Importance of 𝜒p +The 𝜒p parameter encodes information on the spin component in the +orbital plane. Its impact on gravitational-wave signals is much lower +than that of 𝜒eff, and therefore its measurement is less precise. To +understand the impact of 𝜒p on our results, we re-ran the analysis +without this parameter. The results are shown in Table 5 and in Fig. 4 +with empty markers. Fig. 4 shows that, if we do not include 𝜒p, +the models M and M_B21 have almost the same log-likelihood, and +even the F model yields a positive log-likelihood. Furthermore, the +analysis without 𝜒p results in significantly larger values of L for the +kick model GM20. Our results demonstrate that the measured 𝜒p +of GWTC-3 BBHs carries substantial information, despite the large +uncertainties. +5 DISCUSSION +The spin magnitude of BHs is largely uncertain, mostly because +we do not fully understand angular momentum transport in massive +stars. Here, we have taken a number of spin models bracketing the +MNRAS 000, 1–10 (2015) + +8 +C. Périgois et al. +Figure 4. Values of the log-likelihood L defined in Eq. 17 for the four different models Geneva (G), MESA (M), Fuller (F), and Maxwellian (Max), with/without +the tidal spin-up mechanism (B21). Blue crosses: GM20; dark pluses: 𝜎150; red circles: 𝜎265. +main uncertainties, we have implemented them into our population- +synthesis code mobse, and compared them against GWTC-3 data +within a hierarchical Bayesian framework. +The data do not support models in which the entire BH population +has vanishingly small spins (model F). This result is mainly driven +by the 𝜒p parameter. This is in agreement with, e.g., the comple- +mentary analysis presented in Callister et al. (2022). They employed +a variety of complementary methods to measure the distribution of +spin magnitudes and orientations of BBH mergers, and concluded +that the existence of a sub-population of BHs with vanishing spins +is not required by current data. Callister et al. (2022) find that the +fraction of non-spinning BHs can comprise up to ∼ 60 − 70% of the +total population. In our F_B21 models, we have ∼ 99.6% of BHs +with 𝜒 < 0.01. +Recently, Roulet et al. (2021) and Galaudage et al. (2021) claimed +the existence of a sub-population of zero-spin BHs. From our anal- +ysis, we cannot exclude the existence of such sub-population, as the +F model with B21 correction (F_B21) still represents a good match +of the data. Similarly to Belczynski et al. (2020) and Gerosa et al. +(2018), we find that models with large spins (G, G_B21) are less +favoured by the data, but they are still acceptable if we allow for large +kicks. +Overall, we find a preference for large natal kicks. This result goes +into the same direction as the work by Callister et al. (2021). Actually, +this preference for large natal kicks is degenerate with the adopted +formation channel. Had we included the dynamical formation chan- +nel in dense star clusters, we would have added a sub-population of +isotropically oriented spins (see, e.g., Figure 8 of Mapelli et al. 2022). +In a forthcoming study, we will extend our analysis to a multi-channel +analysis. While it is unlikely that BBH mergers only originate from +one single channel, adding more formation channels to a hierarchical +Bayesian analysis dramatically increases the number of parameters, +making it more difficult to reject some portions of the parameter +space. +6 SUMMARY +The origin of BH spins is still controversial, and angular momen- +tum transport inside massive stars is one of the main sources of +uncertainty. Here, we apply hierarchical Bayesian inference to derive +constraints on spin models from the 59 most confident BBH merger +events in GWTC-3. We consider five parameters: chirp mass, mass +ratio, redshift, effective spin, and precessing spin. +For model selection, we use a set of binary population synthe- +sis simulations spanning different assumptions for black hole spins +and natal kicks. In particular, our spin models account for relatively +inefficient (G), efficient (Max and M), and very efficient angular- +momentum transport (F). A higher efficiency of angular momentum +transport is associated with lower BH spins. In particular, model F +predicts vanishingly small spins for the entire BH population. For +each of our models, we also include the possibility that some BHs +are tidally spun-up (B21). We considered three different natal kick +models: according to models 𝜎265 and 𝜎150, we randomly draw +the kicks from a Maxwellian curve with 𝜎 = 265 and 150 km s−1, +respectively; in the third model (G20), we also derive the kicks from +a Maxwellian curve with 𝜎 = 265 km s−1, but the kick magnitude is +then modulated by the ratio between the mass of the ejecta and the +mass of the BH. +We summarize our main results as follows. +• The data from GWTC-3 do not support models in which the +entire BH population has vanishingly small spins (model F). +• In contrast, models in which most spins are vanishingly small, +but that also include a sub-population of tidally spun-up BHs (model +F_B21) are a good match to the data. +• The models in which angular momentum transport is relatively +inefficient (G and G_21) yield log-likelihood values that are much +lower than models with efficient angular momentum transport (M, +M_B21, Max, and Max_B21). +• Models with large BH kicks (𝜎150 and 𝜎265 ) are favoured by +our analysis with respect to low-kick models (G20). +• Our results show that the precessing spin parameter 𝜒p plays a +crucial impact to constrain the spin distribution of BBH mergers. +ACKNOWLEDGEMENTS +MM, CP, FS and YB acknowledge financial support from the +European Research Council for the ERC Consolidator grant DE- +MOBLACK, under contract no. 770017. This research made use of +MNRAS 000, 1–10 (2015) + +300 +GM20 no Xp ++ +o150 no Xp +g265 no Xp +GM20 +g150 +0265 +250 +200 ++ ++ ++ +150 +L +100 +50 +X +0 +-50 +8 +8 +8 +-100 +M +F +Max +G_B21 +M_B21 +F B21 +Max_B21BBH spins: model selection with GWTC-3 +9 +NumPy (Harris et al. 2020), and SciPy (Virtanen et al. 2020). For +the plots we used Matplotlib (Hunter 2007). +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. The latest public version of mobse can be +downloaded from this repository. CosmoRate can be downlowaded +from this link. +REFERENCES +Aasi J., et al., 2015, Classical and Quantum Gravity, 32, 074001 +Abadie J., et al., 2010, Classical and Quantum Gravity, 27, 173001 +Abbott B. P., et al., 2016a, Phys. Rev. Lett., 116, 241103 +Abbott B. P., et al., 2016b, ApJ, 833, L1 +Abbott B. P., et al., 2019, Physical Review X, 9, 031040 +Abbott R., et al., 2020, Phys. Rev. D, 102, 043015 +Abbott R., et al., 2021a, arXiv e-prints, p. arXiv:2108.01045 +Abbott R., et al., 2021b, arXiv e-prints, p. arXiv:2111.03606 +Abbott R., et al., 2021c, arXiv e-prints, p. arXiv:2111.03634 +Abbott R., et al., 2021d, Physical Review X, 11, 021053 +Acernese F., et al., 2015, Classical and Quantum Gravity, 32, 024001 +Aerts C., Mathis S., Rogers T. M., 2019, ARA&A, 57, 35 +Aghanim N., et al., 2020, A&A, 641, A6 +Arca Sedda M., Benacquista M., 2019, MNRAS, 482, 2991 +Arca Sedda M., Mapelli M., Spera M., Benacquista M., Giacobbo N., 2020, +ApJ, 894, 133 +Atri P., et al., 2019, MNRAS, 489, 3116 +Baibhav V., Gerosa D., Berti E., Wong K. W. K., Helfer T., Mould M., 2020, +Phys. Rev. D, 102, 043002 +Barrett J. W., Gaebel S. M., Neijssel C. J., Vigna-Gómez A., Stevenson S., +Berry C. P. L., Farr W. M., Mandel I., 2018, MNRAS, 477, 4685 +Bavera S. S., et al., 2020, A&A, 635, A97 +Bavera S. S., Zevin M., Fragos T., 2021, Research Notes of the American +Astronomical Society, 5, 127 +Belczynski K., Bulik T., Fryer C. L., Ruiter A., Valsecchi F., Vink J. S., Hurley +J. R., 2010, ApJ, 714, 1217 +Belczynski K., et al., 2020, A&A, 636, A104 +Bouffanais Y., Mapelli M., Gerosa D., Di Carlo U. N., Giacobbo N., Berti E., +Baibhav V., 2019, ApJ, 886, 25 +Bouffanais Y., Mapelli M., Santoliquido F., Giacobbo N., Iorio G., Costa G., +2021a, MNRAS, 505, 3873 +Bouffanais Y., Mapelli M., Santoliquido F., Giacobbo N., Di Carlo U. N., +Rastello S., Artale M. C., Iorio G., 2021b, MNRAS, 507, 5224 +Briel M. M., Stevance H. F., Eldridge J. J., 2022, arXiv e-prints, p. +arXiv:2206.13842 +Broekgaarden F. S., et al., 2022a, MNRAS, 516, 5737 +Broekgaarden F. S., Stevenson S., Thrane E., 2022b, ApJ, 938, 45 +Callister T. A., Farr W. M., Renzo M., 2021, ApJ, 920, 157 +Callister T. A., Miller S. J., Chatziioannou K., Farr W. M., 2022, arXiv +e-prints, p. arXiv:2205.08574 +Cantiello M., Mankovich C., Bildsten L., Christensen-Dalsgaard J., Paxton +B., 2014, ApJ, 788, 93 +Chattopadhyay D., Hurley J., Stevenson S., Raidani A., 2022, MNRAS, 513, +4527 +Claeys J. S. W., Pols O. R., Izzard R. G., Vink J., Verbunt F. W. M., 2014, +A&A, 563, A83 +Costa G., Girardi L., Bressan A., Marigo P., Rodrigues T. S., Chen Y., Lanza +A., Goudfrooij P., 2019, MNRAS, 485, 4641 +Dall’Amico M., Mapelli M., Di Carlo U. N., Bouffanais Y., Rastello S., +Santoliquido F., Ballone A., Arca Sedda M., 2021, MNRAS, 508, 3045 +Di Carlo U. N., Giacobbo N., Mapelli M., Pasquato M., Spera M., Wang L., +Haardt F., 2019, MNRAS, 487, 2947 +Ekström S., et al., 2012, A&A, 537, A146 +Eldridge J. J., Stanway E. R., 2016, MNRAS, 462, 3302 +Farr W. M., Sravan N., Cantrell A., Kreidberg L., Bailyn C. D., Mandel I., +Kalogera V., 2011, ApJ, 741, 103 +Farr W. M., Stevenson S., Miller M. C., Mandel I., Farr B., Vecchio A., 2017, +Nature, 548, 426 +Farr B., Holz D. E., Farr W. M., 2018, ApJ, 854, L9 +Fishbach M., Holz D. E., 2017, ApJ, 851, L25 +Fishbach M., Kalogera V., 2022, ApJ, 929, L26 +Fishbach M., Holz D. E., Farr W. M., 2018, The Astrophysical Journal, 863, +L41 +Fragos T., McClintock J. E., 2015, ApJ, 800, 17 +Fryer C. L., Kalogera V., 2001, ApJ, 554, 548 +Fryer C. L., Belczynski K., Wiktorowicz G., Dominik M., Kalogera V., Holz +D. E., 2012, ApJ, 749, 91 +Fryer C. L., Olejak A., Belczynski K., 2022, ApJ, 931, 94 +Fuller J., Ma L., 2019, ApJ, 881, L1 +Fuller J., Piro A. L., Jermyn A. S., 2019, MNRAS, 485, 3661 +Galaudage S., Talbot C., Nagar T., Jain D., Thrane E., Mandel I., 2021, ApJ, +921, L15 +Gehan C., Mosser B., Michel E., Samadi R., Kallinger T., 2018, A&A, 616, +A24 +Gerosa D., Kesden M., Berti E., O’Shaughnessy R., Sperhake U., 2013, Phys. +Rev. D, 87, 104028 +Gerosa D., Berti E., O’Shaughnessy R., Belczynski K., Kesden M., Wysocki +D., Gladysz W., 2018, Phys. Rev. D, 98, 084036 +Giacobbo N., Mapelli M., 2018, MNRAS, 480, 2011 +Giacobbo N., Mapelli M., 2019, MNRAS, 482, 2234 +Giacobbo N., Mapelli M., 2020, ApJ, 891, 141 +Giacobbo N., Mapelli M., Spera M., 2018, MNRAS, 474, 2959 +Harris C. R., et al., 2020, Nature, 585, 357 +Heger A., Fryer C. L., Woosley S. E., Langer N., Hartmann D. H., 2003, ApJ, +591, 288 +Hobbs G., Lorimer D. R., Lyne A. G., Kramer M., 2005, MNRAS, 360, 974 +Hotokezaka K., Piran T., 2017, ApJ, 842, 111 +Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 +Hurley J. R., Pols O. R., Tout C. A., 2000, MNRAS, 315, 543 +Hurley J. R., Tout C. A., Pols O. R., 2002, MNRAS, 329, 897 +Kalogera V., 2000, ApJ, 541, 319 +Kimball C., Talbot C., Berry C. P. L., Carney M., Zevin M., Thrane E., +Kalogera V., 2020, ApJ, 900, 177 +Kroupa P., 2001, MNRAS, 322, 231 +Kushnir D., Zaldarriaga M., Kollmeier J. A., Waldman R., 2016, MNRAS, +462, 844 +Limongi M., Chieffi A., 2018, ApJS, 237, 13 +Loredo T. J., 2004, in Fischer R., Preuss R., Toussaint U. V., eds, American +Institute of Physics Conference Series Vol. 735, Bayesian Inference and +Maximum Entropy Methods in Science and Engineering: 24th Interna- +tional Workshop on Bayesian Inference and Maximum Entropy Methods +in Science and Engineering. pp 195–206 (arXiv:astro-ph/0409387), +doi:10.1063/1.1835214 +Maccarone T. J., Kundu A., Zepf S. E., Rhode K. L., 2007, Nature, 445, 183 +Madau P., Fragos T., 2017, ApJ, 840, 39 +Maeder A., Meynet G., 2000, ARA&A, 38, 143 +Mandel I., Farr W. M., Gair J. R., 2019, MNRAS, 486, 1086 +Mandel I., Müller B., Riley J., de Mink S. E., Vigna-Gómez A., Chattopadhyay +D., 2021, MNRAS, 500, 1380 +Mapelli M., Zampieri L., Ripamonti E., Bressan A., 2013, MNRAS, 429, +2298 +Mapelli M., Giacobbo N., Ripamonti E., Spera M., 2017, MNRAS, 472, 2422 +Mapelli M., Spera M., Montanari E., Limongi M., Chieffi A., Giacobbo N., +Bressan A., Bouffanais Y., 2020, ApJ, 888, 76 +Mapelli M., et al., 2021, MNRAS, 505, 339 +Mapelli M., Bouffanais Y., Santoliquido F., Arca Sedda M., Artale M. C., +2022, MNRAS, 511, 5797 +Marchant P., Langer N., Podsiadlowski P., Tauris T. M., Moriya T. J., 2016, +A&A, 588, A50 +Miller-Jones J. C. A., et al., 2021, Science, 371, 1046 +Miller S., Callister T. A., Farr W. M., 2020, ApJ, 895, 128 +MNRAS 000, 1–10 (2015) + +10 +C. Périgois et al. +Mosser B., et al., 2012, A&A, 548, A10 +Motta S. E., Belloni T. M., Stella L., Muñoz-Darias T., Fender R., 2014, +MNRAS, 437, 2554 +Olejak A., Belczynski K., 2021, ApJ, 921, L2 +Olejak A., Fryer C. L., Belczynski K., Baibhav V., 2022, MNRAS, 516, 2252 +Olsen S., Venumadhav T., Mushkin J., Roulet J., Zackay B., Zaldarriaga M., +2022, Phys. Rev. D, 106, 043009 +Özel F., Psaltis D., Narayan R., McClintock J. E., 2010, ApJ, 725, 1918 +Patton R. A., Sukhbold T., Eldridge J. J., 2022, MNRAS, 511, 903 +Perna R., Artale M. C., Wang Y.-H., Mapelli M., Lazzati D., Sgalletta C., +Santoliquido F., 2022, MNRAS, 512, 2654 +Qin Y., Fragos T., Meynet G., Andrews J., Sørensen M., Song H. F., 2018, +A&A, 616, A28 +Qin Y., Fragos T., Meynet G., Marchant P., Kalogera V., Andrews J., Sørensen +M., Song H. F., 2019, IAU Symposium, 346, 426 +Repetto S., Igoshev A. P., Nelemans G., 2017, MNRAS, 467, 298 +Reynolds C. S., 2021, ARA&A, 59, 117 +Rodriguez C. L., Zevin M., Pankow C., Kalogera V., Rasio F. A., 2016, ApJ, +832, L2 +Roulet J., Zaldarriaga M., 2019, MNRAS, 484, 4216 +Roulet J., Chia H. S., Olsen S., Dai L., Venumadhav T., Zackay B., Zaldarriaga +M., 2021, Phys. Rev. D, 104, 083010 +Sana H., et al., 2012, Science, 337, 444 +Santoliquido F., Mapelli M., Bouffanais Y., Giacobbo N., Di Carlo U. N., +Rastello S., Artale M. C., Ballone A., 2020, ApJ, 898, 152 +Santoliquido F., Mapelli M., Giacobbo N., Bouffanais Y., Artale M. C., 2021, +MNRAS, 502, 4877 +Shafee R., McClintock J. E., Narayan R., Davis S. W., Li L.-X., Remillard +R. A., 2006, ApJ, 636, L113 +Spera M., Mapelli M., 2017, MNRAS, 470, 4739 +Spruit H. C., 2002, A&A, 381, 923 +Stegmann J., Antonini F., 2021, Phys. Rev. D, 103, 063007 +Stevenson S., 2022, ApJ, 926, L32 +Stevenson S., Clarke T. A., 2022, MNRAS, +Stevenson S., Ohme F., Fairhurst S., 2015, ApJ, 810, 58 +Stevenson S., Berry C. P. L., Mandel I., 2017, MNRAS, 471, 2801 +Talbot C., Thrane E., 2017, Phys. Rev. D, 96, 023012 +Tauris T. M., Langer N., Podsiadlowski P., 2015, MNRAS, 451, 2123 +Tauris T. M., et al., 2017, ApJ, 846, 170 +Taylor S. R., Gerosa D., 2018, Phys. Rev. D, 98, 083017 +Thrane E., Talbot C., 2019, Publ. Astron. Soc. Australia, 36, e010 +Tiwari V., Fairhurst S., Hannam M., 2018, ApJ, 868, 140 +Venumadhav T., Zackay B., Roulet J., Dai L., Zaldarriaga M., 2020, Phys. +Rev. D, 101, 083030 +Vink J. S., de Koter A., Lamers H. J. G. L. M., 2001, A&A, 369, 574 +Virtanen P., et al., 2020, Nature Methods, 17, 261 +Vitale S., Lynch R., Sturani R., Graff P., 2017, Classical and Quantum Gravity, +34, 03LT01 +Wysocki D., Gerosa D., O’Shaughnessy R., Belczynski K., Gladysz W., Berti +E., Kesden M., Holz D. E., 2018, Phys. Rev. D, 97, 043014 +Wysocki D., Lange J., O’Shaughnessy R., 2019, Phys. Rev. D, 100, 043012 +Zahn J. P., 1992, A&A, 265, 115 +Zaldarriaga M., Kushnir D., Kollmeier J. A., 2018, MNRAS, 473, 4174 +Zevin M., Pankow C., Rodriguez C. L., Sampson L., Chase E., Kalogera V., +Rasio F. A., 2017, ApJ, 846, 82 +Zevin M., et al., 2021, ApJ, 910, 152 +de Mink S. E., Mandel I., 2016, MNRAS, 460, 3545 +van Son L. A. C., de Mink S. E., Chruslinska M., Conroy C., Pakmor R., +Hernquist L., 2022, arXiv e-prints, p. arXiv:2209.03385 +APPENDIX A: SAMPLE OF GRAVITATIONAL-WAVE +EVENTS +Table A1 lists all the gravitational-wave event candidates we used in +our study. From GWTC-3, we selected all the event candidates with +𝑝astro > 0.9 and FAR< 0.25 yr−1, excluding the following three +systems: +• the binary neutron star GW170807; +• the (possible) neutron star–BH binary system GW190814; +• the BBH GW190521 (𝑚1 = 98.4+33.6 +−21.7M⊙, 𝑚2 = 57.2+27.1 +−30.1M⊙ +Abbott et al. 2021b), which can form only via dynamical interactions +in our models (e.g., Di Carlo et al. 2019; Dall’Amico et al. 2021; +Mapelli et al. 2021). +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–10 (2015) + +BBH spins: model selection with GWTC-3 +11 +Table A1. Catalogue of BBH events adopted in this study +Name +Mc [M⊙] +q +𝜒eff +𝜒p +z +GW150914 +28.6+1.7 +−1.5 +0.86+0.12 +−0.2 +-0.01+0.12 +−0.13 +0.34+0.45 +−0.25 +0.09+0.03 +−0.03 +GW151012 +15.2+2.1 +−1.2 +0.59+0.36 +−0.35 +0.05+0.31 +−0.2 +0.33+0.45 +−0.25 +0.21+0.09 +−0.09 +GW151226 +8.9+0.3 +−0.3 +0.56+0.38 +−0.33 +0.18+0.2 +−0.12 +0.49+0.39 +−0.32 +0.09+0.04 +−0.04 +GW170104 +21.4+2.2 +−1.8 +0.65+0.3 +−0.23 +-0.04+0.17 +−0.21 +0.36+0.42 +−0.27 +0.2+0.08 +−0.08 +GW170608 +7.9+0.2 +−0.2 +0.69+0.28 +−0.36 +0.03+0.19 +−0.07 +0.36+0.45 +−0.27 +0.07+0.02 +−0.02 +GW170729 +35.4+6.5 +−4.8 +0.68+0.28 +−0.28 +0.37+0.21 +−0.25 +0.44+0.35 +−0.28 +0.49+0.19 +−0.21 +GW170809 +24.9+2.1 +−1.7 +0.68+0.28 +−0.24 +0.08+0.17 +−0.17 +0.35+0.43 +−0.26 +0.2+0.05 +−0.07 +GW170814 +24.1+1.4 +−1.1 +0.83+0.15 +−0.23 +0.07+0.12 +−0.12 +0.48+0.41 +−0.36 +0.12+0.03 +−0.04 +GW170818 +26.5+2.1 +−1.7 +0.76+0.21 +−0.25 +-0.09+0.18 +−0.21 +0.49+0.37 +−0.34 +0.21+0.07 +−0.07 +GW170823 +29.2+4.6 +−3.6 +0.74+0.23 +−0.3 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+0.24+0.07 +−0.08 +GW200128_022011 +32.0+7.5 +−5.5 +0.8+0.18 +−0.3 +0.12+0.24 +−0.25 +0.57+0.34 +−0.4 +0.56+0.28 +−0.28 +GW200129_065458 +27.2+2.1 +−2.3 +0.85+0.12 +−0.41 +0.11+0.11 +−0.16 +0.52+0.42 +−0.37 +0.18+0.05 +−0.07 +GW200202_154313 +7.5+0.2 +−0.2 +0.72+0.24 +−0.31 +0.04+0.13 +−0.06 +0.28+0.4 +−0.22 +0.09+0.03 +−0.03 +GW200208_130117 +27.7+3.6 +−3.1 +0.73+0.23 +−0.29 +-0.07+0.22 +−0.27 +0.38+0.41 +−0.29 +0.4+0.15 +−0.14 +GW200209_085452 +26.7+6.0 +−4.2 +0.78+0.19 +−0.31 +-0.12+0.24 +−0.3 +0.51+0.39 +−0.37 +0.57+0.25 +−0.26 +GW200219_094415 +27.6+5.6 +−3.8 +0.77+0.21 +−0.32 +-0.08+0.23 +−0.29 +0.48+0.4 +−0.35 +0.57+0.22 +−0.22 +GW200224_222234 +31.1+3.2 +−2.6 +0.82+0.16 +−0.26 +0.1+0.15 +−0.15 +0.49+0.37 +−0.36 +0.32+0.08 +−0.11 +GW200225_060421 +14.2+1.5 +−1.4 +0.73+0.23 +−0.28 +-0.12+0.17 +−0.28 +0.53+0.34 +−0.38 +0.22+0.09 +−0.1 +GW200302_015811 +23.4+4.7 +−3.0 +0.53+0.36 +−0.2 +0.01+0.25 +−0.26 +0.37+0.45 +−0.28 +0.28+0.16 +−0.12 +GW200311_115853 +26.6+2.4 +−2.0 +0.82+0.16 +−0.27 +-0.02+0.16 +−0.2 +0.45+0.4 +−0.35 +0.23+0.05 +−0.07 +GW200316_215756 +8.8+0.6 +−0.6 +0.6+0.34 +−0.38 +0.13+0.27 +−0.1 +0.29+0.38 +−0.2 +0.22+0.08 +−0.08 +MNRAS 000, 1–10 (2015) + diff --git a/Q9AzT4oBgHgl3EQfW_zX/content/tmp_files/load_file.txt b/Q9AzT4oBgHgl3EQfW_zX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d1d5064b72ce3004633a4ac33e15d4c8a7d0741 --- /dev/null +++ b/Q9AzT4oBgHgl3EQfW_zX/content/tmp_files/load_file.txt @@ -0,0 +1,2283 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf,len=2282 +page_content='MNRAS 000, 1–10 (2015) Preprint 5 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0 Binary black hole spins: model selection with GWTC-3 Carole Périgois1,2,★, , Michela Mapelli1,2,3,†, , Filippo Santoliquido1,2,3, , Yann Bouffanais1,2, , and Roberta Rufolo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 1Physics and Astronomy Department Galileo Galilei, University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy 2INFN - Padova, Via Marzolo 8, I–35131 Padova, Italy 3INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The origin of the spins of stellar-mass black holes is still controversial, and angular momentum transport inside massive stars is one of the main sources of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Here, we apply hierarchical Bayesian inference to derive constraints on spin models from the 59 most confident binary black hole merger events in the third gravitational-wave transient catalogue (GWTC-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We consider up to five parameters: chirp mass, mass ratio, redshift, effective spin, and precessing spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For model selection, we use a set of binary population synthesis simulations spanning drastically different assumptions for black hole spins and natal kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In particular, our spin models range from maximal to minimal efficiency of angular momentum transport in stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We find that, if we include the precessing spin parameter into our analysis, models predicting only vanishingly small spins are in tension with GWTC-3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' On the other hand, models in which most spins are vanishingly small, but that also include a sub-population of tidally spun-up black holes are a good match to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Our results show that the precessing spin parameter has a crucial impact on model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Key words: black hole physics – gravitational waves – binaries: general – stars: black holes 1 INTRODUCTION The third observing run (O3) of the Advanced LIGO (Aasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2015) and Virgo (Acernese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2015) detectors has brought the number of compact binary merger observations up to 90 events with a prob- ability of astrophysical origin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019, 2021d,a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In particular, the 63 confident detections of binary black hole (BBH) mergers (with a false alarm rate FAR< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 yr−1) lead to more ac- curate constraints on the mass and spin distribution of these systems (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The intrinsic distribution of primary black hole (BH) masses in- ferred by the LIGO–Virgo–KAGRA collaboration (hereafter, LVK) shows several sub-structures, including a main peak at ≈ 10 M⊙, a secondary peak at ≈ 30 − 40 M⊙, and a long tail extending up to ∼ 80 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The inferred distribution of mass ratios has a strong preference for equal-mass systems, but several BBHs are confidently unequal-mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=',GW190517 Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Focusing on BH spins, we can safely exclude that all BHs are maximally spinning (Farr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Typical spin magnitudes in BBHs are small, with ∼ 50% of BHs having 𝜒 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Wysocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021d), although not all BHs in the LVK sample have zero spin (Roulet & Zaldarriaga 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For example, GW151226 (Ab- bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016a) and GW190517 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021c) confidently possess spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' LVK data also support some mild evidence for spin- orbit misalignment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021d,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' ★ E-mail: caroleperigois@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='com (CP) † E-mail:michela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='mapelli@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='it Venumadhav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Callister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' These results provide crucial insights to understand BBH forma- tion and evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gerosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Talbot & Thrane 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Fishbach & Holz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Vitale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Zevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Farr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Taylor & Gerosa 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Arca Sedda & Benacquista 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Roulet & Zaldarriaga 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Wysocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Bouffanais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019, 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Kimball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Baibhav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Arca Sedda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Zevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Moreover, the mass and spin of BHs (BHs) carry the memory of their progenitor stars and therefore are a key to unravel the details of massive star evolution and collapse (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fryer & Kalogera 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Fragos & McClintock 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Eldridge & Stanway 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' de Mink & Mandel 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Spera & Mapelli 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Olejak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Chattopadhyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' van Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Briel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Stevenson & Clarke 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Broekgaarden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In particular, the spin magnitude of a stellar-origin BH should retain the imprint of the spin of the core of its progenitor star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Fuller & Ma 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Olejak & Belczynski 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Stevenson 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Several models have been proposed to infer the spin magnitude of the BH from that of the progenitor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The main open question concerns the efficiency of angular momentum transport within a star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Maeder & Meynet 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Cantiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' If angular momentum is efficiently transferred from the core to the outer layers, mass loss by stellar winds can dissipate most of © 2015 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='01312v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='HE] 3 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Périgois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' it, leading to a low-spinning stellar core and then to a low-spinning BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' If instead the core retains most of its initial angular momentum until the final collapse, the BH will be fast spinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In the shellular model (Zahn 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Ekström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Limongi & Chieffi 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019), angular momentum is mainly transported by meridional currents and shear instabilities, leading to relatively inefficient spin dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In contrast, according to the Tayler-Spruit dynamo mechanism (Spruit 2002), differential rotation induces the formation of an unstable magnetic field configuration, leading to an efficient transport of angular momentum via magnetic torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Building upon the Tayler-Spruit mechanism, Fuller & Ma (2019) derived a new model with an even more efficient angular momentum dissipation, predicting that the core of a single massive star might end its life with almost no rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Electromagnetic observations yield controversial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Aster- oseismology favours slowly rotating cores in the late evolutionary stages, but the vast majority of stars with an asteroseismic esti- mate of the spin are low-mass stars (Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Gehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Continuum-fitting derived spins of BHs in high-mass X-ray binaries are extremely high (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Reynolds 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Fishbach & Kalogera 2022), but such measurements might be affected by substantial observational biases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Reynolds 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Finally, BH spins inferred from quasi peri- odic oscillations yield notably smaller values than continuum fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For example, the estimate of the dimensionless spin of the BH in GRO J1655–40 is 𝜒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='290 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='003 from continuum fitting (Shafee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2006) and quasi-periodic oscillations (Motta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2014), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In a binary system, the evolution of the spin is further affected by tidal forces and accretion, which tend to spin up a massive star, whereas non-conservative mass transfer and common-envelope ejec- tion enhance mass loss, leading to more efficient spin dissipation (Kushnir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hotokezaka & Piran 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Zaldarriaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For example, the model by Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020) shows that the second-born BH can be highly spinning if its progenitor was tidally spin up when it was a Wolf-Rayet star orbiting about the first-born BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Furthermore, the orientation of the BH spin with respect to the orbital angular momentum of the binary system encodes information about binary evolution processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In a tight binary system, tides and mass transfer tend to align the stellar spins with the orbital angular momentum (Gerosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018, but see Stegmann & Antonini 2021 for a possible spin flip process induced by mass transfer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' If the binary system is in the field, the supernova kick is the main mechanism that can misalign the spin of a compact object with respect to the orbital angular momentum, by tilting the orbital plane (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Finally, the spins of BHs in dynamically formed binary systems are expected to be isotropically distributed, because close encounters in a dense stellar cluster reset any previous signature of alignment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Here, we perform a model-selection hierarchical Bayesian analysis on confident LVK BBHs (𝑝astro > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='9 and 𝐹𝐴𝑅 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 yr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We consider models of field BBHs for three of the most used angular- momentum transport models: (i) the shellular model as implemented in the Geneva stellar evolution code (Ekström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2012), (ii) the Tayler-Spruit dynamo model as implemented in the mesa code (Cantiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2014), and (iii) the model by Fuller & Ma (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hereafter, we will refer to these three models simply as GENEVA (G), MESA (M) and FULLER (F) models, following the description in Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For each of these models, we consider an additional variation accounting for the Wolf-Rayet (WR) star tidal spin-up mechanism described by Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Also, we account for spin tilts induced by core-collapse supernova explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Section 2 presents our population-synthesis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Section 3 describes the hierarchical Bayesian framework we used and discusses the LVK events used in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We lay down the results in Section 4, and summarize our conclusions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2 ASTROPHYSICAL MODELS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 mobse and natal kicks We simulated our binary systems with the code mobse (Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Giacobbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' mobse is a custom and upgraded version of bse (Hurley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2000, 2002), in which we introduced metallicity- dependent stellar winds for OB (Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2001), WR (Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2010), and luminous blue variable stars (Giacobbo & Mapelli 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' mobse includes a formalism for electron-capture (Giacobbo & Mapelli 2019), core-collapse (Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2012), and (pulsational) pair-instability supernovae (Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Here, we adopt the rapid core-collapse supernova prescription, which enforces a gap between the maximum mass of neutron stars and the minimum mass of BHs (2–5 M⊙, Özel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Farr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We model natal kicks of neutron stars and BHs according to three different models, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 1: A unified kick model, in which both neutron stars and BHs receive a kick 𝑣kick ∝ 𝑚ej/𝑚rem, where 𝑚ej is the mass of the ejecta and 𝑚rem the mass of the compact remnant (Giacobbo & Mapelli 2020, hereafter GM20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This model naturally produces low-kicks for electron-capture, stripped and ultra-stripped supernovae (Tauris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2015, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hereafter, we call this model GM20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A model in which compact-object kicks are drawn from a Maxwellian curve with one-dimensional root-mean-square 𝜎 = 265 km s−1, consistent with observations of Galactic pulsars (Hobbs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This realistically represents the upper limit for BH natal kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hereafter, we name this model 𝜎265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A model in which compact-object kicks are drawn from a Maxwellian curve with 𝜎 = 150 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This value of 𝜎 is more similar to what suggested from indirect measurements of Galactic BH kicks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Repetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Atri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hereafter, we refer to this model as 𝜎150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For more details about mobse, see Giacobbo & Mapelli (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' mobse is an open-source code and can be downloaded from this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2 Spin magnitude We have implemented four models for the spin magnitude in mobse, the first three from Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020), and the fourth from Bouffanais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Given the large uncertainties on angular momentum transport, we do not claim that these four models are a complete description of the underlying physics: our models must be regarded as toy models, which bracket the current uncertainties on BH spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' MNRAS 000, 1–10 (2015) BBH spins: model selection with GWTC-3 3 0 50 100 150 200 250 300 350 400 VCM [km s 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='05 PDF 150 265 GM20 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Probability distribution function (PDF) of the binary kick velocities in the centre of mass (𝑉CM), for our sample of simulated BBH mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The centre-of-mass kick velocity takes into account both the first and the second supernova event in each binary system (Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Dashed dark-cyan line: model GM20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' solid black line: 𝜎150;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' dotted red line: 𝜎265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This figure only shows the kick velocity of the stellar progenitors of BBHs that merge within the lifetime of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 𝑏 𝑚1 (M⊙) 𝑚2 (M⊙) 𝑎low 𝑍 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='58 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='13 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='010 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='578 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='004, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='010) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='434 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0012, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='004) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='666 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0012 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Parameters adopted in model G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 Geneva (G) model In the Geneva (hereafer, G) model, the dimensionless natal spin magnitude of a BH (𝜒) can be approximated as: 𝜒 = ���� ���� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='85 𝑀CO ≤ 𝑚1 𝑎 𝑀CO + 𝑏 𝑚1 < 𝑀CO < 𝑚2 𝑎low 𝑀CO, ≥ 𝑚2 (1) where 𝑎 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='088 for all models, 𝑀CO is the final carbon-oxygen mass of the progenitor star, while the values of 𝑏, 𝑚1, 𝑚2, and 𝑎low depend on metallicity, as indicated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This model springs from a fit by Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020) to some evolutionary tracks by the Geneva group (Ekström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2012), in which angular momentum transport is relatively inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2 MESA (M) model In the M model, we use the fits done by Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020) to a set of stellar tracks run with the mesa code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' mesa models the trans- port of angular momentum according to the Tayler-Spruit magnetic dynamo (Spruit 2002, see also Cantiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This yields a dimensionless natal BH spin 𝜒 = � 𝑎1 𝑀CO + 𝑏1 if 𝑀CO ≤ 𝑚1 𝑎2 𝑀CO + 𝑏2 if 𝑀CO > 𝑚1, (2) 𝑎1 𝑏1 𝑎2 𝑏2 𝑚1 (M⊙) 𝑍 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='115 – – ∞ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='105 – – ∞ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='004, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='165 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='09 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0012, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='004) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='125 – – ∞ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0012 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Parameters adopted in model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' where 𝑎1, 𝑏1, and 𝑚1 are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='3 Fuller (F) model Fuller & Ma (2019) predict that angular momentum transport can be even more efficient than the one predicted by the Tayler-Spruit dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020) summarize the results of the model by Fuller & Ma (2019) simply as 𝜒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='01 for all single stars and metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='4 Maxwellian model (Max) Finally, we also introduce a toy model in which we represent the spin of a BH as a random number drawn from a Maxwellian curve with one-dimensional root-means square 𝜎𝜒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 and truncated to 𝜒max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This model has been first introduced by Bouffanais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2019), because it is a good match to the distribution arising from LVK data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019, 2021d,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hereafter, we will indicate this Maxwellian toy model as Max, for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='3 Tidal spin up The progenitor star of the second-born BH can be substantially spun-up by tidal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In the scenario explored by Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020), a common-envelope or an efficient stable mass transfer episode can lead to the formation of a BH–WR binary system, in which the WR star is the result of mass stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The orbital period of this BH–WR binary system can be sufficiently short to lead to efficient tidal synchronisation and spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The WR star is then efficiently spun-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' If the WR star then collapses to a BH directly, the final spin of the BH will retain the imprint of the final WR spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Based on the simulations by Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020), Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021) derive a fitting formula to describe the spin-up of the WR star and the final spin of the second-born BH: 𝜒 = � 𝛼WR log2 10 (𝑃/[day]) + 𝛽WR log10 (𝑃/day) if𝑃 ≤ 1 d 0 otherwise, (3) where 𝑃 is the orbital period of the BH–WR sytem, 𝛼WR = 𝑓 � 𝑀WR, 𝑐𝛼 1 , 𝑐𝛼 2 , 𝑐𝛼 3 � and 𝛽WR = 𝑓 � 𝑀𝑊 𝑅, 𝑐𝛽 1 , 𝑐𝛽 2 , 𝑐𝛽 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In this def- inition, 𝑓 (𝑀WR, 𝑐1, 𝑐2, 𝑐3) = −𝑐1 𝑐2 + exp (−𝑐3𝑀WR/[M⊙]) , (4) where 𝑀WR is the mass of the WR star, while the coefficients 𝑐1, 𝑐2 and 𝑐3 have been determined through non-linear least-square minimization and can be found in Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In mobse, we can use these fits for the spin of the second-born BH, while still adopting one of the models presented in the previous subsections (G, M, F, and Max) for the first-born BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' MNRAS 000, 1–10 (2015) 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Périgois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Model Name Spin Magnitude𝑎 B21𝑏 Kick Model𝑐 G Geneva (G) no GM20, 𝜎265, 𝜎150 G_B21 Geneva (G) yes GM20, 𝜎265, 𝜎150 M MESA (M) no GM20, 𝜎265, 𝜎150 M_B21 MESA (M) yes GM20, 𝜎265, 𝜎150 F Fuller (F) no GM20, 𝜎265, 𝜎150 F_B21 Fuller (F) yes GM20, 𝜎265, 𝜎150 Max Maxwellian (Max) no GM20, 𝜎265, 𝜎150 Max_B21 Maxwellian (Max) yes GM20, 𝜎265, 𝜎150 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Description of the runs performed for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 𝑎Model for the spin magnitude (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 𝑏Correction of the spin magnitude accounting for tidal spin up, as described in B21 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 𝑐Model for the natal kick (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='4 Spin orientation We assume that natal kicks are the only source of misalignment between the orbital angular momentum vector of the binary system and the direction of BH spins (Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Gerosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Furthermore, we conservatively assume that accretion onto the first-born BH cannot change the direction of its spin (Maccarone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For simplicity, we also neglect the spin-flip process recently described by (Stegmann & Antonini 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Under such assumptions, we can derive the angle between the direction of the spins of the two compact objects and that of the orbital angular momentum of the binary system as (Gerosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016) cos 𝛿 = cos (𝜈1) cos (𝜈2) + sin (𝜈1) sin (𝜈2) cos (𝜙), (5) where 𝜈𝑖 is the angle between the new (�𝐿new) and the old (�𝐿old) orbital angular momentum after a supernova (𝑖 = 1, 2 corre- sponding to the first and second supernova), so that cos (𝜈) = �𝐿new · �𝐿old/(𝐿new 𝐿old), while 𝜙 is the phase of the projection of the orbital angular momentum into the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5 Setup of mobse runs Hereafter, we consider eight possible models for the spins (see also Table 3): the first four models (hereafter, G, M, F, and Max) adopt the Geneva, Mesa, Fuller and Maxwellian models for both the first- and second-born BHs, the other four models (hereafter, G_B21, M_B21, F_B21, and Max_B21) adopt the fits by Bavera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021) for the second-born BH and the Geneva, Mesa, Fuller and Maxwellian models for the first-born BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For each of these eight spin models we consider three different kick models: the GM20, 𝜎265, and 𝜎150 models discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Finally, for each of these 24 models, we considered 12 metallicities (𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0004, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0008, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0012, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0016, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='004, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='006, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='008, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='012, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='016, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For each metallicity, we ran 107 (2 × 107) binary systems if 𝑍 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='002 (𝑍 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hence, for each model we ran 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='8 × 108 binary systems, for a total of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='32 × 109 binary systems encompassing the eight models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We sampled the initial conditions for each binary system as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We have randomly drawn the zero-age main sequence mass of the primary stars from a Kroupa (Kroupa 2001) initial mass func- tion in the range 5 − 150 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The initial orbital parameters (semi- major axis, orbital eccentricity and mass ratio) of binary stars have been randomly drawn as already described in Santoliquido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In particular, we derive the mass ratios 𝑞 ≡ 𝑚2/𝑚1 (with 𝑚2 ≤ 𝑚1) as F (𝑞) ∝ 𝑞−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 with 𝑞 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1, 1], the orbital period 𝑃 from F (Π) ∝ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='55 with Π = log10 (𝑃/d) ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='15, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5] and the eccentricity 𝑒 from F (𝑒) ∝ 𝑒−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='42 with 0 ≤ 𝑒 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='9 (Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' As to the main binary evolution parameters, here we use 𝛼 = 1 for common envelope, while the parameter 𝜆 depends on the stel- lar structure as described in Claeys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The other binary evolution parameters are set-up as described in Santoliquido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='6 Merger rate density We estimate the evolution of BBH mergers with redshift by using our semi-analytic code CosmoRate (Santoliquido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' With CosmoRate, we convolve our mobse catalogues (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5) with an observation-based metallicity-dependent star formation rate (SFR) density evolution of the Universe, SFRD(𝑧, 𝑍), in order to estimate the merger rate density of BBHs as RBBH(𝑧) = ∫ 𝑧 𝑧max �∫ 𝑍max 𝑍min SFRD(𝑧′, 𝑍) F (𝑧′, 𝑧, 𝑍) d𝑍 � d𝑡(𝑧′) d𝑧′ d𝑧′, (6) where d𝑡(𝑧′) d𝑧′ = [𝐻0 (1 + 𝑧′)]−1 [(1 + 𝑧′)3Ω𝑀 + ΩΛ]−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (7) In the above equation, 𝐻0 is the Hubble constant, Ω𝑀 and ΩΛ are the matter and energy density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We adopt the values in Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The term F (𝑧′, 𝑧, 𝑍) is given by: F (𝑧′, 𝑧, 𝑍) = 1 MTOT(𝑍) dN (𝑧′, 𝑧, 𝑍) d𝑡(𝑧) , (8) where MTOT(𝑍) is the total simulated initial stellar mass, and dN (𝑧′, 𝑧, 𝑍)/d𝑡(𝑧) is the rate of BBHs forming from stars with ini- tial metallicity 𝑍 at redshift 𝑧′ and merging at 𝑧, extracted from our mobse catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In CosmoRate, SFRD(𝑧, 𝑍) is given by SFRD(𝑧′, 𝑍) = 𝜓(𝑧′) 𝑝(𝑧′, 𝑍), (9) where 𝜓(𝑧′) is the cosmic SFR density at formation redshift 𝑧′, and 𝑝(𝑧′, 𝑍) is the log-normal distribution of metallicities 𝑍 at fixed formation redshift 𝑧′, with average 𝜇(𝑧′) and spread 𝜎𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Here, we take both 𝜓(𝑧) and 𝜇(𝑧) from Madau & Fragos (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Finally, we assume a metallicity spread 𝜎𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='7 Hyper-parametric model description For each of our models (Table 3), described by their hyper-parameters 𝜆, we predict the distributions of BBH mergers d𝑁 d𝜃 (𝜆) = 𝑁𝜆 𝑝(𝜃|𝜆), (10) where 𝜃 are the merger parameters, and 𝑁𝜆 is the total number of mergers predicted by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Assuming an instrumental horizon redshift 𝑧max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5, 𝑁𝜆 can be calculated as 𝑁𝜆 = ∫ 𝑧max 0 R(𝑧) d𝑉c d𝑧 𝑇obs (1 + 𝑧) d𝑧, (11) where d𝑉c d𝑧 is the comoving volume and𝑇obs the observation duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' To model the population of merging BBHs, we have chosen five observable parameters 𝜃 = {Mc, 𝑞, 𝑧, 𝜒eff, 𝜒p}, where Mc = (𝑚1 𝑚2)3/5/(𝑚1 + 𝑚2)1/5 is the chirp mass in the source frame with 𝑚1 (𝑚2) the masses of the primary (secondary) BH of the binary, MNRAS 000, 1–10 (2015) BBH spins: model selection with GWTC-3 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Predicted detectable distribution of chirp mass, for each kick model: GM20 (solid dark-cyan line), 𝜎150 (dotted black line) and 𝜎265 (dashed red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For detectable distribution we mean the distribution of simulated BBHs with sufficiently high signal-to-noise ratio (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The shaded gray area is the distribution we obtain by stacking the posterior samples of the 59 confident detections from GWTC-3 (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 𝑞 = 𝑚2/𝑚1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' and 𝑧 is the redshift of the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In addition, we used two spin parameters: the effective spin (𝜒eff) and the precessing spin (𝜒p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The effective spin 𝜒eff is the mass-weighted projection of the two individual BH spins on the binary orbital angular momentum �𝐿 𝜒eff = ( �𝜒1 + 𝑞 �𝜒2) 1 + 𝑞 �𝐿 𝐿 , (12) where �𝜒1,2 = �𝑠1,2 𝑐/(𝐺 𝑚2 1,2) is the dimensionless spin parameter of the two BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The precessing spin 𝜒p is defined as 𝜒p = max �𝜒1,⊥, 𝐴 𝜒2,⊥ � , (13) where 𝜒1,⊥ (𝜒2,⊥) is the spin component of the primary (secondary) BH perpendicular to the orbital angular momentum vector �𝐿, and 𝐴 = (4 𝑞 + 3) 𝑞/(4 + 3 𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' To compute the distributions 𝑝(𝜃|𝜆), we constructed a catalogue of 106 sources for all possible combinations of hyper-parameters 𝜆, us- ing the merger rate density and the metallicity given by CosmoRate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' From these catalogues we derived continuous estimations of 𝑝(𝜃|𝜆) by making use of a Gaussian kernel density estimation assuming a bandwidth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 3 HIERARCHICAL BAYESIAN INFERENCE Given a set H = {ℎ𝑘}𝑁obs 𝑘=1 of 𝑁obs GW observations, the posterior distribution of a set of hyper-parameters 𝜆 associated to an astrophys- ical model can be described as an in-homogeneous Poisson distribu- tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Loredo 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Thrane & Talbot 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Bouffanais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019, 2021a,b): 𝑝(𝜆, 𝑁𝜆|H) = 𝑒−𝜇𝜆 𝜋(𝜆, 𝑁𝜆) 𝑁obs � 𝑘=1 𝑁𝜆 ∫ L𝑘 (ℎ𝑘 |𝜃) 𝑝(𝜃|𝜆) d𝜃, (14) where 𝑁obs is the number of events observed by the LVK, with an ensemble of parameters 𝜃, 𝑁𝜆 is the number of predicted mergers by the model (as calculated in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 11), 𝜇𝜆 the number of predicted observations given a model and a detector, 𝜋(𝜆, 𝑁𝜆) are the prior distributions on 𝜆 and 𝑁𝜆, and L𝑘 ({ℎ}𝑘 |𝜃) is the likelihood of the 𝑘th observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The predicted number of events 𝜇𝜆 can be written in terms of detection efficiency 𝛽(𝜆) for a given model: 𝜇𝜆 = 𝑁𝜆 𝛽(𝜆), with 𝛽(𝜆) = ∫ 𝜃 𝑝(𝜃|𝜆) 𝑝det(𝜃) d𝜃, (15) where 𝑝det(𝜃) is the detection probability for a set of parameters 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This probability can be inferred by computing the optimal signal to noise ratio (SNR) of the sources and comparing it to a detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In our case we chose as reference a threshold 𝜌thr = 8 in the LIGO Livingston detector, for which we approximated the sensi- tivity using the measurements for the three runs separately (Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Wysocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The values for the event’s log-likelihood were derived from the posterior and prior samples released by the LVK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Hence, the integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 14 is approximated with a Monte Carlo approach as I𝑘 = ∫ L𝑘 (ℎ𝑘 |𝜃) 𝑝(𝜃|𝜆) d𝜃 ≈ 1 𝑁𝑘𝑠 𝑁 𝑘 𝑠 ∑︁ 𝑖=1 𝑝(𝜃𝑘 𝑖 |𝜆) 𝜋𝑘 (𝜃𝑘 𝑖 ) , (16) where 𝜃𝑘 𝑖 is the 𝑖th posterior sample of the 𝑘th detection and 𝑁𝑘𝑠 is the total number of posterior samples for the 𝑘th detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' To compute the prior term in the denominator, we also used Gaussian kernel density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Finally, we can also choose to neglect the information coming from the number of sources predicted by the model when estimating the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' By doing so, we can have some insights on the impact of the rate on the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In practice, this can be done by marginalising eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 14 over 𝑁𝜆 using a prior 𝜋(𝑁𝜆) ∼ 1/𝑁𝜆 (Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2018), which yields to the following expression for a model log-likelihood L = 𝑝(𝜆|{ℎ}𝑘) ∼ 𝜋(𝜆) 𝑁obs � 𝑘=1 � I𝑘 𝛽(𝜆) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (17) We adopted the formalism described in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 14–17 to perform a hierarchical Bayesian inference to compare the astrophysical models presented Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2 with the third gravitational-wave transient catalogue (GWTC-3), the most updated catalogue of gravitational-wave events from the LVK (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' GWTC-3 contains 90 event candidates with probability of astrophysical origin 𝑝astro > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' From GWTC-3, we extract 59 confident detections of BBHs with a false alarm rate FAR < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In this sub-sample, we do not include binary neutron stars and neutron star – BH systems, and we also exclude the other BBH candidates with an higher FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Our chosen FAR threshold ensures a sufficiently pure sample for our analysis (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A list of the events used in this study is available in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For the observable parameters 𝜃, we use the choice described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='7, namely 𝜃 = {Mc, 𝑞, 𝑧, 𝜒eff, 𝜒p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 Chirp mass The chirp mass distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2) does not depend on the spin model, by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Therefore, we only show different natal kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Models 𝜎150 and 𝜎265 show a similar distribution of chirp masses with two peaks of similar importance, one at Mc ≈ 8 M⊙ and the MNRAS 000, 1–10 (2015) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='175 GM20 g150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='150 0265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='125 LVKevents 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='100 PDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='000 5 10 15 20 25 30 35 40 45 Mc [Mo]6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Périgois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Predicted detectable distribution of 𝜒p (left) and 𝜒eff (right) for all of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Different colours refer to the spin model: G, M, F and Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Solid (dashed) lines include (do not include) the tidal spin-up model by B21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' From top to bottom: GM20, 𝜎150, and 𝜎265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The shaded gray areas are the distributions we obtain by stacking the posterior samples of the 59 confident detections from GWTC-3 (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' other (broader) peak at Mc ≈ 15 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In contrast, model GM20 has a much stronger preference for low-mass BHs, with a dominant peak at Mc ≈ 8 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The reason for this difference is that all BHs in tight binary systems receive slow natal kicks in model GM20 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This happens because stars in tight binary systems lose their envelope during mass transfer episodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' hence, the mass of supernova ejecta (𝑚ej) is small, triggering low kicks in model GM20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Figure 2 also compares the detectable distribution of our models with the stacked posterior samples from the confident BBH detections in GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This figure highlights two main differences between the population synthesis models and the posterior samples: the peak at Mc ≈ 15 M⊙ is stronger in the models than it is in the data, while the data present a more significant excess at Mc ≈ 25 − 30 M⊙ than the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Finally, the peak at Mc ≈ 9 M⊙ in the data approximately matches the peak at Mc ≈ 8 M⊙ in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The main features of our population synthesis models (in particular, the peaks at Mc ≈ 8 − 10 M⊙ and Mc ≈ 15 − 20 M⊙) are also common to other population-synthesis models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' van Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022) and mostly spring from the core-collapse SN prescriptions by Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Alternative core-collapse SN models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' MNRAS 000, 1–10 (2015) G M F Max LVK events G B21 M B21 F B21 Max B21 GM20 101 PDF 10-1 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='00 Xp XeffBBH spins: model selection with GWTC-3 7 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Olejak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022) produce different features and deserve further investigation (Iorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2 Spin parameters Figure 3 shows the detectable distribution of spin parameters 𝜒p and 𝜒eff for all of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' By construction, large spins are much more common in models G and G_B21, while models F and F_B21 have a strong predominance of vanishingly small spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Models M, M_B21, Max and Max_B21 are intermediate between the other two extreme models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Including or not the correction by B21 has negligible impact on the distribution of 𝜒p and 𝜒eff for models G, because of the predom- inance of large spin magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In contrast, introducing the spin-up correction by B21 has a key impact on models F, because it is the only way to account for mild to large spins in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The correction by B21 is important also for models M and Max, being responsible for the large-spin wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Finally, our model with slow kicks (GM20) results in a distribution of 𝜒p that is more peaked at zero (for models G, M and Max) with respect to the other two kick models (𝜎150 and 𝜎265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In fact, the supernova kicks in model GM20 are not large enough to appreciably misalign BH spins (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A similar effect is visible in the distribution of 𝜒eff: model 𝜎265 produces a distribution of 𝜒eff that is less asymmetric about the zero with respect to models 𝜎150 and especially GM20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='3 Model Selection Figure 4 and Table 4 report the values of the log-likelihood log L defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We can quantify the difference between two models A and B by computing the average absolute difference in percentage ΔlogL(A, B) = � 2 ��logLA 𝑖 − logLB 𝑖 �� logLA 𝑖 + logLB 𝑖 � 𝑣𝑎𝑟 , (18) on the non-A,B variation 𝑣𝑎𝑟 (𝑣𝑎𝑟 would be kick(spin) if A and B are spin(kick) models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For example to compare the two mod- els G and G_B21, A and B become G_B21 and G and 𝑣𝑎𝑟 = {GM20, 𝜎150, 𝜎265}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The tidal spin-up mechanism (B21) affects the spin of a small part of the population of each model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' However, it im- proves the likelihood of the F and M models significantly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', ΔlogL(M_B21, M) = 89%, Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This improvement of the log- likelihood can be explained by the presence of higher values of 𝜒p and 𝜒eff in the distribution of populations M_B21 and F_B21 compared to M and F (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The F model yields L(F) = −∞ if we do not include the tidal spin-up correction, regardless of the kick model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This indicates that the LVK data do not support vanishingly small BH spins for the entire BBH population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' However, it is sufficient to inject a tiny sub- population of spinning BHs, by switching on the B21 correction, and the F model becomes one of the best considered models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In fact, the F_B21 models only includes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='4% of BHs with 𝜒 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='01 and achieves log L > 200 (for spin models 𝜎150 and 𝜎265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The G and G_B21 spin models exhibit lower log-likelihood values than the others for all kicks models: logL ⩽ 150 for 𝜎150 and 𝜎265, and logL < 0 for GM20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This happens because the distribution of 𝜒eff has non-negligible support for extreme values 𝜒eff < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5 and 𝜒eff > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The kick models 𝜎150 and 𝜎265 show similar results (ΔlogL(𝜎150, 𝜎265) < 3%) for every spin assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Also, Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Log-likelihood L (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 17) estimated with five merger parameters 𝜃 = � Mc , 𝑧 , 𝜒eff , 𝑞 , 𝜒p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Model Name GM20 𝜎150 𝜎265 G 1 149 145 G_B21 12 150 141 M 0 162 171 M_B21 36 232 232 F ∞ ∞ ∞ F_B21 88 250 242 Max 92 255 254 Max_B21 106 257 250 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Log-likelihood L (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 17) estimated with four merger parameters 𝜃 = {Mc , 𝑧 , 𝜒eff , 𝑞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Here, we ignore 𝜒p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Model Name GM20 𝜎150 𝜎265 G 35 146 147 G_B21 47 149 154 M 141 192 190 M_B21 130 199 180 F 85 146 138 F_B21 185 207 180 Max 161 208 155 Max_B21 160 206 200 for all spin assumptions, the GM20 kick model scores a signifi- cantly lower likelihood than the other models 𝜎150 and 𝜎265 with ΔlogL(𝜎150, GM20) ∼ ΔlogL(𝜎265, GM20) ∼ 150%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This re- sult can be explained by the high peak of model GM20 at low chirp masses (Mc ∼ 8M⊙, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2) and by the low value of 𝜒p compared to the other kick models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Models Max and Max_B21 are possibly the best match to the data, but this is not surprising, because they were built as a toy model to vi- sually match the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Among the astrophysically-motivated models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', after excluding the Max model), M, M_B21 and F_B21 (with kick models 𝜎150 and 𝜎265) are the most favoured by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This might be interpreted as a support for the Tayler-Spruit instability mechanism (adopted in models M) and for the tidal spin-up model by B21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='4 Importance of 𝜒p The 𝜒p parameter encodes information on the spin component in the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Its impact on gravitational-wave signals is much lower than that of 𝜒eff, and therefore its measurement is less precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' To understand the impact of 𝜒p on our results, we re-ran the analysis without this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The results are shown in Table 5 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 4 with empty markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 4 shows that, if we do not include 𝜒p, the models M and M_B21 have almost the same log-likelihood, and even the F model yields a positive log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Furthermore, the analysis without 𝜒p results in significantly larger values of L for the kick model GM20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Our results demonstrate that the measured 𝜒p of GWTC-3 BBHs carries substantial information, despite the large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 5 DISCUSSION The spin magnitude of BHs is largely uncertain, mostly because we do not fully understand angular momentum transport in massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Here, we have taken a number of spin models bracketing the MNRAS 000, 1–10 (2015) 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Périgois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Values of the log-likelihood L defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 17 for the four different models Geneva (G), MESA (M), Fuller (F), and Maxwellian (Max), with/without the tidal spin-up mechanism (B21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Blue crosses: GM20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' dark pluses: 𝜎150;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' red circles: 𝜎265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' main uncertainties, we have implemented them into our population- synthesis code mobse, and compared them against GWTC-3 data within a hierarchical Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The data do not support models in which the entire BH population has vanishingly small spins (model F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This result is mainly driven by the 𝜒p parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This is in agreement with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', the comple- mentary analysis presented in Callister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' They employed a variety of complementary methods to measure the distribution of spin magnitudes and orientations of BBH mergers, and concluded that the existence of a sub-population of BHs with vanishing spins is not required by current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Callister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2022) find that the fraction of non-spinning BHs can comprise up to ∼ 60 − 70% of the total population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In our F_B21 models, we have ∼ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='6% of BHs with 𝜒 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Recently, Roulet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021) and Galaudage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021) claimed the existence of a sub-population of zero-spin BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' From our anal- ysis, we cannot exclude the existence of such sub-population, as the F model with B21 correction (F_B21) still represents a good match of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Similarly to Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2020) and Gerosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2018), we find that models with large spins (G, G_B21) are less favoured by the data, but they are still acceptable if we allow for large kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Overall, we find a preference for large natal kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This result goes into the same direction as the work by Callister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Actually, this preference for large natal kicks is degenerate with the adopted formation channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Had we included the dynamical formation chan- nel in dense star clusters, we would have added a sub-population of isotropically oriented spins (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Figure 8 of Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In a forthcoming study, we will extend our analysis to a multi-channel analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' While it is unlikely that BBH mergers only originate from one single channel, adding more formation channels to a hierarchical Bayesian analysis dramatically increases the number of parameters, making it more difficult to reject some portions of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 6 SUMMARY The origin of BH spins is still controversial, and angular momen- tum transport inside massive stars is one of the main sources of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Here, we apply hierarchical Bayesian inference to derive constraints on spin models from the 59 most confident BBH merger events in GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We consider five parameters: chirp mass, mass ratio, redshift, effective spin, and precessing spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For model selection, we use a set of binary population synthe- sis simulations spanning different assumptions for black hole spins and natal kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In particular, our spin models account for relatively inefficient (G), efficient (Max and M), and very efficient angular- momentum transport (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A higher efficiency of angular momentum transport is associated with lower BH spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In particular, model F predicts vanishingly small spins for the entire BH population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For each of our models, we also include the possibility that some BHs are tidally spun-up (B21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We considered three different natal kick models: according to models 𝜎265 and 𝜎150, we randomly draw the kicks from a Maxwellian curve with 𝜎 = 265 and 150 km s−1, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' in the third model (G20), we also derive the kicks from a Maxwellian curve with 𝜎 = 265 km s−1, but the kick magnitude is then modulated by the ratio between the mass of the ejecta and the mass of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' We summarize our main results as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The data from GWTC-3 do not support models in which the entire BH population has vanishingly small spins (model F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' In contrast, models in which most spins are vanishingly small, but that also include a sub-population of tidally spun-up BHs (model F_B21) are a good match to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The models in which angular momentum transport is relatively inefficient (G and G_21) yield log-likelihood values that are much lower than models with efficient angular momentum transport (M, M_B21, Max, and Max_B21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Models with large BH kicks (𝜎150 and 𝜎265 ) are favoured by our analysis with respect to low-kick models (G20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Our results show that the precessing spin parameter 𝜒p plays a crucial impact to constrain the spin distribution of BBH mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' ACKNOWLEDGEMENTS MM, CP, FS and YB acknowledge financial support from the European Research Council for the ERC Consolidator grant DE- MOBLACK, under contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 770017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This research made use of MNRAS 000, 1–10 (2015) 300 GM20 no Xp + o150 no Xp g265 no Xp GM20 g150 0265 250 200 + + + 150 L 100 50 X 0 50 8 8 8 100 M F Max G_B21 M_B21 F B21 Max_B21BBH spins: model selection with GWTC-3 9 NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020), and SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' For the plots we used Matplotlib (Hunter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' The latest public version of mobse can be downloaded from this repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' CosmoRate can be downlowaded from this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' REFERENCES Aasi J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2015, Classical and Quantum Gravity, 32, 074001 Abadie J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2010, Classical and Quantum Gravity, 27, 173001 Abbott B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2016a, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 116, 241103 Abbott B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2016b, ApJ, 833, L1 Abbott B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, Physical Review X, 9, 031040 Abbott R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 102, 043015 Abbott R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021a, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='01045 Abbott R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021b, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='03606 Abbott R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021c, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='03634 Abbott R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021d, Physical Review X, 11, 021053 Acernese F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2015, Classical and Quantum Gravity, 32, 024001 Aerts C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mathis S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rogers T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, ARA&A, 57, 35 Aghanim N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, A&A, 641, A6 Arca Sedda M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Benacquista M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 482, 2991 Arca Sedda M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Spera M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Benacquista M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, ApJ, 894, 133 Atri P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 489, 3116 Baibhav V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gerosa D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Wong K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Helfer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mould M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 102, 043002 Barrett J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gaebel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Neijssel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Vigna-Gómez A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, MNRAS, 477, 4685 Bavera S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, A&A, 635, A97 Bavera S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zevin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fragos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, Research Notes of the American Astronomical Society, 5, 127 Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bulik T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fryer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Ruiter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Valsecchi F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Vink J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Hurley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2010, ApJ, 714, 1217 Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, A&A, 636, A104 Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gerosa D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Di Carlo U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Baibhav V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, ApJ, 886, 25 Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Santoliquido F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Iorio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Costa G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021a, MNRAS, 505, 3873 Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Santoliquido F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Di Carlo U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rastello S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Artale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Iorio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021b, MNRAS, 507, 5224 Briel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Stevance H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Eldridge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='13842 Broekgaarden F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022a, MNRAS, 516, 5737 Broekgaarden F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Thrane E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022b, ApJ, 938, 45 Callister T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Renzo M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, ApJ, 920, 157 Callister T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Miller S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chatziioannou K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='08574 Cantiello M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mankovich C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bildsten L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Christensen-Dalsgaard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Paxton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2014, ApJ, 788, 93 Chattopadhyay D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Hurley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Raidani A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, MNRAS, 513, 4527 Claeys J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Pols O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Izzard R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Vink J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Verbunt F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2014, A&A, 563, A83 Costa G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Girardi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bressan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Marigo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rodrigues T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Lanza A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Goudfrooij P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 485, 4641 Dall’Amico M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Di Carlo U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rastello S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Santoliquido F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Ballone A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Arca Sedda M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, MNRAS, 508, 3045 Di Carlo U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Pasquato M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Spera M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Wang L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Haardt F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 487, 2947 Ekström S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2012, A&A, 537, A146 Eldridge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Stanway E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2016, MNRAS, 462, 3302 Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sravan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Cantrell A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kreidberg L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bailyn C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2011, ApJ, 741, 103 Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Miller M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Vecchio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, Nature, 548, 426 Farr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Holz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, ApJ, 854, L9 Fishbach M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Holz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, ApJ, 851, L25 Fishbach M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, ApJ, 929, L26 Fishbach M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Holz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, The Astrophysical Journal, 863, L41 Fragos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', McClintock J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2015, ApJ, 800, 17 Fryer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2001, ApJ, 554, 548 Fryer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Wiktorowicz G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Dominik M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Holz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2012, ApJ, 749, 91 Fryer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Olejak A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, ApJ, 931, 94 Fuller J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Ma L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, ApJ, 881, L1 Fuller J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Piro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Jermyn A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 485, 3661 Galaudage S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Talbot C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Nagar T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Jain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Thrane E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, ApJ, 921, L15 Gehan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mosser B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Michel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Samadi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kallinger T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, A&A, 616, A24 Gerosa D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kesden M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', O’Shaughnessy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sperhake U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2013, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 87, 104028 Gerosa D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', O’Shaughnessy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kesden M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Wysocki D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gladysz W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 98, 084036 Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, MNRAS, 480, 2011 Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 482, 2234 Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, ApJ, 891, 141 Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Spera M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, MNRAS, 474, 2959 Harris C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, Nature, 585, 357 Heger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fryer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Woosley S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Langer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Hartmann D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2003, ApJ, 591, 288 Hobbs G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Lorimer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Lyne A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kramer M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2005, MNRAS, 360, 974 Hotokezaka K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Piran T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, ApJ, 842, 111 Hunter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2007, Computing in Science & Engineering, 9, 90 Hurley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Pols O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Tout C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2000, MNRAS, 315, 543 Hurley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Tout C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Pols O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2002, MNRAS, 329, 897 Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2000, ApJ, 541, 319 Kimball C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Talbot C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Carney M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zevin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Thrane E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, ApJ, 900, 177 Kroupa P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2001, MNRAS, 322, 231 Kushnir D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kollmeier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Waldman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2016, MNRAS, 462, 844 Limongi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chieffi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, ApJS, 237, 13 Loredo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2004, in Fischer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Preuss R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Toussaint U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', eds, American Institute of Physics Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 735, Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 24th Interna- tional Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' pp 195–206 (arXiv:astro-ph/0409387), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1835214 Maccarone T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kundu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zepf S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rhode K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2007, Nature, 445, 183 Madau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fragos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, ApJ, 840, 39 Maeder A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Meynet G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2000, ARA&A, 38, 143 Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gair J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 486, 1086 Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Müller B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Riley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', de Mink S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Vigna-Gómez A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chattopadhyay D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, MNRAS, 500, 1380 Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zampieri L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Ripamonti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bressan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2013, MNRAS, 429, 2298 Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Ripamonti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Spera M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, MNRAS, 472, 2422 Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Spera M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Montanari E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Limongi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chieffi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bressan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, ApJ, 888, 76 Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, MNRAS, 505, 339 Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Santoliquido F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Arca Sedda M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Artale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, MNRAS, 511, 5797 Marchant P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Langer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Podsiadlowski P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Tauris T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Moriya T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2016, A&A, 588, A50 Miller-Jones J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, Science, 371, 1046 Miller S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Callister T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Farr W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, ApJ, 895, 128 MNRAS 000, 1–10 (2015) 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Périgois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mosser B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2012, A&A, 548, A10 Motta S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belloni T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Stella L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Muñoz-Darias T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fender R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2014, MNRAS, 437, 2554 Olejak A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, ApJ, 921, L2 Olejak A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fryer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Baibhav V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, MNRAS, 516, 2252 Olsen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Venumadhav T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mushkin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Roulet J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zackay B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 106, 043009 Özel F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Psaltis D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Narayan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', McClintock J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2010, ApJ, 725, 1918 Patton R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sukhbold T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Eldridge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, MNRAS, 511, 903 Perna R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Artale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Wang Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Lazzati D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sgalletta C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Santoliquido F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, MNRAS, 512, 2654 Qin Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fragos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Meynet G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Andrews J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sørensen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Song H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, A&A, 616, A28 Qin Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fragos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Meynet G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Marchant P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Andrews J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sørensen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Song H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, IAU Symposium, 346, 426 Repetto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Igoshev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Nelemans G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, MNRAS, 467, 298 Reynolds C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, ARA&A, 59, 117 Rodriguez C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zevin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Pankow C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rasio F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2016, ApJ, 832, L2 Roulet J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, MNRAS, 484, 4216 Roulet J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chia H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Olsen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Dai L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Venumadhav T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zackay B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 104, 083010 Sana H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2012, Science, 337, 444 Santoliquido F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Di Carlo U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rastello S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Artale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Ballone A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, ApJ, 898, 152 Santoliquido F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Giacobbo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Bouffanais Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Artale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, MNRAS, 502, 4877 Shafee R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', McClintock J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Narayan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Davis S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Li L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Remillard R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2006, ApJ, 636, L113 Spera M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mapelli M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, MNRAS, 470, 4739 Spruit H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2002, A&A, 381, 923 Stegmann J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Antonini F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 103, 063007 Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, ApJ, 926, L32 Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Clarke T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, MNRAS, Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Ohme F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fairhurst S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2015, ApJ, 810, 58 Stevenson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, MNRAS, 471, 2801 Talbot C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Thrane E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 96, 023012 Tauris T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Langer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Podsiadlowski P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2015, MNRAS, 451, 2123 Tauris T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, ApJ, 846, 170 Taylor S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gerosa D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 98, 083017 Thrane E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Talbot C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Australia, 36, e010 Tiwari V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Fairhurst S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Hannam M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, ApJ, 868, 140 Venumadhav T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zackay B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Roulet J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Dai L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 101, 083030 Vink J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', de Koter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Lamers H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2001, A&A, 369, 574 Virtanen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2020, Nature Methods, 17, 261 Vitale S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Lynch R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sturani R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Graff P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, Classical and Quantum Gravity, 34, 03LT01 Wysocki D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gerosa D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', O’Shaughnessy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Belczynski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Gladysz W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Berti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kesden M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Holz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 97, 043014 Wysocki D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Lange J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', O’Shaughnessy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2019, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' D, 100, 043012 Zahn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 1992, A&A, 265, 115 Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kushnir D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kollmeier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2018, MNRAS, 473, 4174 Zevin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Pankow C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rodriguez C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Sampson L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chase E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Kalogera V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Rasio F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2017, ApJ, 846, 82 Zevin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2021, ApJ, 910, 152 de Mink S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Mandel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2016, MNRAS, 460, 3545 van Son L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', de Mink S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Chruslinska M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Conroy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Pakmor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='03385 APPENDIX A: SAMPLE OF GRAVITATIONAL-WAVE EVENTS Table A1 lists all the gravitational-wave event candidates we used in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' From GWTC-3, we selected all the event candidates with 𝑝astro > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='9 and FAR< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 yr−1, excluding the following three systems: the binary neutron star GW170807;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' the (possible) neutron star–BH binary system GW190814;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' the BBH GW190521 (𝑚1 = 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='4+33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='6 −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='7M⊙, 𝑚2 = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2+27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1 −30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='1M⊙ Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021b), which can form only via dynamical interactions in our models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=', Di Carlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Dall’Amico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' MNRAS 000, 1–10 (2015) BBH spins: model selection with GWTC-3 11 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content=' Catalogue of BBH events adopted in this study Name Mc [M⊙] q 𝜒eff 𝜒p z GW150914 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='7 −1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='03 GW151012 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='2+2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} +page_content='08 MNRAS 000, 1–10 (2015)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AzT4oBgHgl3EQfW_zX/content/2301.01312v1.pdf'} diff --git a/Q9E3T4oBgHgl3EQfDAmJ/content/tmp_files/2301.04282v1.pdf.txt b/Q9E3T4oBgHgl3EQfDAmJ/content/tmp_files/2301.04282v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..13d0acf993f6496a28924d3746dea29a15165d8f --- /dev/null +++ b/Q9E3T4oBgHgl3EQfDAmJ/content/tmp_files/2301.04282v1.pdf.txt @@ -0,0 +1,1551 @@ +arXiv:2301.04282v1 [hep-ph] 11 Jan 2023 +Speed of sound in QCD matter at finite +temperature and density +Guo-yun Shao1*, Xin-ran Yang1, Chong-long Xie1 +and Wei-bo He2 +1School of Physics, Xi’an Jiaotong University, Xi’an, 710049, +China. +2School of Physics, Peking University, Beijing, 100871, China. +*Corresponding author(s). E-mail(s): gyshao@mail.xjtu.edu.cn; +Abstract +The speed of sound in QCD matter at finite temperature and den- +sity is investigated within the Polyakov loop improved Nambu–Jona- +Lasinio (PNJL) model. The spinodal structure associated with the +chiral first-order chiral phase transition is considered to describe the +continuous variation of the speed of sound. The behaviors of the +squared sound speed in different phases, including the stable, metastable +and unstable phases, are derived. The relation between speed of +sound and QCD phase transitions is systematically explored. In par- +ticular, the boundary of vanishing sound velocity is derived in the +temperature-density phase diagram, and the region where the sound +wave equation being broken is pointed out. Some interesting fea- +tures of speed of sound under different definitions are also discussed. +Keywords: Speed of sound, Quark matter, Chiral phase transition +1 Introduction +Quark-gluon plasma (QGP) can be created in heavy-ion collision (HIC) experi- +ments at relativistic energies. A crucial topic relevant is to explore the equation +of state (EOS) and phase transition from QGP to hadronic matter. The hydro- +dynamic simulation provides a method to study the EOS of QGP [1–3]. During +1 + +2 +Speed of sound in QCD matter at finite temperature and density +the space-time evolution of QCD matter, the speed of sound is one of the cru- +cial physical quantities. Its dependence on environment (temperature, density, +chemical potential, etc.) carries important information in describing the evo- +lution of the fireball and final observables. Recently, the studies in Refs. [4–6] +show that the speed of sound as a function of charged particle multiplic- +ity ⟨dNch/dη⟩ can be extracted from heavy-ion collision data. In Ref. [7] the +authors try to build a connection between the sound speed and baryon number +cumulants to study the QCD phase structure. +The speed of sound in neutron star has also received a lot of attention (e.g., +Refs. [8–10]). The density dependent behavior of sound velocity influences the +mass-radius relation, the tidal deformability, and provides a sensitive probe of +the EOS of neutron star matter. To obtain a two solar mass neutron star, some +studies find that it is essential for neutron star matter to have a density range +where the EOS is very stiff and the corresponding squared speed of sound +is significantly larger than 1/3 [11–17]. The study in Ref. [18] indicates that +the speed of sound is crucial for the gravitational wave frequencies induced +by the g-mode oscillation of a neutron star. It is also interesting to study the +gravitational wave induced by the cosmic QCD phase transition in which the +speed of sound plays a significant role. +As an important quantum in describing the evolution of strongly inter- +acting matter, the relation between the speed of sound and QCD phase +transition is worth exploring. The speed of sound has been calculated, e.g., +in lattice QCD [19–23], (P)NJL model [3, 24–28], quark-meson coupling +model [29, 30], hadron resonance gas (HRG) model [31, 32], field correlator +method (FCM) [33, 34] and quasiparticle model [35]. In previous studies, the +main focus is put on the region of high temperature and vanishing or small +chemical potential. In Ref. [36], we give an intensive study on the speed of +sound in QCD matter in the full temperature-chemical potential phase dia- +gram. The numerical results indicate that the dependence of sound speed on +temperature and chemical potential is indicative of QCD phase transition. +However, only the sound speed in the stable phase is considered in Ref. [36]. +There are still some crucial issues that need to be clarified. First of all, the +spinodal structure may be involved in heavy-ion collision experiments with +the decrease of collision energy [37–47]. A complete evolution of sound speed +in the metastable and unstable phases needs to be explored to give a distinct +description of the fireball expansion. Secondly, it is found that the sound speed +takes small values at the CEP and on the boundaries of the first-order phase +transition near the CEP [36]. A question aroused is that where is the boundary +of vanishing speed of sound. Furthermore, the behavior of sound speed in the +temperature and density phase diagram is still not explored. +On the other hand, the values of speed of sound under different conditions +are involved in dealing with different problems in nuclear physics, such as +the gravitational signal from cosmic QCD phase transition [48, 49], the bulk +viscosity of strongly interacting matter [3], the equation of state of neutron +star matter [8–10] and the evolution of QGP in HIC experiments. In this + +Speed of sound in QCD matter at finite temperature and density +3 +work, we will give a systematic study on the relation between speed of sound +and QCD phase transitions at finite temperature and density under several +different constraint conditions. This work is helpful in dealing with the physics +problems mentioned above. +The paper is organized as follows. In Sec. II, we derive the formulae of speed +of sound under different definitions in the temperature and density space, and +then briefly introduce the 2+1 flavor PNJL quark model. In Sec. III, we present +the numerical results of squared sound speed and discuss the relations with +the QCD phase structure. A summary is finally given in Sec. IV. +2 Speed of sound and the PNJL quark model +The general definition of speed of sound is +c2 +X = +�∂p +∂ǫ +� +X +. +(1) +A specifying constant quantity X is required to describe the propagation of +the compression wave through a medium. To indicate the different profiles of +QCD matter, X can be chosen as s/ρB, s, ρB, T, µB. Different definitions of +speed of sound are taken in practice in dealing with different physics issues. +For a fireball created in relativistic heavy-ion collisions, it evolves with +a constant entropy density per baryon s/ρB if it is taken as an ideal fluid. +Therefore, it is meaningful to calculate the speed of sound along the isentropic +curve +c2 +s/ρB = +�∂p +∂ǫ +� +s/ρB +. +(2) +The dependence of c2 +s/ρB on parameters, e.g., temperature and density, can +indicate the variation of sound speed during the evolution and provide +important knowledge of interaction, phase transition and the EOS of QGP. +The speed of sound with constant baryon number density or entropy +density are taken in describing the intermediate process of a hydrodynamic +evolution [3], +c2 +ρB = +�∂p +∂ǫ +� +ρB +and +c2 +s = +�∂p +∂ǫ +� +s +. +(3) +For example, the temporal derivatives of temperature and chemical potential +are functions of c2 +ρB and c2 +s, as +∂0µB = −c2 +sµB ∇ · u, +(4) +and +∂0T = −c2 +ρBT ∇ · u, +(5) +where u denotes the space component of four-velocity. The values of c2 +ρB and +c2 +s are directly connected to the bulk viscosity coefficient. + +4 +Speed of sound in QCD matter at finite temperature and density +It is also interesting to calculate the sound speed with a fixed temperature +or chemical potential +c2 +T = +�∂p +∂ǫ +� +T +, +c2 +µB = +�∂p +∂ǫ +� +µB +, +(6) +In Ref. [7] the authors estimate c2 +T as a function of the logarithmic deriva- +tive with respect to the baryon density of QCD matter, and try to build a +connection with the baryon number cumulants to aid in detecting the QCD +critical endpoint. Besides, c2 +T is also usually taken to study the speed of sound +in neutron star matter. +In this study, we will explore the speed of sound under different definitions +in the full temperature-density space. Since the general definitions can only +be used to calculate the sound speed on special trajectories, it is necessary to +derive the corresponding formulae in terms of T and ρB. With the fundamental +thermodynamic relations, the sound speed formulae under different constraint +conditions can be derived as +c2 +s/ρB = +s2+ρ2 +B +�� +∂µB +∂ρB +� +T +� ∂s +∂T +� +ρB − +� +∂µB +∂T +� +ρB +� +∂s +∂ρB +� +T +� ++sρB +�� +∂µB +∂T +� +ρB +− +� +∂s +∂ρB +� +T +� +(T s + µBρB) +� ∂s +∂T +� +ρB +, (7) +c2 +s = +ρB +�� ∂s +∂T +� +ρB +� +∂µB +∂ρB +� +T − +� +∂s +∂ρB +� +T +� +∂µB +∂T +� +ρB +� +− s +� +∂s +∂ρB +� +T +µB +� ∂s +∂T +� +ρB +, +(8) +c2 +ρB = +s + ρB +� +∂µB +∂T +� +ρB +T +� ∂s +∂T +� +ρB +, +c2 +T = +ρB +� +∂µB +∂ρB +� +T +T +� +∂s +∂ρB +� +T + µB +, +(9) +and +c2 +µB = +s +� +∂µB +∂ρB +� +T +T +�� ∂s +∂T +� +ρB +� +∂µB +∂ρB +� +T − +� +∂µB +∂T +� +ρB +� +∂s +∂ρB +� +T +� +− µB +� +∂µB +∂T +� +ρB +. +(10) +The details for deriving these formulae are affiliated in the appendix A. The +above formulae are only correct for isospin symmetric matter. The corre- +sponding formulae will be much more complicated for isospin asymmetric +matter. +To demonstrate the relation between the speed of sound under different +definition and QCD phase structure, we take the 2+1 flavor PNJL quark model + +Speed of sound in QCD matter at finite temperature and density +5 +in the calculation. The Lagrangian density is given by +L = ¯q(iγµDµ+γ0ˆµ− ˆm0)q+G +8 +� +k=0 +� +(¯qλkq)2+(¯qiγ5λkq)2� +−K +� +detf(¯q(1 + γ5)q) + detf(¯q(1 − γ5)q) +� +−U(Φ[A], ¯Φ[A], T ), +(11) +where q denotes the quark fields with three flavors, u, d, and s; ˆm0 = +diag(mu, md, ms) in flavor space; G and K are the four-point and six-point +interacting constants, respectively. The ˆµ = diag(µu, µd, µs) are the quark +chemical potentials. +The covariant derivative in the Lagrangian is defined as Dµ = ∂µ−iAµ. The +gluon background field Aµ = δ0 +µA0 is supposed to be homogeneous and static, +with A0 = gAα +0 +λα +2 , where λα +2 is SU(3) color generators. The effective potential +U(Φ[A], ¯Φ[A], T ) is expressed with the traced Polyakov loop Φ = (TrcL)/NC +and its conjugate ¯Φ = (TrcL†)/NC. The Polyakov loop L is a matrix in color +space +L(⃗x) = Pexp +� +i +� β +0 +dτA4(⃗x, τ) +� +, +(12) +where β = 1/T is the inverse of temperature and A4 = iA0. +The Polyakov-loop effective potential is +U(Φ, ¯Φ, T ) +T 4 += −a(T ) +2 +¯ΦΦ + b(T )ln +� +1 − 6¯ΦΦ + 4(¯Φ3 + Φ3) − 3(¯ΦΦ)2� +,(13) +where +a(T )=a0+a1 +�T0 +T +� ++a2 +�T0 +T +�2 and +b(T )=b3 +�T0 +T +�3. +(14) +The parameters ai, bi listed in Table. 1 are fitted according to the lattice +simulation of QCD thermodynamics in pure gauge sector. The T0 = 210 MeV +is implemented in the calculation. +Table 1 Parameters in the Polyakov-loop potential [50] +a0 +a1 +a2 +b3 +3.51 +-2.47 +15.2 +-1.75 +The constituent quark mass in the mean field approximation can be derived +as +Mi = mi − 4Gφi + 2Kφjφk +(i ̸= j ̸= k), +(15) +where φi stands for quark condensate of the flavor i. + +6 +Speed of sound in QCD matter at finite temperature and density +The thermodynamical potential of bulk quark matter is derived as +Ω = −2T +� +i=u,d,s +� +d3p +(2π)3 (Q1 + Q2) − 2 +� +Λ +d3p +(2π)3 3(Eu + Ed + Es) ++2G +� +φu +2 + φd +2 + φs +2� +− 4Kφu φd φs + U(¯Φ, Φ, T ) +(16) +where Q1 = ln(1 + 3Φe−(Ei−µi)/T + 3¯Φe−2(Ei−µi)/T + e−3(Ei−µi)/T ), Q2 = +ln(1 + 3¯Φe−(Ei+µi)/T + 3Φe−2(Ei+µi)/T + e−3(Ei+µi)/T ), and Ei = +� +⃗p 2 + M 2 +i +is the dispersion relation. µi = µB/3 is taken for u, d, s quark flavors. The pres- +sure p and energy density ǫ can be derived using the thermodynamic relations +in the grand canonical ensemble as +P = −Ω, +ǫ = −P + T s + +� +µiρi, +(17) +where s is the entropy density and ρi the quark number density of flavor i. +For given T and baryon density ρB, the values of φu, φd, φs, Φ, ¯Φ and µB +are determined by solving the equations by minimizing the thermodynamical +potential +∂Ω +∂φu += ∂Ω +∂φd += ∂Ω +∂φs += ∂Ω +∂Φ = ∂Ω +∂ ¯Φ = 0, +(18) +and the relevant constraint condition. Other physical quantities can be then +derived using thermodynamic relations. The numerical results of speed of +sound under different conditions can be then derived according to Eqs. (7)-(10). +In the numerical calculation, a cut-off Λ is implemented in 3-momentum +space for divergent integrations. We take the model parameters obtained +in [51]: Λ = 602.3 MeV, GΛ2 = 1.835, KΛ5 = 12.36, mu,d = 5.5 and +ms = 140.7 MeV, determined by fitting fπ = 92.4 MeV, Mπ = 135.0 MeV, +mK = 497.7 MeV and mη = 957.8 MeV. +3 Numerical results and discussions +In this section, we present the numerical results of the speed of sound under +different constraint conditions and discuss the relations with the QCD phase +transitions. +3.1 Sound velocity at constant s/ρB +Firstly, we plot the QCD phase diagram, including the first-order phase +transition (black solid line) and the spinodal structure (blue dashed line), +which separate the phase diagram into the stable, metastable and unstable +phases. The first-order phase transition line is obtained according to the ther- +modynamic conditions for two-phase equilibrium, i.e., T1 = T2, µ1 = µ2, +P1 = P2 for two stable phases. The spinodal line is derived with the mechan- +ical unstable condition. The corresponding inflection points of pressure as a +function of density can be determined for a given T . For more details to + +Speed of sound in QCD matter at finite temperature and density +7 +derive the phase boundaries, one can refer to Refs. [52, 53]. The spinodal +phase decomposition plays a dominant role in the experimental exploration of +the first-order nuclear liquid-gas transition[54, 55]. It has inspired the antic- +ipation to identify the first-order chiral transition in high-energy heavy-ion +collisions through the spinodal phase separation [37–46]. The recent simula- +tion suggests that the spinodal instability can be triggered within a certain +energy range [44, 46] . We also demonstrate in Fig. 1 the isentropic curves +with s/ρB = 0.1, 1, 3, 5, 5.9, 10, 50, 100, 300 in the T − ρB plane to indicate the +evolutionary trajectories of an ideal fluid at different collision energies. + metastable phase + unstable phase + 1st-order transition line + spinodal line +0 +1 +2 +3 +4 +5 +6 +50 +100 +150 +200 +250 +300 +CEP +0.1 +1 +5.9 +10 +50 +100 +T(MeV) +r +B +/r +0 +300 +5 +3 +Fig. +1 +QCD +phase +diagram +and +the +isentropic +curves +for +s/ρB += +0.1, 1, 3, 5, 5.9, 10, 50, 100, 300 in T − ρB plane. +We present in Fig. 2 the equations of state with chosen parameter s/ρB = +0.1, 1, 3, 5, 5.9 that pass through the first-order phase transition. For each curve +with s/ρB < 5.9, there are two inflection points where ∂p/∂ǫ changes the +sign. The red dashed curve in Fig. 2 is the connections of these inflection +points, which lies in the spinodal boundary associated with the first-order +phase transition, i.e., in the interior of the unstable phase. We will present +the relation clearly in the contour map of speed of sound in the T − ρB panel +soon. The inflection points also correspond to the locations where the sound +velocity vanishes in the phase diagram. +The square of speed of sound can be directly derived with the definition +given in Eq. (2). In Fig. 3, we plot the curve of squared sound speed c2 +s/ρB +as functions of energy density along the evolutionary trajectories for s/ρB = +0.1, 1, 3, 5, 5.9. Figure 3 shows that there exists one peak and two valleys on +each curve, which indicates that the speed of sound are closely related to +temperature and density. In particular, for the case of c2 +s/ρB < 0 in Fig. 3, it +corresponds to ∂p/∂ǫ < 0 for a fixed s/ρB, as shown in Fig. 2. + +8 +Speed of sound in QCD matter at finite temperature and density + inflection points + s/r +B +=0.1 + s/r +B +=1.0 + s/r +B +=3.0 + s/r +B +=5.0 + s/r +B +=5.9 +Pressure(MeVfm +-3 +) +e(MeVfm +-3 +) +Fig. 2 +Equations of state of QCD matter for s/ρB = 0.1, 1, 3, 5, 5.9. The squares on each +curve are the inflection points where ( ∂p +∂ǫ )s/ρB changes the sign. The red dashed line is the +profile of these inflection points. +0 +500 +1000 +1500 +2000 +2500 +3000 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +C +2 +s/r +B +e(MeVfm +-3 +) + s/r +B +=0.1 + s/r +B +=1 + s/r +B +=3 + s/r +B +=5 + s/r +B +=5.9 +Fig. 3 +Values of c2 +s/ρB as functions of energy density for s/ρB = 0.1, 1, 3, 5, 5.9. +To show more clearly the relation between the speed of sound and QCD +phase transitions, we present the contour map and 3D map of c2 +s/ρB as functions +of T and ρB in Fig. 4 and Fig. 5, respectively. The phase structure includ- +ing the chiral crossover, chiral first-order, spinodal and deconfinement phase +transitions. The chiral crossover line and deconfinement line are determined +by requiring ∂φi/∂T and ∂Φi/∂T taking extreme values for a given chemical +potential. +The two figures indicate that the region around the peaks of c2 +s/ρB in Fig. 3 +is located in the region where the chrial symmetry of u, d quark is approxi- +mately restored already but still confined. The valley at high energy density + +000008 +a00eoo +1001 +1400 +002S00 +30040 +02 +oaSO +30O0 +00 +-SOSpeed of sound in QCD matter at finite temperature and density +9 +side in Fig. 3 lies in the region where the chiral condensate of strange quark +changes quickly. After the chiral restoration of strange quark the sound speed +increases again towards high density. The valley in the low density side is +located in the spinodal region of the first-order phase transition, which is +closely related to the chiral condensate of u, d quark. For each value of s/ρB +smaller than 5.9, there exist a range of ( ∂p +∂ǫ )s/ρB taking negative values. The +red dashed line is the boundary of vanishing sound velocity with c2 +s/ρB = 0 +derived with Eq. (7) . +0 +2 +4 +6 +8 +10 +12 +14 +50 +100 +150 +200 +250 +300 +T(MeV) +B +/ +0 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +c +2 +s/ +B + chiral crossover + chiral first order + spinodal line + CEP + deconfinement line +Fig. 4 +Contour map of c2 +s/ρB in the T − ρB plane. The red dashed line is the boundary +of vanishing sound velocity . +Figure 4 and 5 also show that, at the high temperature or very high density, +with the restoration of chiral symmetry c2 +s/ρB approaches to 1/3, the value +of noninteracting gas. The value of c2 +s/ρB at lower density descends with the +decrease of temperature. A rapid decrease occurs in the chiral crossover region +of u, d quark, as shown in Fig. 5. It indicates that the value of speed of sound +is sensitive to the change of dynamical quark mass. In the low-density region, +a minimum of c2 +s/ρB appears near the deconfinement phase transition for a +given density (chemical potential). A similar behavior exists in lattice QCD +at zero chemical potential [19–23]. However, such a feature does not appear in +the NJL model which cannot describe the confinement-deconfinement phase +transition [3, 25–27], which indicates that the color confinement also plays an +important role on the speed of sound near the crossover phase transition line. +The value of c2 +s/ρB is relatively smaller in the region of low temperature and +density. The red dashed line is the boundary of vanishing sound velocity. Inside +this boundary c2 +s/ρB < 0, it means physically that ( ∂p +∂ǫ )s/ρB < 0. In this region, +the mechanically stable condition is broken and the corresponding sound wave +equation becomes a decay function. A perturbance can not be propagated like +a sound wave in this situation. It can be seen that such a region lies in the +interior of the unstable phase of the spinodal structure. Figure 4 and 5 also + +10 +Speed of sound in QCD matter at finite temperature and density +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +0 +50 +100 +150 +200 +250 +-0.1 +0.0 +0.1 +0.2 +0.3 +T(MeV) +C +2 +s/r +B +r +B +/r +0 +Fig. 5 +3D map of c2 +s/ρB as functions of tempeature and density. The black solid line is +the first-order chiral phase transition line. The blue dashed line is the spinodal line. The red +dashed line is the boundary of vanishing sound velocity. +indicate that the speed of sound at the critical endpoint is small but not zero +in the mean field approximation. +The value of c2 +s/ρB reflects the speed of sound in an ideal fluid which can be +approximately realized in heavy-ion collision experiments. On the other hand, +s/ρB is connected with the collision energy. If the value of c2 +s/ρB at a fixed +energy can be extracted from the charged particle multiplicity ⟨dNch/dη⟩ [4– +6], we can access the information of phase transition using the relation between +c2 +s/ρB and QCD phase diagram. Furthermore, combining with the beam energy +scan experiments, it provides a possible way to diagnose the QCD phase struc- +ture. It is also inspiring for study on the gravitational signal from the cosmic +QCD phase transition in which a constant speed of sound +� +1/3 is usually +taken in literature. +3.2 Sound velocity at constant ρB and s +We present in Fig. 6 the contour map of c2 +ρB in the T − ρB panel. Besides +at the high-temperature side, this figure shows that c2 +ρB take relatively larger +values in the region of low temperature and density. The value is even larger +than 1/3, in particular, in the metastable phase and unstable phase. There also +exists a wide region (inside the red line filled with the blue color) of c2 +ρB < 0. +More physically, it means that the ( ∂p +∂ǫ )ρB < 0 in this region, i.e., the pressure +decreases with the increase of energy density along the line of constant density. +The red line shows the boundary of vanishing sound speed at constant baryon +density. + +Speed of sound in QCD matter at finite temperature and density +11 +0 +2 +4 +6 +8 +10 +12 +14 +50 +100 +150 +200 +250 +300 +c +2 +B +T(MeV) +B +/ +0 +-0.35 +-0.30 +-0.25 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 + chiral crossover + chiral first order + spinodal line + CEP + deconfinement line +Fig. 6 +Contour map of c2 +ρB in the T − ρB plane. The red dashed line is the boundary of +vanishing sound velocity. +2 +4 +6 +8 +10 +12 +14 +50 +100 +150 +200 +250 +300 +c +2 +s +T(MeV) +B +/ +0 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +s=0.5 +s=0.1 +s=1.0 +s=5.0 + chiral crossover + chiral first order + spinodal line + CEP + deconfinement line +0 +Fig. 7 +Contour map of c2 +s in the T − ρB plane. The red dashed line is the boundary of +vanishing sound velocity. The green curves shows the paths of s = 0.1, 0.5, 1.0, 5. +The contour map of c2 +s is presented in Fig. 7. The behavior of c2 +s at low +temperature and high density is similar with that of c2 +s/ρB, because the curves +at constant s and s/ρB in the T − ρB diagram are both roughly parallel with +the density axis, as indicated in Fig. 1. and Fig. 7. However, the curves at +constant s at low density are almost perpendicular to those at constant s/ρB, +the resulting behaviors of c2 +s and c2 +s/ρB are quite different in the corresponding +region. A distinct characteristic is the location of vanishing sound speed. The +red dashed line in Fig. 7 is the boundary of c2 +s = 0. ( ∂p +∂ǫ )s takes minus val- +ues inside the boundary, which includes a wide range of the first-order phase +transition and a region around the CEP. + +12 +Speed of sound in QCD matter at finite temperature and density +0 +2 +4 +6 +8 +10 +12 +14 +50 +100 +150 +200 +250 +T (MeV) +B +/ +0 +Fig. 8 +Regions of ( ∂µB +∂T )s/ρB < 0 in the T − ρB panel. +The region of c2 +s < 0 in Fig. 7 and c2 +ρB < 0 in Fig. 6 are connected with the +formula +�∂µB +∂T +� +s/ρB += µB +T +( ∂p +∂ǫ )s +( ∂p +∂ǫ )ρB += µB +T +c2 +s +c2ρB +. +(19) +When the condition ( ∂µB +∂T )s/ρB < 0 is fulfilled, one of the two physical quan- +tities c2 +s and c2 +ρB takes a negative value. We show in Fig. 8 the regions of +( ∂µB +∂T )s/ρB < 0. The numerical results indicate that there indeed exists the +regions where ( ∂µB +∂T )s/ρB < 0 in the T − ρB diagram. Comparing Fig. 8 with +the negative value regions in Fig. 6 and Fig. 7, we can conclude that these +numerical results confirm the formula in Eqs. (19). +The behaviors of c2 +ρB and c2 +s in the phase diagram can be used to study the +fluid properties of quark gluon plasma. The values of c2 +ρB and c2 +s are important +parameters to indicate the intermediated process in the evolution of a fluid. +Eqs. (4) and (5) clearly show that c2 +ρB and c2 +s are connected with the temporal +derivatives of temperature and chemical potential, respectively. Moreover, c2 +ρB +and c2 +s are related to the bulk viscosity of a fluid. In particular they directly +connect with the bulk viscosity coefficient. Exploring the relation between the +bulk viscosity and phase transition is attractive to study the dissipation in the +evolution of QGP. A further research in this respect is undergoing. +3.3 Sound velocity at constant T and µB +The contour maps of c2 +T at constant temperature and c2 +µB at constant chem- +ical potential in the T − ρB panel are demonstrated in Fig. 9 and Fig. 10, +respectively. The two figures show that c2 +T and c2 +µB are both close to 1/3 at +high temperature. +The contour of c2 +T looks in general like that of c2 +s/ρB, because the curves +of constant s/ρB in the T − ρB panel are almost parallel to the density axis +in a wide range. The relative larger deviation lies in the range with densities +smaller than the boundary of the first-order transition on the low-density side. + +Speed of sound in QCD matter at finite temperature and density +13 +The deviation produces different behaviors between ( ∂p +∂ǫ )T and ( ∂p +∂ǫ )s/ρB at low +density. A crutial point is that the inflections points of ( ∂p +∂ǫ )T and ( ∂p +∂ǫ )s/ρB are +different, i.e., the boundary of vanishing sound speed are different for the two +cases. +0 +2 +4 +6 +8 +10 +12 +14 +50 +100 +150 +200 +250 +300 +c +2 +T +T(MeV) +B +/ +0 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 + chiral crossover + chiral first order + spinodal line + CEP + deconfinement line +Fig. 9 Contour map of c2 +T in the T − ρB plane. The red dashed line is the boundary of +vanishing sound velocity. +0 +2 +4 +6 +8 +10 +12 +14 +50 +100 +150 +200 +250 +300 +c +2 +B +T(MeV) +B +/ +0 +-0.35 +-0.30 +-0.25 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 + chiral crossover + chiral first order + spinodal line + CEP + deconfinement line +Fig. 10 +Contour map of c2 +µB in the T − ρB plane. The red dashed line is the boundary of +vanishing sound velocity. +Both the figure 9 and 10 show that the boundary of zero sound speed at +constant temperature or chemical potential is just the spinodal line associated +with the first-order phase transition. However the boundary of zero sound +velocity at constant s/ρB is in the interior of the spinodal structure as shown + +14 +Speed of sound in QCD matter at finite temperature and density +in Fig. 4. The negative values of c2 +T and c2 +µB both appear in the unstable phase +of the spinodal structure, since ( ∂p +∂ǫ )T and ( ∂p +∂ǫ )µB are negative in this region. +From the behavior of c2 +T derived above, we can approximately deduce the +speed of sound in the quark core of a massive neutron star. Since the richness +of lepton including electron and muon approaches to zero at high density of a +hybrid neutron star, and the richness of u, d, s quark tends to be equivalent [56], +quite similar to the situation in this study at high density and low temperature. +Therefore, it may be concluded to a certain degree that the squared speed of +sound in the quark core of a massive neutron star gradually approaches to 1/3 +at high density. A further study on the speed of sound in neutron star matter +with a hadron-quark phase transition will be conducted with the combination +of observation data. +4 Summary +In this work, we studied the speed of sound in QCD matter at finite temper- +ature and density in the PNJL model. We derived the behavior of speed of +sound under different definitions in the T − ρB phase diagram including the +stable phase, metastable and unstable phases associated with the first-order +phase transition. We systematically discussed the relations between the speed +of sound and QCD phase structure. +The numerical results indicate that the squared speed of sounds under +different definitions are all approaching to 1/3 at high temperature. However, +the behaviors in the phase-transition region are closely related to the non- +perturbative interaction and the phase structure. From the perspective of idea +fluid evolution, more attention are put on the speed of sound under isentropic +condition. The calculation indicates that c2 +s/ρB is nonzero at the CEP under +the mean field approximation, and the boundary of vanishing sound velocity +is further derived. +We also obtained the contour maps of c2 +X (X = s, ρB, T, µB), and analyzed +their relations with the QCD phase transition, as well as the relations between +different definitions. For each definition of speed of sound, we find that there +exist one or several regions with ( ∂p +∂ǫ )X < 0, even in the stable phase for X = s +and ρB. +The different definitions of speed of sound are involved in some impor- +tant physics problems in nuclear physics, such as the evolution of quark gluon +plasma, the cosmic QCD phase transition, the bulk viscosity of strongly inter- +acting matter and the equation of state of a hybrid neutron star. The indepth +investigations on these physics issues will be performed in the future. +Acknowledgments. +This work is supported by the National Natural Sci- +ence Foundation of China under Grant No. 11875213. +Data Availability. +Data sharing not applicable to this article as no datasets +were generated or analyzed during the current study. + +Speed of sound in QCD matter at finite temperature and density +15 +References +[1] H. C. Song, S. A. Bass, U. Heinz, T. Hirano, and C. Shen, 106, 192301 +(2011). +[2] H. C. Song, S. A. Bass, U. Heinz, Phys. Rev. C 83, 024912 (2011). +[3] P. Deb, G. P. Kadam, and H. Mishra, Phys. Rev. D 94, 094002 (2016). +[4] F. G. Gardim, G. Giacalone, M. Luzum, and J. Y. Ollitrault, Nat. Phys. +16, 615 (2020). +[5] D. Sahu, S. Tripathy, R. Sahoo, and A. R. Dash, Eur. Phys. J. A 56, 187 +(2021). +[6] D. Biswas, K. Deka, A. Jaiswal, and S. Roy, Phys. Rev. C 102, 014912 +(2020). +[7] A. Sorensen, D. Oliinychenko, V. Koch, and L. McLerran, Phys. Rev. +Lett. 127, 042303 (2021). +[8] B. Reed and C. J. Horowitz, Phys. Rev. C 101, 045803 (2020). +[9] A. Kanakis-Pegios, P. S. Koliogiannis, and Ch. C. Moustakidis, Phys. Rev. +C 102, 055801 (2020). +[10] S. Han and M. Prakash, Astrophys. J. 899, 164 (2020). +[11] I. Tews, J. Carlson, S. Gandolfi, and S. Reddy, Astrophys. J. 860, 149 +(2018). +[12] S. K. Greif, G. Raaijmakers, K. Hebeler, A. Schwenk, and A. L. Watts, +Mon. Not. R. Astron. Soc. 485, 5363 (2019). +[13] M. M. Forbes, S. Bose, S. Reddy, D. Zhou, A. Mukherjee, and S. De, +Phys. Rev. D 100, 083010 (2019). +[14] C. Drischler, S. Han, J. M. Lattimer, M. Prakash, S. Reddy, and T. Zhao, +Phys. Rev. C 103, 045808 (2021). +[15] R. Essick, I. Tews, P. Landry, S. Reddy, and D. E. Holz, Phys. Rev. C +102, 055803 (2020). +[16] S. Han, M. A. A. Mamun, S. Lalit, C. Constantinou, and M. Prakash, +Phys. Rev. D 100, 103022 (2019). +[17] T. Kojo, AAPPS Bull. 31, 11 (2021). + +16 +Speed of sound in QCD matter at finite temperature and density +[18] P. Jaikumar, A. Semposki, M. Prakash, and C. Constantinou, Phys. Rev. +D 103, 123009 (2021). +[19] O. Philipsen, Prog. Part. Nucl. Phys. 70, 55 (2013). +[20] S. Bors´anyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. +Pasztor, C. Ratti, and K. K. Szabo, Phys. Rev. Lett. 125, 052001 (2020). +[21] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, K. K. Szabo, Nature (London) +443, 675 (2006). +[22] S. Bors´anyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, and K. K. Sabz´o, +Phys. Lett. B 730, 99 (2014). +[23] A. Bazavov et al. (hotQCD Collaboration), Phys. Rev. D. 90, 094503 +(2014). +[24] M. Motta, R. Stiele, W. M. Alberico, A. Beraudo, Eur. Phys. J. C 80, +770 (2020). +[25] S. K. Ghosh, T. K. Mukherjee, M. G. Mustafa, and R. Ray, Phys. Rev. +D 73, 114007 (2006). +[26] R. Marty, E. Bratkovskaya, W. Cassing, J. Aichelin, and H. Berrehrah, +Phys. Rev. C 88, 045204 (2013). +[27] K. Saha, S. Ghosh, S. Upadhaya, S. Maity, Phys. Rev. D 97, 116020 +(2018). +[28] Y. P. Zhao, Phys. Rev. D 101, 096006, (2020). +[29] B. J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D 81, 074013 +(2010). +[30] A. Abhishek, H. Mishra, and S. Ghosh, Phys. Rev. D 97, 014005 (2018). +[31] R. Venugopalan and M. Prakash, Nucl. Phys. A546, 718 (1992). +[32] M. Bluhm, P. Alba, W. Alberico, A. Beraudo, and C. Ratti, Nucl. Phys. +A929, 157 (2014). +[33] Z. V. Khaidukov, M. S. Lukashov, and Yu. A. Simonov, Phys. Rev. D 98, +074031 (2018). +[34] Z. V. Khaidukov and Yu. A. Simonov, Phys. Rev. D 100, 076009 (2019). +[35] V. Mykhaylova and C. Sasaki, Phys. Rev. D 103, 014007 (2021). + +Speed of sound in QCD matter at finite temperature and density +17 +[36] W. B. He, G. Y. Shao, X. Y. Gao, X. R. Yang, and C. L. Xie, Phys. Rev. +D 105, 094024 (2022). +[37] I. N. Mishustin, Phys. Rev. Lett. 82, 4779 (1999). +[38] J. Randrup, Phys. Rev. Lett. 92, 122301 (2004). +[39] V. Koch, A. Majumder, and J. Randrup, Phys. Rev. C 72, 064903 (2005). +[40] C. Sasaki, B. Friman, and K. Redlich, Phys. Rev. Lett. 99, 232301 (2007). +[41] C. Sasaki, B. Friman, and K. Redlich, Phys. Rev. D 77, 034024 (2008). +[42] J. Randrup, Phys. Rev. C 79, 054911 (2009); Phys. Rev. C 82, 034902 +(2010). +[43] J. Steinheimer and J. Randrup, Phys. Rev. Lett. 109, 212301 (2012). +[44] J. Steinheimer and J. Randrup, Eur. Phys. J. A 52, 239 (2016). +[45] F. Li and C. M. Ko, Phys. Rev. C 93, 035205 (2016). +[46] J. Steinheimer and V. Koch, Phys. Rev. C. 96, 034907 (2017). +[47] G. Y. Shao, X. Y. Gao, and W. B. He, Eur. Phys. J. A 56, 115 (2020). +[48] M. Ahmadvand, K. Bitaghsir Fadafan, Phys. Lett. B 779, 1 (2018). +[49] Tuomas V. I. Tenkanen and Jorinde van de Vis, J. High Energ. Phys. +2022, 302 (2022). +[50] S. R¨oßner, C. Ratti, and W. Weise, Phys. Rev. D 75, 034007 (2007). +[51] P. Rehberg, S. P. Klevansky, and J. H¨ufner, Phys. Rev. C 53, 410 (1996). +[52] P. Costa, M. C. Ruivo, C. A. de Sousa, and H. Hansen, Symmetry 2, 1338 +(2010). +[53] G. Y. Shao, Z. D. Tang, X. Y. Gao, and W. B. He, Eur. Phys. J . C 78, +138 (2018). +[54] P. Chomaz, M. Colonna, and J. Randrup, Phys. Rep. 389 263 (2004). +[55] G. Y. Shao, X. Y. Gao, and W. B. He, Phys.Rev.D 101, 074029 (2020). +[56] G. Y. Shao, M. Colonna, M. Di Toro, Y. X. Liu, and B. Liu, Phys.Rev.D +87, 096012 (2013). + +18 +Speed of sound in QCD matter at finite temperature and density +A Derivations of the formulae of speed of +sound under different definitions in the +temperature and density space +The general definition of speed of sound is +c2 +X = +�∂p +∂ǫ +� +X +, +(20) +where X is a physics quantum fixed in the calculation of sound speed. In prac- +tice, the squared speed of sound c2 +X (X = s/ρB, s, ρB, T, µB) under different +conditions are taken in dealing with different physics problems. With the basic +definitions of speed of sound, calculation can only be done along some special +paths. +To calculate the speed of sound under different definitions in the whole +T − ρB space, it is necessary to derive the corresponding formulae in terms of +temperature and density. Using the Jacobian formula in thermodynamics, we +can derive +c2 +X (T, ρB) = +�∂p +∂ǫ +� +X += ∂(p, X) +∂(ǫ, X) = +∂(p,X) +∂(T,ρB) +∂(ǫ,X) +∂(T,ρB) += +� +∂p +∂T +� +ρB +� +∂X +∂ρB +� +T − +� +∂p +∂ρB +� +T +� ∂X +∂T +� +ρB +� ∂ǫ +∂T +� +ρB +� +∂X +∂ρB +� +T − +� +∂ǫ +∂ρB +� +T +� ∂X +∂T +� +ρB +(21) +According to the thermodynamic characteristic function in the giant canon- +ical ensemble, it is convenient to get the following relations for isospin +symmetric matter +� ∂p +∂T +� +ρB += s + ρB +�∂µB +∂T +� +ρB +, +(22) +� ∂p +∂ρB +� +T += ρB +�∂µB +∂ρB +� +T +, +(23) +� ∂ǫ +∂T +� +ρB += T +� ∂s +∂T +� +ρB +, +(24) +� ∂ǫ +∂ρB +� +T += T +� ∂s +∂ρB +� +T ++ µB, +(25) +�∂(s/ρB) +∂T +� +ρB += 1 +ρB +� ∂s +∂T +� +ρB +, +(26) +and +�∂(s/ρB) +∂ρB +� +T += 1 +ρB +� ∂s +∂ρB +� +T +− s +ρ2 +B +(27) + +Speed of sound in QCD matter at finite temperature and density +19 +For the different constraint conditions, X = s/ρB, s, ρB, T, µB, we can +derived the corresponding formulae of speed of sound as follows +c2 +s/ρB = +s2+ρ2 +B +�� +∂µB +∂ρB +� +T +� ∂s +∂T +� +ρB − +� +∂µB +∂T +� +ρB +� +∂s +∂ρB +� +T +� ++sρB +�� +∂µB +∂T +� +ρB +− +� +∂s +∂ρB +� +T +� +(T s + µBρB) +� ∂s +∂T +� +ρB +, +(28) +c2 +s = +ρB +�� ∂s +∂T +� +ρB +� +∂µB +∂ρB +� +T − +� +∂s +∂ρB +� +T +� +∂µB +∂T +� +ρB +� +− s +� +∂s +∂ρB +� +T +µB +� ∂s +∂T +� +ρB +, +(29) +c2 +ρB = +s + ρB +� +∂µB +∂T +� +ρB +T +� ∂s +∂T +� +ρB +, +(30) +c2 +T = +ρB +� +∂µB +∂ρB +� +T +T +� +∂s +∂ρB +� +T + µB +, +(31) +and +c2 +µB = +s +� +∂µB +∂ρB +� +T +T +�� ∂s +∂T +� +ρB +� +∂µB +∂ρB +� +T − +� +∂µB +∂T +� +ρB +� +∂s +∂ρB +� +T +� +− µB +� +∂µB +∂T +� +ρB +. +(32) + diff --git a/Q9E3T4oBgHgl3EQfDAmJ/content/tmp_files/load_file.txt b/Q9E3T4oBgHgl3EQfDAmJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e57aa1f19d1abc7fb9e4369945ce2a6495a68ff8 --- /dev/null +++ b/Q9E3T4oBgHgl3EQfDAmJ/content/tmp_files/load_file.txt @@ -0,0 +1,853 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf,len=852 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='04282v1 [hep-ph] 11 Jan 2023 Speed of sound in QCD matter at finite temperature and density Guo-yun Shao1*, Xin-ran Yang1, Chong-long Xie1 and Wei-bo He2 1School of Physics, Xi’an Jiaotong University, Xi’an, 710049, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2School of Physics, Peking University, Beijing, 100871, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' E-mail(s): gyshao@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Abstract The speed of sound in QCD matter at finite temperature and den- sity is investigated within the Polyakov loop improved Nambu–Jona- Lasinio (PNJL) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The spinodal structure associated with the chiral first-order chiral phase transition is considered to describe the continuous variation of the speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The behaviors of the squared sound speed in different phases, including the stable, metastable and unstable phases, are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The relation between speed of sound and QCD phase transitions is systematically explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In par- ticular, the boundary of vanishing sound velocity is derived in the temperature-density phase diagram, and the region where the sound wave equation being broken is pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Some interesting fea- tures of speed of sound under different definitions are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Keywords: Speed of sound, Quark matter, Chiral phase transition 1 Introduction Quark-gluon plasma (QGP) can be created in heavy-ion collision (HIC) experi- ments at relativistic energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A crucial topic relevant is to explore the equation of state (EOS) and phase transition from QGP to hadronic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The hydro- dynamic simulation provides a method to study the EOS of QGP [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' During 1 2 Speed of sound in QCD matter at finite temperature and density the space-time evolution of QCD matter, the speed of sound is one of the cru- cial physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Its dependence on environment (temperature, density, chemical potential, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=') carries important information in describing the evo- lution of the fireball and final observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Recently, the studies in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [4–6] show that the speed of sound as a function of charged particle multiplic- ity ⟨dNch/dη⟩ can be extracted from heavy-ion collision data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [7] the authors try to build a connection between the sound speed and baryon number cumulants to study the QCD phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The speed of sound in neutron star has also received a lot of attention (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [8–10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The density dependent behavior of sound velocity influences the mass-radius relation, the tidal deformability, and provides a sensitive probe of the EOS of neutron star matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' To obtain a two solar mass neutron star, some studies find that it is essential for neutron star matter to have a density range where the EOS is very stiff and the corresponding squared speed of sound is significantly larger than 1/3 [11–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The study in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [18] indicates that the speed of sound is crucial for the gravitational wave frequencies induced by the g-mode oscillation of a neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' It is also interesting to study the gravitational wave induced by the cosmic QCD phase transition in which the speed of sound plays a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' As an important quantum in describing the evolution of strongly inter- acting matter, the relation between the speed of sound and QCD phase transition is worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The speed of sound has been calculated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=', in lattice QCD [19–23], (P)NJL model [3, 24–28], quark-meson coupling model [29, 30], hadron resonance gas (HRG) model [31, 32], field correlator method (FCM) [33, 34] and quasiparticle model [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In previous studies, the main focus is put on the region of high temperature and vanishing or small chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [36], we give an intensive study on the speed of sound in QCD matter in the full temperature-chemical potential phase dia- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The numerical results indicate that the dependence of sound speed on temperature and chemical potential is indicative of QCD phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' However, only the sound speed in the stable phase is considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' There are still some crucial issues that need to be clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' First of all, the spinodal structure may be involved in heavy-ion collision experiments with the decrease of collision energy [37–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A complete evolution of sound speed in the metastable and unstable phases needs to be explored to give a distinct description of the fireball expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Secondly, it is found that the sound speed takes small values at the CEP and on the boundaries of the first-order phase transition near the CEP [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A question aroused is that where is the boundary of vanishing speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Furthermore, the behavior of sound speed in the temperature and density phase diagram is still not explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' On the other hand, the values of speed of sound under different conditions are involved in dealing with different problems in nuclear physics, such as the gravitational signal from cosmic QCD phase transition [48, 49], the bulk viscosity of strongly interacting matter [3], the equation of state of neutron star matter [8–10] and the evolution of QGP in HIC experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In this Speed of sound in QCD matter at finite temperature and density 3 work, we will give a systematic study on the relation between speed of sound and QCD phase transitions at finite temperature and density under several different constraint conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' This work is helpful in dealing with the physics problems mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' II, we derive the formulae of speed of sound under different definitions in the temperature and density space, and then briefly introduce the 2+1 flavor PNJL quark model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' III, we present the numerical results of squared sound speed and discuss the relations with the QCD phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A summary is finally given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2 Speed of sound and the PNJL quark model The general definition of speed of sound is c2 X = �∂p ∂ǫ � X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (1) A specifying constant quantity X is required to describe the propagation of the compression wave through a medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' To indicate the different profiles of QCD matter, X can be chosen as s/ρB, s, ρB, T, µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Different definitions of speed of sound are taken in practice in dealing with different physics issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' For a fireball created in relativistic heavy-ion collisions, it evolves with a constant entropy density per baryon s/ρB if it is taken as an ideal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Therefore, it is meaningful to calculate the speed of sound along the isentropic curve c2 s/ρB = �∂p ∂ǫ � s/ρB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (2) The dependence of c2 s/ρB on parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=', temperature and density, can indicate the variation of sound speed during the evolution and provide important knowledge of interaction, phase transition and the EOS of QGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The speed of sound with constant baryon number density or entropy density are taken in describing the intermediate process of a hydrodynamic evolution [3], c2 ρB = �∂p ∂ǫ � ρB and c2 s = �∂p ∂ǫ � s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (3) For example, the temporal derivatives of temperature and chemical potential are functions of c2 ρB and c2 s, as ∂0µB = −c2 sµB ∇ · u, (4) and ∂0T = −c2 ρBT ∇ · u, (5) where u denotes the space component of four-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The values of c2 ρB and c2 s are directly connected to the bulk viscosity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 4 Speed of sound in QCD matter at finite temperature and density It is also interesting to calculate the sound speed with a fixed temperature or chemical potential c2 T = �∂p ∂ǫ � T , c2 µB = �∂p ∂ǫ � µB , (6) In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [7] the authors estimate c2 T as a function of the logarithmic deriva- tive with respect to the baryon density of QCD matter, and try to build a connection with the baryon number cumulants to aid in detecting the QCD critical endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Besides, c2 T is also usually taken to study the speed of sound in neutron star matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In this study, we will explore the speed of sound under different definitions in the full temperature-density space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Since the general definitions can only be used to calculate the sound speed on special trajectories, it is necessary to derive the corresponding formulae in terms of T and ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' With the fundamental thermodynamic relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' the sound speed formulae under different constraint conditions can be derived as c2 s/ρB = s2+ρ2 B �� ∂µB ∂ρB � T � ∂s ∂T � ρB − � ∂µB ∂T � ρB � ∂s ∂ρB � T � +sρB �� ∂µB ∂T � ρB − � ∂s ∂ρB � T � (T s + µBρB) � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (7) c2 s = ρB �� ∂s ∂T � ρB � ∂µB ∂ρB � T − � ∂s ∂ρB � T � ∂µB ∂T � ρB � − s � ∂s ∂ρB � T µB � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (8) c2 ρB = s + ρB � ∂µB ∂T � ρB T � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' c2 T = ρB � ∂µB ∂ρB � T T � ∂s ∂ρB � T + µB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (9) and c2 µB = s � ∂µB ∂ρB � T T �� ∂s ∂T � ρB � ∂µB ∂ρB � T − � ∂µB ∂T � ρB � ∂s ∂ρB � T � − µB � ∂µB ∂T � ρB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (10) The details for deriving these formulae are affiliated in the appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The above formulae are only correct for isospin symmetric matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The corre- sponding formulae will be much more complicated for isospin asymmetric matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' To demonstrate the relation between the speed of sound under different definition and QCD phase structure, we take the 2+1 flavor PNJL quark model Speed of sound in QCD matter at finite temperature and density 5 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The Lagrangian density is given by L = ¯q(iγµDµ+γ0ˆµ− ˆm0)q+G 8 � k=0 � (¯qλkq)2+(¯qiγ5λkq)2� −K � detf(¯q(1 + γ5)q) + detf(¯q(1 − γ5)q) � −U(Φ[A], ¯Φ[A], T ), (11) where q denotes the quark fields with three flavors, u, d, and s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' ˆm0 = diag(mu, md, ms) in flavor space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' G and K are the four-point and six-point interacting constants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The ˆµ = diag(µu, µd, µs) are the quark chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The covariant derivative in the Lagrangian is defined as Dµ = ∂µ−iAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The gluon background field Aµ = δ0 µA0 is supposed to be homogeneous and static, with A0 = gAα 0 λα 2 , where λα 2 is SU(3) color generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The effective potential U(Φ[A], ¯Φ[A], T ) is expressed with the traced Polyakov loop Φ = (TrcL)/NC and its conjugate ¯Φ = (TrcL†)/NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The Polyakov loop L is a matrix in color space L(⃗x) = Pexp � i � β 0 dτA4(⃗x, τ) � , (12) where β = 1/T is the inverse of temperature and A4 = iA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The Polyakov-loop effective potential is U(Φ, ¯Φ, T ) T 4 = −a(T ) 2 ¯ΦΦ + b(T )ln � 1 − 6¯ΦΦ + 4(¯Φ3 + Φ3) − 3(¯ΦΦ)2� ,(13) where a(T )=a0+a1 �T0 T � +a2 �T0 T �2 and b(T )=b3 �T0 T �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (14) The parameters ai, bi listed in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 1 are fitted according to the lattice simulation of QCD thermodynamics in pure gauge sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The T0 = 210 MeV is implemented in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Table 1 Parameters in the Polyakov-loop potential [50] a0 a1 a2 b3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='47 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='75 The constituent quark mass in the mean field approximation can be derived as Mi = mi − 4Gφi + 2Kφjφk (i ̸= j ̸= k), (15) where φi stands for quark condensate of the flavor i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 6 Speed of sound in QCD matter at finite temperature and density The thermodynamical potential of bulk quark matter is derived as Ω = −2T � i=u,d,s � d3p (2π)3 (Q1 + Q2) − 2 � Λ d3p (2π)3 3(Eu + Ed + Es) +2G � φu 2 + φd 2 + φs 2� − 4Kφu φd φs + U(¯Φ, Φ, T ) (16) where Q1 = ln(1 + 3Φe−(Ei−µi)/T + 3¯Φe−2(Ei−µi)/T + e−3(Ei−µi)/T ), Q2 = ln(1 + 3¯Φe−(Ei+µi)/T + 3Φe−2(Ei+µi)/T + e−3(Ei+µi)/T ), and Ei = � ⃗p 2 + M 2 i is the dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' µi = µB/3 is taken for u, d, s quark flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The pres- sure p and energy density ǫ can be derived using the thermodynamic relations in the grand canonical ensemble as P = −Ω, ǫ = −P + T s + � µiρi, (17) where s is the entropy density and ρi the quark number density of flavor i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' For given T and baryon density ρB, the values of φu, φd, φs, Φ, ¯Φ and µB are determined by solving the equations by minimizing the thermodynamical potential ∂Ω ∂φu = ∂Ω ∂φd = ∂Ω ∂φs = ∂Ω ∂Φ = ∂Ω ∂ ¯Φ = 0, (18) and the relevant constraint condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Other physical quantities can be then derived using thermodynamic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The numerical results of speed of sound under different conditions can be then derived according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (7)-(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In the numerical calculation, a cut-off Λ is implemented in 3-momentum space for divergent integrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We take the model parameters obtained in [51]: Λ = 602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='3 MeV, GΛ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='835, KΛ5 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='36, mu,d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='5 and ms = 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='7 MeV, determined by fitting fπ = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='4 MeV, Mπ = 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 MeV, mK = 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='7 MeV and mη = 957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='8 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3 Numerical results and discussions In this section, we present the numerical results of the speed of sound under different constraint conditions and discuss the relations with the QCD phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 Sound velocity at constant s/ρB Firstly, we plot the QCD phase diagram, including the first-order phase transition (black solid line) and the spinodal structure (blue dashed line), which separate the phase diagram into the stable, metastable and unstable phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The first-order phase transition line is obtained according to the ther- modynamic conditions for two-phase equilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=', T1 = T2, µ1 = µ2, P1 = P2 for two stable phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The spinodal line is derived with the mechan- ical unstable condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The corresponding inflection points of pressure as a function of density can be determined for a given T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' For more details to Speed of sound in QCD matter at finite temperature and density 7 derive the phase boundaries, one can refer to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The spinodal phase decomposition plays a dominant role in the experimental exploration of the first-order nuclear liquid-gas transition[54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' It has inspired the antic- ipation to identify the first-order chiral transition in high-energy heavy-ion collisions through the spinodal phase separation [37–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The recent simula- tion suggests that the spinodal instability can be triggered within a certain energy range [44, 46] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We also demonstrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 1 the isentropic curves with s/ρB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1, 1, 3, 5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9, 10, 50, 100, 300 in the T − ρB plane to indicate the evolutionary trajectories of an ideal fluid at different collision energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' metastable phase unstable phase 1st-order transition line spinodal line 0 1 2 3 4 5 6 50 100 150 200 250 300 CEP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9 10 50 100 T(MeV) r B /r 0 300 5 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 1 QCD phase diagram and the isentropic curves for s/ρB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1, 1, 3, 5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9, 10, 50, 100, 300 in T − ρB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2 the equations of state with chosen parameter s/ρB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1, 1, 3, 5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9 that pass through the first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' For each curve with s/ρB < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9, there are two inflection points where ∂p/∂ǫ changes the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2 is the connections of these inflection points, which lies in the spinodal boundary associated with the first-order phase transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=', in the interior of the unstable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We will present the relation clearly in the contour map of speed of sound in the T − ρB panel soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The inflection points also correspond to the locations where the sound velocity vanishes in the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The square of speed of sound can be directly derived with the definition given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3, we plot the curve of squared sound speed c2 s/ρB as functions of energy density along the evolutionary trajectories for s/ρB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1, 1, 3, 5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Figure 3 shows that there exists one peak and two valleys on each curve, which indicates that the speed of sound are closely related to temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In particular, for the case of c2 s/ρB < 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3, it corresponds to ∂p/∂ǫ < 0 for a fixed s/ρB, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 8 Speed of sound in QCD matter at finite temperature and density inflection points s/r B =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 s/r B =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 s/r B =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 s/r B =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 s/r B =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9 Pressure(MeVfm 3 ) e(MeVfm 3 ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2 Equations of state of QCD matter for s/ρB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1, 1, 3, 5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The squares on each curve are the inflection points where ( ∂p ∂ǫ )s/ρB changes the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the profile of these inflection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='5 C 2 s/r B e(MeVfm 3 ) s/r B =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 s/r B =1 s/r B =3 s/r B =5 s/r B =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3 Values of c2 s/ρB as functions of energy density for s/ρB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1, 1, 3, 5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' To show more clearly the relation between the speed of sound and QCD phase transitions, we present the contour map and 3D map of c2 s/ρB as functions of T and ρB in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The phase structure includ- ing the chiral crossover, chiral first-order, spinodal and deconfinement phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The chiral crossover line and deconfinement line are determined by requiring ∂φi/∂T and ∂Φi/∂T taking extreme values for a given chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The two figures indicate that the region around the peaks of c2 s/ρB in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3 is located in the region where the chrial symmetry of u, d quark is approxi- mately restored already but still confined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The valley at high energy density 000008 a00eoo 1001 1400 002S00 30040 02 oaSO 30O0 00 SOSpeed of sound in QCD matter at finite temperature and density 9 side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3 lies in the region where the chiral condensate of strange quark changes quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' After the chiral restoration of strange quark the sound speed increases again towards high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The valley in the low density side is located in the spinodal region of the first-order phase transition, which is closely related to the chiral condensate of u, d quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' For each value of s/ρB smaller than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='9, there exist a range of ( ∂p ∂ǫ )s/ρB taking negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity with c2 s/ρB = 0 derived with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (7) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 50 100 150 200 250 300 T(MeV) B / 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='30 c 2 s/ B chiral crossover chiral first order spinodal line CEP deconfinement line Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 4 Contour map of c2 s/ρB in the T − ρB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Figure 4 and 5 also show that, at the high temperature or very high density, with the restoration of chiral symmetry c2 s/ρB approaches to 1/3, the value of noninteracting gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The value of c2 s/ρB at lower density descends with the decrease of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A rapid decrease occurs in the chiral crossover region of u, d quark, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' It indicates that the value of speed of sound is sensitive to the change of dynamical quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In the low-density region, a minimum of c2 s/ρB appears near the deconfinement phase transition for a given density (chemical potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A similar behavior exists in lattice QCD at zero chemical potential [19–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' However, such a feature does not appear in the NJL model which cannot describe the confinement-deconfinement phase transition [3, 25–27], which indicates that the color confinement also plays an important role on the speed of sound near the crossover phase transition line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The value of c2 s/ρB is relatively smaller in the region of low temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Inside this boundary c2 s/ρB < 0, it means physically that ( ∂p ∂ǫ )s/ρB < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In this region, the mechanically stable condition is broken and the corresponding sound wave equation becomes a decay function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A perturbance can not be propagated like a sound wave in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' It can be seen that such a region lies in the interior of the unstable phase of the spinodal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Figure 4 and 5 also 10 Speed of sound in QCD matter at finite temperature and density 0 2 4 6 8 10 12 14 16 18 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='3 T(MeV) C 2 s/r B r B /r 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 5 3D map of c2 s/ρB as functions of tempeature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The black solid line is the first-order chiral phase transition line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The blue dashed line is the spinodal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' indicate that the speed of sound at the critical endpoint is small but not zero in the mean field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The value of c2 s/ρB reflects the speed of sound in an ideal fluid which can be approximately realized in heavy-ion collision experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' On the other hand, s/ρB is connected with the collision energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' If the value of c2 s/ρB at a fixed energy can be extracted from the charged particle multiplicity ⟨dNch/dη⟩ [4– 6], we can access the information of phase transition using the relation between c2 s/ρB and QCD phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Furthermore, combining with the beam energy scan experiments, it provides a possible way to diagnose the QCD phase struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' It is also inspiring for study on the gravitational signal from the cosmic QCD phase transition in which a constant speed of sound � 1/3 is usually taken in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='2 Sound velocity at constant ρB and s We present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 6 the contour map of c2 ρB in the T − ρB panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Besides at the high-temperature side, this figure shows that c2 ρB take relatively larger values in the region of low temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The value is even larger than 1/3, in particular, in the metastable phase and unstable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' There also exists a wide region (inside the red line filled with the blue color) of c2 ρB < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' More physically, it means that the ( ∂p ∂ǫ )ρB < 0 in this region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=', the pressure decreases with the increase of energy density along the line of constant density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red line shows the boundary of vanishing sound speed at constant baryon density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Speed of sound in QCD matter at finite temperature and density 11 0 2 4 6 8 10 12 14 50 100 150 200 250 300 c 2 B T(MeV) B / 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='35 chiral crossover chiral first order spinodal line CEP deconfinement line Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 6 Contour map of c2 ρB in the T − ρB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2 4 6 8 10 12 14 50 100 150 200 250 300 c 2 s T(MeV) B / 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='30 s=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='5 s=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1 s=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 s=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0 chiral crossover chiral first order spinodal line CEP deconfinement line 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 7 Contour map of c2 s in the T − ρB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The green curves shows the paths of s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The contour map of c2 s is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The behavior of c2 s at low temperature and high density is similar with that of c2 s/ρB, because the curves at constant s and s/ρB in the T − ρB diagram are both roughly parallel with the density axis, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' However, the curves at constant s at low density are almost perpendicular to those at constant s/ρB, the resulting behaviors of c2 s and c2 s/ρB are quite different in the corresponding region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A distinct characteristic is the location of vanishing sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 7 is the boundary of c2 s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' ( ∂p ∂ǫ )s takes minus val- ues inside the boundary, which includes a wide range of the first-order phase transition and a region around the CEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 12 Speed of sound in QCD matter at finite temperature and density 0 2 4 6 8 10 12 14 50 100 150 200 250 T (MeV) B / 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 8 Regions of ( ∂µB ∂T )s/ρB < 0 in the T − ρB panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The region of c2 s < 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 7 and c2 ρB < 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 6 are connected with the formula �∂µB ∂T � s/ρB = µB T ( ∂p ∂ǫ )s ( ∂p ∂ǫ )ρB = µB T c2 s c2ρB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (19) When the condition ( ∂µB ∂T )s/ρB < 0 is fulfilled, one of the two physical quan- tities c2 s and c2 ρB takes a negative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 8 the regions of ( ∂µB ∂T )s/ρB < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The numerical results indicate that there indeed exists the regions where ( ∂µB ∂T )s/ρB < 0 in the T − ρB diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 8 with the negative value regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 7, we can conclude that these numerical results confirm the formula in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The behaviors of c2 ρB and c2 s in the phase diagram can be used to study the fluid properties of quark gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The values of c2 ρB and c2 s are important parameters to indicate the intermediated process in the evolution of a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (4) and (5) clearly show that c2 ρB and c2 s are connected with the temporal derivatives of temperature and chemical potential, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Moreover, c2 ρB and c2 s are related to the bulk viscosity of a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In particular they directly connect with the bulk viscosity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Exploring the relation between the bulk viscosity and phase transition is attractive to study the dissipation in the evolution of QGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A further research in this respect is undergoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='3 Sound velocity at constant T and µB The contour maps of c2 T at constant temperature and c2 µB at constant chem- ical potential in the T − ρB panel are demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The two figures show that c2 T and c2 µB are both close to 1/3 at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The contour of c2 T looks in general like that of c2 s/ρB, because the curves of constant s/ρB in the T − ρB panel are almost parallel to the density axis in a wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The relative larger deviation lies in the range with densities smaller than the boundary of the first-order transition on the low-density side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Speed of sound in QCD matter at finite temperature and density 13 The deviation produces different behaviors between ( ∂p ∂ǫ )T and ( ∂p ∂ǫ )s/ρB at low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A crutial point is that the inflections points of ( ∂p ∂ǫ )T and ( ∂p ∂ǫ )s/ρB are different, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=', the boundary of vanishing sound speed are different for the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 50 100 150 200 250 300 c 2 T T(MeV) B / 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='30 chiral crossover chiral first order spinodal line CEP deconfinement line Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 9 Contour map of c2 T in the T − ρB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 50 100 150 200 250 300 c 2 B T(MeV) B / 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='30 chiral crossover chiral first order spinodal line CEP deconfinement line Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 10 Contour map of c2 µB in the T − ρB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The red dashed line is the boundary of vanishing sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Both the figure 9 and 10 show that the boundary of zero sound speed at constant temperature or chemical potential is just the spinodal line associated with the first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' However the boundary of zero sound velocity at constant s/ρB is in the interior of the spinodal structure as shown 14 Speed of sound in QCD matter at finite temperature and density in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The negative values of c2 T and c2 µB both appear in the unstable phase of the spinodal structure, since ( ∂p ∂ǫ )T and ( ∂p ∂ǫ )µB are negative in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' From the behavior of c2 T derived above, we can approximately deduce the speed of sound in the quark core of a massive neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Since the richness of lepton including electron and muon approaches to zero at high density of a hybrid neutron star, and the richness of u, d, s quark tends to be equivalent [56], quite similar to the situation in this study at high density and low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Therefore, it may be concluded to a certain degree that the squared speed of sound in the quark core of a massive neutron star gradually approaches to 1/3 at high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A further study on the speed of sound in neutron star matter with a hadron-quark phase transition will be conducted with the combination of observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 4 Summary In this work, we studied the speed of sound in QCD matter at finite temper- ature and density in the PNJL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We derived the behavior of speed of sound under different definitions in the T − ρB phase diagram including the stable phase, metastable and unstable phases associated with the first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We systematically discussed the relations between the speed of sound and QCD phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The numerical results indicate that the squared speed of sounds under different definitions are all approaching to 1/3 at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' However, the behaviors in the phase-transition region are closely related to the non- perturbative interaction and the phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' From the perspective of idea fluid evolution, more attention are put on the speed of sound under isentropic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The calculation indicates that c2 s/ρB is nonzero at the CEP under the mean field approximation, and the boundary of vanishing sound velocity is further derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' We also obtained the contour maps of c2 X (X = s, ρB, T, µB), and analyzed their relations with the QCD phase transition, as well as the relations between different definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' For each definition of speed of sound, we find that there exist one or several regions with ( ∂p ∂ǫ )X < 0, even in the stable phase for X = s and ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The different definitions of speed of sound are involved in some impor- tant physics problems in nuclear physics, such as the evolution of quark gluon plasma, the cosmic QCD phase transition, the bulk viscosity of strongly inter- acting matter and the equation of state of a hybrid neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' The indepth investigations on these physics issues will be performed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' This work is supported by the National Natural Sci- ence Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 11875213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Data Availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Speed of sound in QCD matter at finite temperature and density 15 References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bass, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Heinz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Hirano, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Shen, 106, 192301 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bass, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Heinz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 83, 024912 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Deb, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Kadam, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mishra, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 94, 094002 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Gardim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Giacalone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Luzum, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ollitrault, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 16, 615 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Sahu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Tripathy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Sahoo, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Dash, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A 56, 187 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Biswas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Deka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Jaiswal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Roy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 102, 014912 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Sorensen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Oliinychenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Koch, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' McLerran, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 127, 042303 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Reed and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Horowitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 101, 045803 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Kanakis-Pegios, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Koliogiannis, and Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Moustakidis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 102, 055801 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Han and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Prakash, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 899, 164 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [11] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Tews, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Carlson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Gandolfi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Reddy, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 860, 149 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Greif, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Raaijmakers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Hebeler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Schwenk, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Watts, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 485, 5363 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Forbes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Reddy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mukherjee, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' De, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 100, 083010 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Drischler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lattimer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Prakash, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Reddy, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 103, 045808 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Essick, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Tews, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Landry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Reddy, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Holz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 102, 055803 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Han, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mamun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lalit, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Constantinou, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Prakash, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 100, 103022 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Kojo, AAPPS Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 31, 11 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 16 Speed of sound in QCD matter at finite temperature and density [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Jaikumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Semposki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Prakash, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Constantinou, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 103, 123009 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [19] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Philipsen, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 70, 55 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bors´anyi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Fodor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Guenther, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Kara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Katz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Parotto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Pasztor, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ratti, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Szabo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 125, 052001 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Aoki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Endrodi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Fodor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Katz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Szabo, Nature (London) 443, 675 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bors´anyi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Fodor, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Hoelbling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Katz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Krieg, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Sabz´o, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' B 730, 99 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bazavov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (hotQCD Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 90, 094503 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Motta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Stiele, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Alberico, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Beraudo, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 80, 770 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ghosh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mukherjee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mustafa, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ray, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 73, 114007 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Marty, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bratkovskaya, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Cassing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Aichelin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Berrehrah, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 88, 045204 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Saha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Upadhaya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Maity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 97, 116020 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 101, 096006, (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [29] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Schaefer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Wagner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Wambach, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 81, 074013 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Abhishek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mishra, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ghosh, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 97, 014005 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Venugopalan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Prakash, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A546, 718 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bluhm, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Alba, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Alberico, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Beraudo, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ratti, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A929, 157 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [33] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Khaidukov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lukashov, and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Simonov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 98, 074031 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [34] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Khaidukov and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Simonov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 100, 076009 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [35] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mykhaylova and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Sasaki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 103, 014007 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Speed of sound in QCD matter at finite temperature and density 17 [36] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Shao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Yang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Xie, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 105, 094024 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [37] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Mishustin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 82, 4779 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Randrup, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 92, 122301 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [39] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Koch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Majumder, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Randrup, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 72, 064903 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Sasaki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Friman, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Redlich, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 99, 232301 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Sasaki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Friman, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Redlich, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 77, 034024 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Randrup, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 79, 054911 (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 82, 034902 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Steinheimer and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Randrup, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 109, 212301 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Steinheimer and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Randrup, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A 52, 239 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [45] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Li and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 93, 035205 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Steinheimer and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Koch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 96, 034907 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [47] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Shao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Gao, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' He, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A 56, 115 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ahmadvand, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Bitaghsir Fadafan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' B 779, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [49] Tuomas V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Tenkanen and Jorinde van de Vis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 2022, 302 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [50] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' R¨oßner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ratti, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Weise, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D 75, 034007 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [51] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rehberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Klevansky, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' H¨ufner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 53, 410 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [52] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Costa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Ruivo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' de Sousa, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Hansen, Symmetry 2, 1338 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [53] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Shao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Gao, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' He, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' C 78, 138 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [54] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Chomaz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Colonna, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Randrup, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 389 263 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [55] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Shao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Gao, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' He, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='D 101, 074029 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' [56] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Shao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Colonna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Di Toro, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Liu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Liu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='D 87, 096012 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' 18 Speed of sound in QCD matter at finite temperature and density A Derivations of the formulae of speed of sound under different definitions in the temperature and density space The general definition of speed of sound is c2 X = �∂p ∂ǫ � X , (20) where X is a physics quantum fixed in the calculation of sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' In prac- tice, the squared speed of sound c2 X (X = s/ρB, s, ρB, T, µB) under different conditions are taken in dealing with different physics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' With the basic definitions of speed of sound, calculation can only be done along some special paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' To calculate the speed of sound under different definitions in the whole T − ρB space, it is necessary to derive the corresponding formulae in terms of temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' Using the Jacobian formula in thermodynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' we can derive c2 X (T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' ρB) = �∂p ∂ǫ � X = ∂(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' X) ∂(ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' X) = ∂(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='X) ∂(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='ρB) ∂(ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='X) ∂(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content='ρB) = � ∂p ∂T � ρB � ∂X ∂ρB � T − � ∂p ∂ρB � T � ∂X ∂T � ρB � ∂ǫ ∂T � ρB � ∂X ∂ρB � T − � ∂ǫ ∂ρB � T � ∂X ∂T � ρB (21) According to the thermodynamic characteristic function in the giant canon- ical ensemble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' it is convenient to get the following relations for isospin symmetric matter � ∂p ∂T � ρB = s + ρB �∂µB ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (22) � ∂p ∂ρB � T = ρB �∂µB ∂ρB � T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (23) � ∂ǫ ∂T � ρB = T � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (24) � ∂ǫ ∂ρB � T = T � ∂s ∂ρB � T + µB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (25) �∂(s/ρB) ∂T � ρB = 1 ρB � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (26) and �∂(s/ρB) ∂ρB � T = 1 ρB � ∂s ∂ρB � T − s ρ2 B (27) Speed of sound in QCD matter at finite temperature and density 19 For the different constraint conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' X = s/ρB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' ρB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' µB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' we can derived the corresponding formulae of speed of sound as follows c2 s/ρB = s2+ρ2 B �� ∂µB ∂ρB � T � ∂s ∂T � ρB − � ∂µB ∂T � ρB � ∂s ∂ρB � T � +sρB �� ∂µB ∂T � ρB − � ∂s ∂ρB � T � (T s + µBρB) � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (28) c2 s = ρB �� ∂s ∂T � ρB � ∂µB ∂ρB � T − � ∂s ∂ρB � T � ∂µB ∂T � ρB � − s � ∂s ∂ρB � T µB � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (29) c2 ρB = s + ρB � ∂µB ∂T � ρB T � ∂s ∂T � ρB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (30) c2 T = ρB � ∂µB ∂ρB � T T � ∂s ∂ρB � T + µB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (31) and c2 µB = s � ∂µB ∂ρB � T T �� ∂s ∂T � ρB � ∂µB ∂ρB � T − � ∂µB ∂T � ρB � ∂s ∂ρB � T � − µB � ∂µB ∂T � ρB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} +page_content=' (32)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfDAmJ/content/2301.04282v1.pdf'} diff --git a/RNFRT4oBgHgl3EQfKzd9/content/2301.13500v1.pdf b/RNFRT4oBgHgl3EQfKzd9/content/2301.13500v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3ffe8c4c324a930fd8c5d057677f42a4533a11b4 --- /dev/null +++ b/RNFRT4oBgHgl3EQfKzd9/content/2301.13500v1.pdf @@ -0,0 +1,3 @@ +version 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a/T9E0T4oBgHgl3EQf2QLa/content/tmp_files/2301.02711v1.pdf.txt b/T9E0T4oBgHgl3EQf2QLa/content/tmp_files/2301.02711v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1febccc0a10595da533ef6dfe43e32e0b09d9103 --- /dev/null +++ b/T9E0T4oBgHgl3EQf2QLa/content/tmp_files/2301.02711v1.pdf.txt @@ -0,0 +1,888 @@ +Autonomy for Ferries and Harbour Buses: +a Collision Avoidance Perspective +Thomas T. Enevoldsen ∗ Mogens Blanke ∗ Roberto Galeazzi ∗ +∗ Automation and Control Group, Department of Electrical and +Photonics Engineering, Technical University of Denmark, Kgs. +Lyngby, DK 2800, Denmark (e-mail: {tthen,mobl,roga}@dtu.dk). +Abstract: This paper provides a collision avoidance perspective to maritime autonomy, in the +shift towards Maritime Autonomous Surface Ships (MASS). In particular, the paper presents +the developments related to the Greenhoper, a Danish autonomous harbour bus. The collision +and grounding avoidance scheme, called the Short Horizon Planner (SHP), is described and +discussed in detail. Furthermore, the required autonomy stack for facilitating safe and rule- +compliant collision avoidance is presented. The inherent difficulties relating to adhering to the +COLREGs are outlined, highlighting some of the operational constraints and challenges within +the space of autonomous ferries and harbour buses. Finally, collision and grounding avoidance +is demonstrated using a simulation of the entire proposed autonomy stack. +Keywords: Autonomous surface vehicles, Guidance, Navigarion and Control, Collision and +grounding avoidance, COLREGs compliance +1. INTRODUCTION +As the world’s population continues to expand and the +green transition accelerates, there is a growing demand for +improved logistic and mobility capabilities. Autonomous +transportation systems seek to increase efficiency, both in +terms of availability, mobility, safety, and emissions. In +recent times, rapid development has occurred within the +space of maritime autonomy, with the first technological +benefits emerging. In the beginning of 2022, the Japanese +project, MEGURI2040, demonstrated autonomous capa- +bilities onboard a container vessel. During fall 2021, Sea +Machines launched a 1000nm autonomous voyage for their +11m long craft, showcasing the maturity of their technol- +ogy within inner coastal waters. In fall 2022, the Norwe- +gian autonomous ferry, Milliampere, was demonstrated, +highlighting an important case study for inner-city mo- +bility (Brekke et al., 2022). These Maritime Autonomous +Surface Ships (MASS) are prominent candidates to meet +increasing mobility demands, strengthen connectivity to +island societies, and reduce road congestion in major urban +areas by opening their waterways for transport of goods +and people. MASS will extend the availability of existing +waterborne transportation services, and will open for new +mobility on demand services. A major challenge in the shift +towards autonomous vessels and systems is the complete +adherence to the rules, regulations, and practises laid out +by the preceding sailors and navigators. +In Denmark, ShippingLab represents the Danish initia- +tive within autonomous waterborne transport, where the +⋆ This research was sponsored by the Danish Innovation Fund, The +Danish Maritime Fund, Orients Fund, and the Lauritzen Foundation +through the Autonomy part of the ShippingLab project, grant +number 8090-00063B. The electronic navigational charts have been +provided by the Danish Geodata Agency. +goal is to create Denmark’s first autonomous and envi- +ronmentally friendly ship. This effort has resulted in the +Greenhopper, a 12.2m long battery operated double-ended +catamaran, designed and built in Denmark. The vessel will +facilitate the expansion and growth of the city of Aalborg, +located in the northern part of the Danish peninsula, +Jutland. It will cross the Limfjorden, with its journey +lasting 5-7 minutes (580m). +(a) The Greenhopper: a Danish autonomous ferry. +10 +15 +20 +25 +30 +Longitude (dd.ddd) +56 +58 +60 +Latitude (dd.ddd) +9.92 +9.94 +57.050 +57.055 +57.060 +(b) The operational area at Limfjorden, Aalborg, Denmark. The +dashed line is the nominal route, crosses buoys and hatched areas +dredged locations. Darker blues are deeper contours. +Fig. 1. The Greenhopper vessel and its area of operation. +arXiv:2301.02711v1 [cs.RO] 6 Jan 2023 + +24PASS +GreenHopperRecent efforts within collision avoidance for marine au- +tonomy focus on confined and inner coastal waters. In +these waters, there are various efforts that concern com- +puting trajectories in compliance with COLREGs 8 & +13-17. Bergman et al. (2020) demonstrated a two-step +optimisation procedure, where a lattice-based planner +computes suboptimal trajectories based on motion primi- +tives that are refined by solving optimal control problems +(OCP). Enevoldsen et al. (2022) presented a sampling- +based method to calculate minimal route deviations, min- +imising cross-track error and speed loss. Thyri and Breivik +(2022) detailed a collision avoidance scheme that assigns +and uses control barrier functions for preventing ship do- +main violation, and thereby enforcing the COLREGs. +For the MilliAmpere (Brekke et al., 2022), Bitar et al. +(2021) detailed a method consisting of the three aspects of +an autonomous voyage: undocking, transit, and docking. +Docking was dealt with using model predictive control, +whereas the transit phase combined a hybrid A* with +an OCP solver. Thyri et al. (2020) instead cast the +problem as a velocity planning problem, by leveraging +a set of pre-defined feasible paths. The planning phase +then recomputes with respect to dynamic obstacles. The +Dutch project Roboat seeks to implement an autonomous +platform for urban mobility (Wang et al., 2020), where +in (de Vries et al., 2022) the system demonstrates its +capabilities and basic adherence to COLREGs rule 13-15. +For the Rhine river, Koschorrek et al. (2022) presented a +system that used a hybrid A* to find feasible trajectories. +Here, COLREGs are not directly considered because local +law dictates that a ferry must yield for everything. +This paper proposes a collision avoidance scheme, the +Short Horizon Planner (SHP), designed for a fjord cross- +ing ferry, such as the Greenhopper. The SHP considers +the available manoeuvrability for precise obstacle avoid- +ance, while partially adhering to the IMO COLREGs +(rules 8 & 13-17). The role, purpose, and responsibility of +the collision avoidance system within the autonomy stack +is detailed and discussed, outlining apparent operational +constraints in both the collision avoidance system and the +remaining stack. The particular operation of the Green- +hopper is detailed, highlighting the interplay between the +SHP and the remaining autonomy stack. +2. SYSTEM MODELLING AND IDENTIFICATION +The Greenhopper is propelled and manoeuvred by two +azimuth thrusters, located fore and aft, at the centre +line. It is equipped with four RGB cameras, eight Long +Wavelength Infrared (LWIR) cameras, four W band and +one X band radar, two 3D lidars, a GNSS, a gyro compass, +an AIS transponder and an IMU. Sensors mounted on the +mast can be seen in Fig. 1a. A Voyage Control System +(VCS) is responsible for executing steering and track con- +trol along a nominal route. The VCS will safely dock, un- +dock, and carry out the voyage in nominal conditions. The +nominal route can be modified by adding supplemental +waypoints to the VCS, as safe navigation requires. +2.1 Surge velocity dynamics +Surge acceleration is the result of the balance between +propeller forces and hull resistance. +0 +200 +400 +600 +800 +1000 +0.0 +0.7 +1.4 +2.1 +2.8 +3.5 +Surge speed (m/s) +0 +200 +400 +600 +800 +1000 +0.00 +0.25 +0.50 +0.75 +1.00 +Thruster RPM (%) +0 +200 +400 +600 +800 +1000 +Time (s) +−100 +0 +100 +Thruster angle (degrees) +(a) Experimental data from the Greenhopper. t = 0s to t = +800s +contains acceleration and deceleration experiments. From t = 800s +and onwards are recorded emergency stops at various speeds. +0 +5 +10 15 20 25 30 35 40 +Time (s) +0.0 +0.7 +1.4 +2.1 +2.8Surge speed (m/s) @ T% = 50% +Experiment +Grey-box Fit +(b) Grey-box estimate of (3) at +50% thrust. +0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 +Surge speed (m/s) +0.0 +7.5 +15.0 +22.5 +30.0 +Stopping distance (m) +(c) Second order fit for the emer- +gency stopping distances. +Fig. 2. Experiments and estimates from the Greenhopper. +With azimuth thrusters fore and aft, along ship thrust is, +Tx = TP,1 cos(φ1) + TP,2 cos(φ2) +(1) +where φi is the azimuth angle of thruster i and TP,i is +propeller thrust. +Hull resistance consists of Stokes friction, linear in u, +and pressure drag that is quadratic in u|u|. With mass +and added mass in the left hand side factor, and thrust +deduction t, surge dynamics reads, +(m − X ˙u) ˙u = (1 − t)Tx − Xuu − Xu|u|u|u|. +(2) +Introducing Tx = βsT% and T% as commanded thrust +percentage, (2) has the form, +˙u = βT% − αu − γu|u| +(3) +with thrust scaling β, linear damping α, and quadratic +damping γ. These are identified from full scale testing in +the following section. +2.2 Grey-box identification and analytical solution +The identification of (3) based on full-scale acceleration +data revealed that due to the low speed regime, the par- +ticular hull form, and the propeller slip stream interaction +with the pontoons of the catamaran hull, the nonlinear +damping coefficient γ is hard to identify. The contribution +of the term is essentially zero, on the basis of the available +experimental data. Therefore, a linear model is pursued, +˙u = βT% − αu, +˙N = cos (ψ) u, +˙E = sin (ψ) u +(4) +where T% and heading angle ψ are known and constant. +The solutions to the linear ODEs are as follows + +� +��� +u(t) +N(t) +E(t) +� +��� = +� +��� +(1 − e−αt) β +αT% +� 1 +α2 e−αt + 1 +αt +� +T%β cos(ψ) +� 1 +α2 e−αt + 1 +αt +� +T%β sin(ψ) +� +��� . +(5) +The analytical solution is used to calculate a trajectory +between two points in the north-east plane, simply by +computing the arrival time tf at the final point. The arrival +time is obtained by setting the left-hand side of (5) to +the desired point and solving for t. Once tf is obtained, +obtaining the trajectory is trivial. +3. PATH PLANNING AND COLLISION AVOIDANCE +The nominal path of the Greenhopper is described by two +waypoints located on the north and south side of the fjord. +In conditions with traffic, the objective is to find a path +that connects either the nominal waypoints, or the current +position of ownship and the goal in a collision-free and safe +manner. +3.1 Spatio-temporal lattice planner +Let X ⊆ R3 be the state space, with x ∈ X and x = +[E, N, t]T . X is divided into two subsets, the free space +Xfree and the obstacle space Xobs, with Xfree = X\Xobs. +The objective is to find a sequence σ of states that +minimises the cost function c(σ), while connecting the +starting state xs and end state xe +σ∗ = arg min +σ∈Σ +{c(σ) | σ(0) = xs , σ(1) = xe , +∀s ∈ [0, 1], σ(s) ∈ Xfree } . +(6) +The obstacle subset Xobs is formed by the union over all +constraints (Enevoldsen et al., 2022), namely +Xobs = X OS +obs ∪ X ENC +obs +∪ X TV +obs , +X TV +obs = +n +� +i=1 +XTV,i(t) (7) +with X OS +obs containing states that violate manoeuvring +constraints, X ENC +obs +the grounding and buoy collision states, +and finally X TV +obs the target vessel constraints, which is +the union of n vessels, such that all n are considered +simultaneously. The spatial constraints are encoded by the +predicted trajectories of each target vessel (XTV,i(t)). +A deterministic planning algorithm is proposed for build- +ing a directed graph. The starting state xs is the position +of ownship in the north-east plane at t = 0, and xe is the +current desired destination, at some unknown final time +t = tf. Consider a grid G = [G1, . . . , Gm]T ∈ Rm×n, +with rows Gi = [gi,1 . . . , gi,n] , where each row represents +the depth towards the goal and the width of potential +deviations, and each element gi,j ∈ G represents a point in +the north-east plane. The grid is pre-processed, in order +to refine it, by modifying points gi,j that violate the envi- +ronmental constraints, as follows ¯G = G\X ENC +obs . A directed +graph T with root xs is built over k = m+1 iterations. Two +sets of nodes are formed, one with all current parent nodes +Cp = {xs}, which always contains the start xs, and a set +for all current child nodes Cc = {xe} that always contains +the end xe. At each iteration, the sets are modified by the +grid rows, which dictates edges that are to be formed +−500 +0 +500 +East (m) +−400 +−200 +0 +200 +400 +North (m) +(a) Full lattice using a 7x5 grid. +−500 +0 +500 +East (m) +−400 +−200 +0 +200 +400 +North (m) +(b) Recomputing the lattice from an arbitrary location. +Fig. 3. The obstacle subset X ENC +obs +consists of the area sur- +rounding the blue polygon (land and shallow waters) +and the interior area of the red circles (buoys). +Cp = {xs}, Cc = {xe, Gk} +if k = 1, +Cp = {xs, Gk−1}, Cc = {xe, Gk} +if 1 < k < m + 1, +(8) +Cp = {xs, Gk−1}, Cc = {xe} +if k = m + 1. +Before adding a given edge to T , the resulting trajectory +between the two nodes is checked to see if it violates any +constraints (Xobs). The trajectory between two nodes is +computed by forward simulating (5). Nodes in collision +from Cc are discarded and omitted from Cp. +3.2 Rules and regulations +Adherence to the rules and practises of safe navigation +is fundamental for MASS. There is a general consensus +that the most essential IMO COLREGs are rules 8 & +13-17. Rules 13-15 dictate the three most common vessel +encounters: overtaking (13), head-on (14) and crossing +(15). Rule 14 & 15 specifies that the give-way vessel must +perform the manoeuvre toward the port side, where rule +13 allows passing on either side in a safe manner. Rule 16 +& 17 dictate the behaviour of the give-way and stand-on +vessel, see (Cockcroft and Lameijer, 2003). +In the literature, there is a consensus that partial ad- +herence to the COLREGs is sufficient to demonstrate +capable and safe navigation. Within confined waters, such +as rivers and urban environments, additional complexities +may arise. Rule 14 and 15 specifically apply between two +power-driven vessels; therefore, if the system encounters + +(a) Overtaking (13) +(b) Head-on (14) +(c) Crossing (15) +Fig. 4. Ship domains for COLREGs-compliance, dimen- +sions are dependant on target vessel ship length. +a sailboat, different rules and obligations apply. Further- +more, certain vessels may be restricted in their manoeu- +vrability. If a vessel is restricted, rule 9 applies, which +states that a vessel less than 20m should give-way for a +restricted vessel, even if according to rule 15 the target +vessel is the give-way vessel. Hansen et al. (2022) used +rule 9 within the decision making loop for a river crossing +ferry, where its highlighted that simply following rule 15 +without considering rule 9 may cause problems. +The complexity of the rule framework strongly depends on +local laws (e.g. the Rhine River (Koschorrek et al., 2022)) +for the particular waters. In Canadian waters, rule 15 is +modified so that any vessel, with minor exceptions, cross- +ing a river must yield to power-driven vessels travelling +along it (CSA, 2001). According to §19 of Danish law on +seafaring (Søfartsstyrelsen, 2020, 1991), only three specific +ferry routes must disregard the usual obligation to rule 15, +and instead yield for any traffic the ferry may impede. +3.3 Ship domains for COLREGs-compliance +If MASS have adequate situation awareness, give-way +and stand-on obligations can be enforced using Lam´e +curves (Enevoldsen et al., 2022), see Fig. 4. The curve for +compliance with crossing and overtaking scenarios is given +by +���� +cos(ψ(t))∆El(t) − sin(ψ(t))∆Nl(t) +aL +���� +p ++ +���� +sin(ψ(t))∆El(t) + cos(ψ(t))∆Nl(t) +bL +���� +p +≤ 1 +(9) +when used in conjunction with a circular constraint, +∆Ec(t)2 + ∆Nc(t)2 ≤ r2 +L +(10) +where aL, bL and rL are scalar values based on the length +of the target vessel and additional safety margins. The +difference in coordinates at time t between the own ship +and a given target vessel is used to evaluate the domains +∆E(t) = E(t)−ETV(t)+Eo, ∆N(t) = N(t)−NTV(t)+No +and ¯ψ(t) = ψTV(t) + ψo, where the offset is used to shift +the elliptical and circular components of the domain, +� +Eo +No +� += +� +cos +� +− ¯ψ +� +− sin +� +− ¯ψ +� +sin +� +− ¯ψ +� +cos +� +− ¯ψ +� +� � +pE +pN +� +(11) +with pE = 0, and pN equal to bL in (9) and aL in (10). +For (9) and (10), the ψo is equal to 0 and π +2 respectively. +3.4 Safety margins +Ship length is commonly used to compute a safety margin +with respect to other vessels. However, vessels navigating +within inner coastal or confined waters are typically either +sailboats or pleasure crafts, which are not obligated to +carry an AIS transponder. Vessels of length greater than +SFU +Sensory +information +SAS +MWS +ANS +ACS +APS +VCS +SHP +HMI +RUT +Fig. 5. Module interconnection within the autonomy stack. +20m are often required to have AIS. Therefore, it is neces- +sary to select an adequate safety margin in the absence of +an accurate ship length estimate. The safety margin is se- +lected according to emergency stop manoeuvres (Fig. 2a), +such that a suitable distance is maintained to the target +vessel, in instances with an erroneously perceived scenario. +For the current transit speed of 3 knots, the required stop- +ping distance is approximately 17.5m, therefore selecting +25m as the minimum ship length is ample distance. +4. THE AUTONOMY STACK +The following section details the composition of the auton- +omy stack. The middleware is introduced and the purpose +and responsibility of each module is outlined. +4.1 Middleware and autonomy stack features +Dittmann and Blanke (2022) investigated the regulatory +framework and system requirements for the development +and commissioning of MASS, highlighting some important +considerations regarding reliability and redundancy. In +addition, design choices and developments related to the +custom middleware solution were detailed. +The autonomous system is composed of various modules, +such that each block is compartmentalised and its interface +clearly defined. Using a modular approach allows each part +of the stack to be developed and tested individually, and +also undergo strict stress and acceptance testing, before +being rolled out and combined with the remaining system. +The middleware facilitates publish–subscribe communica- +tion between modules, such that multiple modules can +subscribe to the same module. Testing and simulation +of various modules within the stack is achieved using a +dedicated middleware simulator (MWS). +4.2 Module functionality and interconnection +The core of the stack consists of the Autonomous Coordi- +nation Supervisor (ACS), Autonomous Navigation Super- +visor (ANS) and Autonomous Platform Supervisor (APS), +which replace the traditional roles employed by the cap- +tain, navigator, and chief engineer (Dittmann et al., 2021). +The ACS coordinates departures and exchanges routes +with the route server (RUT), which stores the destination. +Effective and precise fused perception and sensory infor- +mation is crucial for the remaining autonomous system. + +Human lookouts and navigators are replaced by an elec- +tronic outlook (Blanke et al., 2018) that uses cameras to +detect and classify objects (Sch¨oller et al., 2020). The vi- +sion system is fused with the remaining sensors, producing +a robust and resilient estimate of static and dynamic ob- +stacles (Dagdilelis et al., 2022), all of which is encapsulated +by the Sensor Fusion (SFU) module. The estimated and +fused states of the surrounding vessels can be augmented +by a trajectory prediction scheme that uses information +from the local area (Sch¨oller et al., 2021). However, the +current stack only implements straight-line predictions. +The Situation Awareness Service (SAS) is driven by in- +formation from the SFU, in order to maintain an overview +of the unfolding scenario. Once a vessel violates set CPA +and TCPA limits, the scenario is passed from SAS to the +ANS, i.e. the ’navigator’ is informed about the situation, +and triggers the SHP for a route deviation. This ensures +that the Greenhopper deals with the scenario in a timely +manner and with a reasonable safety margin. Details on +the interaction between the SAS and ANS modules can +be found in Hansen et al. (2020) and Papageorgiou et al. +(2022). A Human-machine Interface (HMI) visualises the +SFU, RUT, SAS and SHP on an electronic navigational +chart, with correct symbolism from IMO. +The SHP, originally introduced in Enevoldsen et al. (2022) +as a generalised collision avoidance scheme for vessels in +confined and inner coastal waters, is in this paper spe- +cialised for crossings, such as those encountered by the +Greenhopper. The underlying planning scheme is imple- +mented as described in Section 3. As part of the autonomy +stack, the SHP is tasked with computing rule-compliant, +collision-, and grounding free passages for a given scenario +at hand. Given an input consisting of own ship naviga- +tional data, predicted target vessel information, perceived +COLREGs scenario (from the SFU-SAS-ANS) and the +current destination (from RUT). The SHP reports within +finite time whether a valid crossing exists and, if so, which +sequence of waypoints must be followed to achieve it. +5. DEMONSTRATION AND DISCUSSION +The autonomy stack and SHP is validated using software- +in-the-loop testing by simulating the sensor fusion output. +The results are visualised on the HMI, and generated +while running the MWS, SAS, ANS, ACS, RUT and SHP +modules. The MWS acts as both a vessel and SFU simu- +lator. Figure 6 shows a scenario in which the Greenhopper +departs the southern harbour, following the nominally +planned route. As the vessel is underway, two target ves- +sels approach from the starboard side. The CPA for the +southernmost target vessel is greater than the safety limit +and is therefore not considered. However, the second target +vessel violates the CPA limit; therefore, once TCPA falls +below the chosen limit, the SHP is triggered and a route +deviation is performed. Executing the deviation allows +the Greenhopper to safely avoid the target vessel before +reaching its destination at the northern point of the fjord. +A fundamental requirement for adhering to the COLREGs +is an adequate estimate of the scenario at hand. Most +systems, including the proposed one, break down if their +assumptions do not hold. Most often, MASS depend on +AIS for identifying vessel type and size, but in waters +(a) t=10s +(b) t=1m40s +(c) t=2m40s +(d) t=2m47s +(e) t=4m16s +(f) t=5m58s +Fig. 6. Simulated demonstration of the SHP with most +of the autonomy stack. The Greenhopper is depart- +ing the southern harbour, following the nominally +planned path. A manoeuvre is required while under- +way, as two other vessels approach from starboard. +as those navigated by the Greenhopper and other au- +tonomous ferries, a vast majority operate without AIS +(due to being a leisure craft or other exemptions). It is +therefore crucial that the perception system can classify +if the perceived vessel is power-driven or not and is ca- +pable of determining whether a vessel is manoeuvrability +restricted. Otherwise, the COLREGs cannot be applied +correctly. For safe navigation, this is a major defect, since +the target vessels are commanded by humans, who expect +that any vessel they encounter adheres and acts according +to the COLREGs, and if a severe risk of collision occurs, +knows how to mitigate or lessen its severity. If the per- +ceived scenario is correct, the proposed collision avoidance +strategy can, in a deterministic fashion and within a finite +time, report whether or not a path deviation exists within +the chosen safety limits. If no solution exists during the +voyage, the issue is raised to the ACS, which must be +capable of correctly dealing with such an emergency sit- +uation, either by calling for help from a Remote Control +Centre (RCC) or by stopping the vessel and signalling the +surroundings that an emergency is unfolding. + +Sn/Wn +5.5 +VESSEL +BRG +359° +RNG +92.52m +1.6 +CPA +91.95 m +5.5 +TCPA +2.4sQ +31 +39 +0.04nm +061° +VESSEL +BRG +193° +RNG +173.86m +6.0 +6.0Sn/Wn +VESSEL +5.5 +BRG +268° +RNG +0.16nm +1.6 +4.9 +5.5 +1007° +31 +3.9 +0.04nm +*061* +VESSEL +BRG +218° +RNG +0.20nm +6.0 +6.0Sn/Wn +5.5 +1.6 +4.9 +3.0 +5.5 +31 +3.9 +0.04nm +1030° +6.0 +09Sn/Wn +5.5 +1.6 +4.9 +3.0 +5.5 +31 +3.9 +0.04nm +VESSEL +BRG +100° +RNG +0.20 nm +CPA +0.11nm +TCPA +2.1 min +6.0 +6.0Sn/Wn +5.5 +VESSEL +1.6 +4.9 +3.0 +BRG +071° +31 +RNG +0.21nm +CPA +26.96 m +TCPA +1.9 min +3.9 +0.04nm +030° +VESSEL +BRG +118° +RNG +0.14nm +CPA +0.11nm +TCPA +1.1 min +6.0 +09Sn/Wn +5.5 +1.6 +VESSEL +4.9 +3.0 +BRG +070° +007° +31 +RNG +0.19nm +CPA +48.92m +TCPA +1.6 min +- +3.9 +0.04nm +060° +VESSEL +BRG +121° +RNG +0.13nm +CPA +0.10 nm +TCPA +1.0 min +6.0 +6.06. CONCLUSION +The paper presented a collision avoidance perspective +to autonomous ferries and harbour buses. A Danish au- +tonomous ferry initiative, the Greenhopper, was intro- +duced, and its autonomy stack was detailed and discussed. +A deterministic collision avoidance strategy was presented, +as well as simple ship domains for enforcing give-way +responsibilities. The importance of a well-functioning and +sufficiently accurate estimate of the unfolding situation +was discussed in great detail. In conclusion, to navigate +according to the COLREGs and safe navigation practises, +the target vessels must be correctly classified in terms of +vessel type and manoeuvrability. +Future work includes field verification of the proposed SHP +in conjunction with the remaining autonomy stack. Once +verified, the final steps towards fully commissioning the +ferry for autonomous operation must be undertaken. +ACKNOWLEDGEMENTS +The authors acknowledge Mette Bennedsen and Jens +Brauchli Jensen from SIMAC for the fruitful discussions +on rules for navigation, Jann-Timothy G. Mayer from +W¨artsil¨a for data collection, and fellow ShippingLab con- +tributors Kjeld Dittman, Nicholas Hansen, Dimitrios Pa- +pageorgio, and Andreas Gamborg for discussions and de- +velopments of the autonomy stack. This research was spon- +sored by the Danish Innovation Fund, The Danish Mar- +itime Fund, Orients Fund, and the Lauritzen Foundation +through the Autonomy part of the ShippingLab project, +grant number 8090-00063B. The electronic navigational +charts have been provided by the Danish Geodata Agency. +REFERENCES +Bergman, K., Ljungqvist, O., Linder, J., and Axehill, +D. (2020). +A colregs-compliant motion planner for +autonomous maneuvering of marine vessels in complex +environments. arXiv preprint arXiv:2012.12145. +Bitar, G., Eriksen, B.O.H., Lekkas, A.M., and Breivik, M. +(2021). Three-phase automatic crossing for a passenger +ferry with field trials. In 2021 European Control Con- +ference (ECC), 2271–2277. IEEE. +Blanke, M., Hansen, S., Stets, J.D., Koester, T., Brøsted, +J., Maurin, A.L., Nykvist, N., Bang, J., and Authority, +D.M. (2018). Outlook for navigation–comparing human +performance with a robotic solution. Proc. of ICMASS. +Brekke, E.F., Eide, E., Eriksen, B.O.H., Wilthil, E.F., +Breivik, M., Skjellaug, E., Helgesen, Ø.K., Lekkas, A.M., +Martinsen, A.B., Thyri, E.H., et al. (2022). milliampere: +an autonomous ferry prototype. In Journal of Physics: +Conf. Series, volume 2311, 012029. +Cockcroft, A.N. and Lameijer, J.N.F. (2003). Guide to the +collision avoidance rules. Elsevier. +CSA (2001). +Collision Regulations, C.R.C., c. 1416, +Canada Shipping Act (CSA), 2001. https://laws-loi +s.justice.gc.ca/PDF/C.R.C., c. 1416.pdf. +Dagdilelis, D., Blanke, M., Andersen, R.H., and Galeazzi, +R. (2022). +Cyber-resilience for marine navigation by +information fusion and change detection. Ocean Engi- +neering, 266, 112605. doi:https://doi.org/10.1016/j.oc +eaneng.2022.112605. +de Vries, J., Trevisan, E., van der Toorn, J., Das, T., Brito, +B., and Alonso-Mora, J. (2022). +Regulations aware +motion planning for autonomous surface vessels in urban +canals. In 2022 Int. Conf. on Robotics and Automation +(ICRA), 3291–3297. doi:10.1109/ICRA46639.2022.981 +1608. +Dittmann, K. and Blanke, M. (2022). +Risk mitigation +by design of autonomous maritime automation systems. +Automatisierungstechnik, 70(5), 469–481. doi:10.1515/ +auto-2021-0151. +Dittmann, K., Hansen, P.N., Papageorgiou, D., and +Blanke, M. (2021). +Autonomy for ships: A sovereign +agents architecture for reliability and safety by design. +IEEE Xplore-Proc. IEEE SysTol. +Enevoldsen, T.T., Blanke, M., and Galeazzi, R. (2022). +Sampling-based collision and grounding avoidance for +marine crafts. +Ocean Engineering, 261, 112078. +doi: +https://doi.org/10.1016/j.oceaneng.2022.112078. +Hansen, P.N., Enevoldsen, T.T., Papageorgiou, D., and +Blanke, M. (2022). Autonomous Navigation in Confined +Waters–A COLREGs Rule 9 Compliant Framework. +arXiv preprint arXiv:2207.08227. +Hansen, P.N., Papageorgiou, D., Blanke, M., Galeazzi, R., +L¨utzen, M., Mogensen, J., Bennedsen, M., and Hansen, +D. (2020). +COLREGs-based Situation Awareness for +Marine Vessels-a Discrete Event Systems Approach. +IFAC-PapersOnLine, 53(2), 14501–14508. doi:10.1016/ +j.ifacol.2020.12.1453. +Koschorrek, P., Kosch, M., Nitsch, M., Abel, D., and +J¨urgens, D. (2022). Towards semi-autonomous opera- +tion of an over-actuated river ferry. Automatisierung- +stechnik, 70(5), 433–443. doi:10.1515/auto-2021-0152. +Papageorgiou, D., Hansen, P.N., Dittmann, K., and +Blanke, M. (2022). Anticipation of ship behaviours in +multi-vessel scenarios. Ocean Engineering, 266, 112777. +doi:https://doi.org/10.1016/j.oceaneng.2022.112777. +Sch¨oller, F.E.T., Blanke, M., Plenge-Feidenhans, M.K., +and Nalpantidis, L. (2020). Vision-based object track- +ing in marine environments using features from neural +network detections. IFAC-PapersOnLine, 53(2), 14517– +14523. +Sch¨oller, F.E., Enevoldsen, T.T., Becktor, J.B., and +Hansen, P.N. (2021). Trajectory prediction for marine +vessels using historical ais heatmaps and long short-term +memory networks. IFAC-PapersOnLine, 54(16), 83–89. +Søfartsstyrelsen (1991). +Bekendtgørelse om sejlads p˚a +Limfjorden mellem Egholm og Kattegat (BEK nr 953 +af 18/12/1991). +Søfartsstyrelsen (2020). Bekendtgørelse om sejlads m.m. i +visse danske farvande (BEK nr 656 af 20/05/2020). +Thyri, E.H. and Breivik, M. (2022). A domain-based and +reactive colav method with a partially colregs-compliant +domain for asvs operating in confined waters. +Field +Robotics, 2, 637–677. +Thyri, E.H., Breivik, M., and Lekkas, A.M. (2020). A path- +velocity decomposition approach to collision avoidance +for autonomous passenger ferries in confined waters. +IFAC-PapersOnLine, 53(2), 14628–14635. +Wang, W., Shan, T., Leoni, P., Fern´andez-Guti´errez, D., +Meyers, D., Ratti, C., and Rus, D. (2020). Roboat II: +A novel autonomous surface vessel for urban environ- +ments. +In 2020 IEEE/RSJ Int. Conf. on Intelligent +Robots and Systems (IROS), 1740–1747. IEEE. + diff --git a/T9E0T4oBgHgl3EQf2QLa/content/tmp_files/load_file.txt b/T9E0T4oBgHgl3EQf2QLa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1edaf708a6dd81a6221a09a7386e00f18d812e9 --- /dev/null +++ b/T9E0T4oBgHgl3EQf2QLa/content/tmp_files/load_file.txt @@ -0,0 +1,588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf,len=587 +page_content='Autonomy for Ferries and Harbour Buses: a Collision Avoidance Perspective Thomas T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Enevoldsen ∗ Mogens Blanke ∗ Roberto Galeazzi ∗ ∗ Automation and Control Group, Department of Electrical and Photonics Engineering, Technical University of Denmark, Kgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Lyngby, DK 2800, Denmark (e-mail: {tthen,mobl,roga}@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Abstract: This paper provides a collision avoidance perspective to maritime autonomy, in the shift towards Maritime Autonomous Surface Ships (MASS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In particular, the paper presents the developments related to the Greenhoper, a Danish autonomous harbour bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The collision and grounding avoidance scheme, called the Short Horizon Planner (SHP), is described and discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Furthermore, the required autonomy stack for facilitating safe and rule- compliant collision avoidance is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The inherent difficulties relating to adhering to the COLREGs are outlined, highlighting some of the operational constraints and challenges within the space of autonomous ferries and harbour buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Finally, collision and grounding avoidance is demonstrated using a simulation of the entire proposed autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Keywords: Autonomous surface vehicles, Guidance, Navigarion and Control, Collision and grounding avoidance, COLREGs compliance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' INTRODUCTION As the world’s population continues to expand and the green transition accelerates, there is a growing demand for improved logistic and mobility capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Autonomous transportation systems seek to increase efficiency, both in terms of availability, mobility, safety, and emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In recent times, rapid development has occurred within the space of maritime autonomy, with the first technological benefits emerging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In the beginning of 2022, the Japanese project, MEGURI2040, demonstrated autonomous capa- bilities onboard a container vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' During fall 2021, Sea Machines launched a 1000nm autonomous voyage for their 11m long craft, showcasing the maturity of their technol- ogy within inner coastal waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In fall 2022, the Norwe- gian autonomous ferry, Milliampere, was demonstrated, highlighting an important case study for inner-city mo- bility (Brekke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' These Maritime Autonomous Surface Ships (MASS) are prominent candidates to meet increasing mobility demands, strengthen connectivity to island societies, and reduce road congestion in major urban areas by opening their waterways for transport of goods and people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' MASS will extend the availability of existing waterborne transportation services, and will open for new mobility on demand services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A major challenge in the shift towards autonomous vessels and systems is the complete adherence to the rules, regulations, and practises laid out by the preceding sailors and navigators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In Denmark, ShippingLab represents the Danish initia- tive within autonomous waterborne transport, where the ⋆ This research was sponsored by the Danish Innovation Fund, The Danish Maritime Fund, Orients Fund, and the Lauritzen Foundation through the Autonomy part of the ShippingLab project, grant number 8090-00063B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The electronic navigational charts have been provided by the Danish Geodata Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' goal is to create Denmark’s first autonomous and envi- ronmentally friendly ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' This effort has resulted in the Greenhopper, a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2m long battery operated double-ended catamaran, designed and built in Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The vessel will facilitate the expansion and growth of the city of Aalborg, located in the northern part of the Danish peninsula, Jutland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' It will cross the Limfjorden, with its journey lasting 5-7 minutes (580m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (a) The Greenhopper: a Danish autonomous ferry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 10 15 20 25 30 Longitude (dd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='ddd) 56 58 60 Latitude (dd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='ddd) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='92 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='94 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='050 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='055 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='060 (b) The operational area at Limfjorden, Aalborg, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The dashed line is the nominal route, crosses buoys and hatched areas dredged locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Darker blues are deeper contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The Greenhopper vessel and its area of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='02711v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='RO] 6 Jan 2023 24PASS GreenHopperRecent efforts within collision avoidance for marine au- tonomy focus on confined and inner coastal waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In these waters, there are various efforts that concern com- puting trajectories in compliance with COLREGs 8 & 13-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Bergman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020) demonstrated a two-step optimisation procedure, where a lattice-based planner computes suboptimal trajectories based on motion primi- tives that are refined by solving optimal control problems (OCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Enevoldsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022) presented a sampling- based method to calculate minimal route deviations, min- imising cross-track error and speed loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Thyri and Breivik (2022) detailed a collision avoidance scheme that assigns and uses control barrier functions for preventing ship do- main violation, and thereby enforcing the COLREGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' For the MilliAmpere (Brekke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2022), Bitar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2021) detailed a method consisting of the three aspects of an autonomous voyage: undocking, transit, and docking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Docking was dealt with using model predictive control, whereas the transit phase combined a hybrid A* with an OCP solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Thyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020) instead cast the problem as a velocity planning problem, by leveraging a set of pre-defined feasible paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The planning phase then recomputes with respect to dynamic obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The Dutch project Roboat seeks to implement an autonomous platform for urban mobility (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2020), where in (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2022) the system demonstrates its capabilities and basic adherence to COLREGs rule 13-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' For the Rhine river, Koschorrek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022) presented a system that used a hybrid A* to find feasible trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Here, COLREGs are not directly considered because local law dictates that a ferry must yield for everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' This paper proposes a collision avoidance scheme, the Short Horizon Planner (SHP), designed for a fjord cross- ing ferry, such as the Greenhopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The SHP considers the available manoeuvrability for precise obstacle avoid- ance, while partially adhering to the IMO COLREGs (rules 8 & 13-17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The role, purpose, and responsibility of the collision avoidance system within the autonomy stack is detailed and discussed, outlining apparent operational constraints in both the collision avoidance system and the remaining stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The particular operation of the Green- hopper is detailed, highlighting the interplay between the SHP and the remaining autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' SYSTEM MODELLING AND IDENTIFICATION The Greenhopper is propelled and manoeuvred by two azimuth thrusters, located fore and aft, at the centre line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' It is equipped with four RGB cameras, eight Long Wavelength Infrared (LWIR) cameras, four W band and one X band radar, two 3D lidars, a GNSS, a gyro compass, an AIS transponder and an IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Sensors mounted on the mast can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A Voyage Control System (VCS) is responsible for executing steering and track con- trol along a nominal route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The VCS will safely dock, un- dock, and carry out the voyage in nominal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The nominal route can be modified by adding supplemental waypoints to the VCS, as safe navigation requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1 Surge velocity dynamics Surge acceleration is the result of the balance between propeller forces and hull resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 Surge speed (m/s) 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='00 Thruster RPM (%) 0 200 400 600 800 1000 Time (s) −100 0 100 Thruster angle (degrees) (a) Experimental data from the Greenhopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' t = 0s to t = 800s contains acceleration and deceleration experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' From t = 800s and onwards are recorded emergency stops at various speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 Time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='8Surge speed (m/s) @ T% = 50% Experiment Grey-box Fit (b) Grey-box estimate of (3) at 50% thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 Surge speed (m/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 Stopping distance (m) (c) Second order fit for the emer- gency stopping distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Experiments and estimates from the Greenhopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' With azimuth thrusters fore and aft, along ship thrust is, Tx = TP,1 cos(φ1) + TP,2 cos(φ2) (1) where φi is the azimuth angle of thruster i and TP,i is propeller thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Hull resistance consists of Stokes friction, linear in u, and pressure drag that is quadratic in u|u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' With mass and added mass in the left hand side factor, and thrust deduction t, surge dynamics reads, (m − X ˙u) ˙u = (1 − t)Tx − Xuu − Xu|u|u|u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2) Introducing Tx = βsT% and T% as commanded thrust percentage, (2) has the form, ˙u = βT% − αu − γu|u| (3) with thrust scaling β, linear damping α, and quadratic damping γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' These are identified from full scale testing in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2 Grey-box identification and analytical solution The identification of (3) based on full-scale acceleration data revealed that due to the low speed regime, the par- ticular hull form, and the propeller slip stream interaction with the pontoons of the catamaran hull, the nonlinear damping coefficient γ is hard to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The contribution of the term is essentially zero, on the basis of the available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Therefore, a linear model is pursued, ˙u = βT% − αu, ˙N = cos (ψ) u, ˙E = sin (ψ) u (4) where T% and heading angle ψ are known and constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The solutions to the linear ODEs are as follows � ��� u(t) N(t) E(t) � ��� = � ��� (1 − e−αt) β αT% � 1 α2 e−αt + 1 αt � T%β cos(ψ) � 1 α2 e−αt + 1 αt � T%β sin(ψ) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (5) The analytical solution is used to calculate a trajectory between two points in the north-east plane, simply by computing the arrival time tf at the final point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The arrival time is obtained by setting the left-hand side of (5) to the desired point and solving for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Once tf is obtained, obtaining the trajectory is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' PATH PLANNING AND COLLISION AVOIDANCE The nominal path of the Greenhopper is described by two waypoints located on the north and south side of the fjord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In conditions with traffic, the objective is to find a path that connects either the nominal waypoints, or the current position of ownship and the goal in a collision-free and safe manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1 Spatio-temporal lattice planner Let X ⊆ R3 be the state space, with x ∈ X and x = [E, N, t]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' X is divided into two subsets, the free space Xfree and the obstacle space Xobs, with Xfree = X\\Xobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The objective is to find a sequence σ of states that minimises the cost function c(σ), while connecting the starting state xs and end state xe σ∗ = arg min σ∈Σ {c(σ) | σ(0) = xs , σ(1) = xe , ∀s ∈ [0, 1], σ(s) ∈ Xfree } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (6) The obstacle subset Xobs is formed by the union over all constraints (Enevoldsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2022), namely Xobs = X OS obs ∪ X ENC obs ∪ X TV obs , X TV obs = n � i=1 XTV,i(t) (7) with X OS obs containing states that violate manoeuvring constraints, X ENC obs the grounding and buoy collision states, and finally X TV obs the target vessel constraints, which is the union of n vessels, such that all n are considered simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The spatial constraints are encoded by the predicted trajectories of each target vessel (XTV,i(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A deterministic planning algorithm is proposed for build- ing a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The starting state xs is the position of ownship in the north-east plane at t = 0, and xe is the current desired destination, at some unknown final time t = tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Consider a grid G = [G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' , Gm]T ∈ Rm×n, with rows Gi = [gi,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' , gi,n] , where each row represents the depth towards the goal and the width of potential deviations, and each element gi,j ∈ G represents a point in the north-east plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The grid is pre-processed, in order to refine it, by modifying points gi,j that violate the envi- ronmental constraints, as follows ¯G = G\\X ENC obs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A directed graph T with root xs is built over k = m+1 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Two sets of nodes are formed, one with all current parent nodes Cp = {xs}, which always contains the start xs, and a set for all current child nodes Cc = {xe} that always contains the end xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' At each iteration, the sets are modified by the grid rows, which dictates edges that are to be formed −500 0 500 East (m) −400 −200 0 200 400 North (m) (a) Full lattice using a 7x5 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' −500 0 500 East (m) −400 −200 0 200 400 North (m) (b) Recomputing the lattice from an arbitrary location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The obstacle subset X ENC obs consists of the area sur- rounding the blue polygon (land and shallow waters) and the interior area of the red circles (buoys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Cp = {xs}, Cc = {xe, Gk} if k = 1, Cp = {xs, Gk−1}, Cc = {xe, Gk} if 1 < k < m + 1, (8) Cp = {xs, Gk−1}, Cc = {xe} if k = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Before adding a given edge to T , the resulting trajectory between the two nodes is checked to see if it violates any constraints (Xobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The trajectory between two nodes is computed by forward simulating (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Nodes in collision from Cc are discarded and omitted from Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2 Rules and regulations Adherence to the rules and practises of safe navigation is fundamental for MASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' There is a general consensus that the most essential IMO COLREGs are rules 8 & 13-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Rules 13-15 dictate the three most common vessel encounters: overtaking (13), head-on (14) and crossing (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Rule 14 & 15 specifies that the give-way vessel must perform the manoeuvre toward the port side, where rule 13 allows passing on either side in a safe manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Rule 16 & 17 dictate the behaviour of the give-way and stand-on vessel, see (Cockcroft and Lameijer, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In the literature, there is a consensus that partial ad- herence to the COLREGs is sufficient to demonstrate capable and safe navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Within confined waters, such as rivers and urban environments, additional complexities may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Rule 14 and 15 specifically apply between two power-driven vessels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' therefore, if the system encounters (a) Overtaking (13) (b) Head-on (14) (c) Crossing (15) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Ship domains for COLREGs-compliance, dimen- sions are dependant on target vessel ship length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' a sailboat, different rules and obligations apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Further- more, certain vessels may be restricted in their manoeu- vrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' If a vessel is restricted, rule 9 applies, which states that a vessel less than 20m should give-way for a restricted vessel, even if according to rule 15 the target vessel is the give-way vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022) used rule 9 within the decision making loop for a river crossing ferry, where its highlighted that simply following rule 15 without considering rule 9 may cause problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The complexity of the rule framework strongly depends on local laws (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' the Rhine River (Koschorrek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2022)) for the particular waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In Canadian waters, rule 15 is modified so that any vessel, with minor exceptions, cross- ing a river must yield to power-driven vessels travelling along it (CSA, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' According to §19 of Danish law on seafaring (Søfartsstyrelsen, 2020, 1991), only three specific ferry routes must disregard the usual obligation to rule 15, and instead yield for any traffic the ferry may impede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='3 Ship domains for COLREGs-compliance If MASS have adequate situation awareness, give-way and stand-on obligations can be enforced using Lam´e curves (Enevoldsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2022), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The curve for compliance with crossing and overtaking scenarios is given by ���� cos(ψ(t))∆El(t) − sin(ψ(t))∆Nl(t) aL ���� p + ���� sin(ψ(t))∆El(t) + cos(ψ(t))∆Nl(t) bL ���� p ≤ 1 (9) when used in conjunction with a circular constraint, ∆Ec(t)2 + ∆Nc(t)2 ≤ r2 L (10) where aL, bL and rL are scalar values based on the length of the target vessel and additional safety margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The difference in coordinates at time t between the own ship and a given target vessel is used to evaluate the domains ∆E(t) = E(t)−ETV(t)+Eo, ∆N(t) = N(t)−NTV(t)+No and ¯ψ(t) = ψTV(t) + ψo, where the offset is used to shift the elliptical and circular components of the domain, � Eo No � = � cos � − ¯ψ � − sin � − ¯ψ � sin � − ¯ψ � cos � − ¯ψ � � � pE pN � (11) with pE = 0, and pN equal to bL in (9) and aL in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' For (9) and (10), the ψo is equal to 0 and π 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='4 Safety margins Ship length is commonly used to compute a safety margin with respect to other vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' However, vessels navigating within inner coastal or confined waters are typically either sailboats or pleasure crafts, which are not obligated to carry an AIS transponder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Vessels of length greater than SFU Sensory information SAS MWS ANS ACS APS VCS SHP HMI RUT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Module interconnection within the autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 20m are often required to have AIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Therefore, it is neces- sary to select an adequate safety margin in the absence of an accurate ship length estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The safety margin is se- lected according to emergency stop manoeuvres (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 2a), such that a suitable distance is maintained to the target vessel, in instances with an erroneously perceived scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' For the current transit speed of 3 knots, the required stop- ping distance is approximately 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5m, therefore selecting 25m as the minimum ship length is ample distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' THE AUTONOMY STACK The following section details the composition of the auton- omy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The middleware is introduced and the purpose and responsibility of each module is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1 Middleware and autonomy stack features Dittmann and Blanke (2022) investigated the regulatory framework and system requirements for the development and commissioning of MASS, highlighting some important considerations regarding reliability and redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In addition, design choices and developments related to the custom middleware solution were detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The autonomous system is composed of various modules, such that each block is compartmentalised and its interface clearly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Using a modular approach allows each part of the stack to be developed and tested individually, and also undergo strict stress and acceptance testing, before being rolled out and combined with the remaining system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The middleware facilitates publish–subscribe communica- tion between modules, such that multiple modules can subscribe to the same module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Testing and simulation of various modules within the stack is achieved using a dedicated middleware simulator (MWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2 Module functionality and interconnection The core of the stack consists of the Autonomous Coordi- nation Supervisor (ACS), Autonomous Navigation Super- visor (ANS) and Autonomous Platform Supervisor (APS), which replace the traditional roles employed by the cap- tain, navigator, and chief engineer (Dittmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The ACS coordinates departures and exchanges routes with the route server (RUT), which stores the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Effective and precise fused perception and sensory infor- mation is crucial for the remaining autonomous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Human lookouts and navigators are replaced by an elec- tronic outlook (Blanke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2018) that uses cameras to detect and classify objects (Sch¨oller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The vi- sion system is fused with the remaining sensors, producing a robust and resilient estimate of static and dynamic ob- stacles (Dagdilelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2022), all of which is encapsulated by the Sensor Fusion (SFU) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The estimated and fused states of the surrounding vessels can be augmented by a trajectory prediction scheme that uses information from the local area (Sch¨oller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' However, the current stack only implements straight-line predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The Situation Awareness Service (SAS) is driven by in- formation from the SFU, in order to maintain an overview of the unfolding scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Once a vessel violates set CPA and TCPA limits, the scenario is passed from SAS to the ANS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' the ’navigator’ is informed about the situation, and triggers the SHP for a route deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' This ensures that the Greenhopper deals with the scenario in a timely manner and with a reasonable safety margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Details on the interaction between the SAS and ANS modules can be found in Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020) and Papageorgiou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A Human-machine Interface (HMI) visualises the SFU, RUT, SAS and SHP on an electronic navigational chart, with correct symbolism from IMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The SHP, originally introduced in Enevoldsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022) as a generalised collision avoidance scheme for vessels in confined and inner coastal waters, is in this paper spe- cialised for crossings, such as those encountered by the Greenhopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The underlying planning scheme is imple- mented as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' As part of the autonomy stack, the SHP is tasked with computing rule-compliant, collision-, and grounding free passages for a given scenario at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Given an input consisting of own ship naviga- tional data, predicted target vessel information, perceived COLREGs scenario (from the SFU-SAS-ANS) and the current destination (from RUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The SHP reports within finite time whether a valid crossing exists and, if so, which sequence of waypoints must be followed to achieve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' DEMONSTRATION AND DISCUSSION The autonomy stack and SHP is validated using software- in-the-loop testing by simulating the sensor fusion output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The results are visualised on the HMI, and generated while running the MWS, SAS, ANS, ACS, RUT and SHP modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The MWS acts as both a vessel and SFU simu- lator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Figure 6 shows a scenario in which the Greenhopper departs the southern harbour, following the nominally planned route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' As the vessel is underway, two target ves- sels approach from the starboard side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The CPA for the southernmost target vessel is greater than the safety limit and is therefore not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' However, the second target vessel violates the CPA limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' therefore, once TCPA falls below the chosen limit, the SHP is triggered and a route deviation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Executing the deviation allows the Greenhopper to safely avoid the target vessel before reaching its destination at the northern point of the fjord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A fundamental requirement for adhering to the COLREGs is an adequate estimate of the scenario at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Most systems, including the proposed one, break down if their assumptions do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Most often, MASS depend on AIS for identifying vessel type and size, but in waters (a) t=10s (b) t=1m40s (c) t=2m40s (d) t=2m47s (e) t=4m16s (f) t=5m58s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Simulated demonstration of the SHP with most of the autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The Greenhopper is depart- ing the southern harbour, following the nominally planned path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A manoeuvre is required while under- way, as two other vessels approach from starboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' as those navigated by the Greenhopper and other au- tonomous ferries, a vast majority operate without AIS (due to being a leisure craft or other exemptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' It is therefore crucial that the perception system can classify if the perceived vessel is power-driven or not and is ca- pable of determining whether a vessel is manoeuvrability restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Otherwise, the COLREGs cannot be applied correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' For safe navigation, this is a major defect, since the target vessels are commanded by humans, who expect that any vessel they encounter adheres and acts according to the COLREGs, and if a severe risk of collision occurs, knows how to mitigate or lessen its severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' If the per- ceived scenario is correct, the proposed collision avoidance strategy can, in a deterministic fashion and within a finite time, report whether or not a path deviation exists within the chosen safety limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' If no solution exists during the voyage, the issue is raised to the ACS, which must be capable of correctly dealing with such an emergency sit- uation, either by calling for help from a Remote Control Centre (RCC) or by stopping the vessel and signalling the surroundings that an emergency is unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Sn/Wn 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 VESSEL BRG 359° RNG 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='52m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='6 CPA 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='95 m 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 TCPA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='4sQ 31 39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='04nm 061° VESSEL BRG 193° RNG 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='86m 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0Sn/Wn VESSEL 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 BRG 268° RNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='16nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 1007° 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='04nm 061* VESSEL BRG 218° RNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='20nm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0Sn/Wn 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='04nm 1030° 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 09Sn/Wn 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='04nm VESSEL BRG 100° RNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='20 nm CPA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='11nm TCPA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1 min 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0Sn/Wn 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 VESSEL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 BRG 071° 31 RNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='21nm CPA 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='96 m TCPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='04nm 030° VESSEL BRG 118° RNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='14nm CPA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='11nm TCPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1 min 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 09Sn/Wn 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='6 VESSEL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 BRG 070° 007° 31 RNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='19nm CPA 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='92m TCPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='6 min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='04nm 060° VESSEL BRG 121° RNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='13nm CPA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='10 nm TCPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 min 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' CONCLUSION The paper presented a collision avoidance perspective to autonomous ferries and harbour buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A Danish au- tonomous ferry initiative, the Greenhopper, was intro- duced, and its autonomy stack was detailed and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A deterministic collision avoidance strategy was presented, as well as simple ship domains for enforcing give-way responsibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The importance of a well-functioning and sufficiently accurate estimate of the unfolding situation was discussed in great detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In conclusion, to navigate according to the COLREGs and safe navigation practises, the target vessels must be correctly classified in terms of vessel type and manoeuvrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Future work includes field verification of the proposed SHP in conjunction with the remaining autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Once verified, the final steps towards fully commissioning the ferry for autonomous operation must be undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors acknowledge Mette Bennedsen and Jens Brauchli Jensen from SIMAC for the fruitful discussions on rules for navigation, Jann-Timothy G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Mayer from W¨artsil¨a for data collection, and fellow ShippingLab con- tributors Kjeld Dittman, Nicholas Hansen, Dimitrios Pa- pageorgio, and Andreas Gamborg for discussions and de- velopments of the autonomy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' This research was spon- sored by the Danish Innovation Fund, The Danish Mar- itime Fund, Orients Fund, and the Lauritzen Foundation through the Autonomy part of the ShippingLab project, grant number 8090-00063B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' The electronic navigational charts have been provided by the Danish Geodata Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' REFERENCES Bergman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Ljungqvist, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Linder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Axehill, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A colregs-compliant motion planner for autonomous maneuvering of marine vessels in complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='12145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Bitar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Eriksen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Lekkas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Breivik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Three-phase automatic crossing for a passenger ferry with field trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In 2021 European Control Con- ference (ECC), 2271–2277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Hansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Stets, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Koester, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Brøsted, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Maurin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Nykvist, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Bang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Authority, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Outlook for navigation–comparing human performance with a robotic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' of ICMASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Brekke, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Eide, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Eriksen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Wilthil, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Breivik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Skjellaug, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Helgesen, Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Lekkas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Martinsen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Thyri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' milliampere: an autonomous ferry prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In Journal of Physics: Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Series, volume 2311, 012029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Cockcroft, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' and Lameijer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Guide to the collision avoidance rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' CSA (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Collision Regulations, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 1416, Canada Shipping Act (CSA), 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' https://laws-loi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='justice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='ca/PDF/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' 1416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Dagdilelis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Andersen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Galeazzi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Cyber-resilience for marine navigation by information fusion and change detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Ocean Engi- neering, 266, 112605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='oc eaneng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='112605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' de Vries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Trevisan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', van der Toorn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Das, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Brito, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Alonso-Mora, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Regulations aware motion planning for autonomous surface vessels in urban canals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In 2022 Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' on Robotics and Automation (ICRA), 3291–3297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1109/ICRA46639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='981 1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Dittmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' and Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Risk mitigation by design of autonomous maritime automation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Automatisierungstechnik, 70(5), 469–481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1515/ auto-2021-0151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Dittmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Hansen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Papageorgiou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Autonomy for ships: A sovereign agents architecture for reliability and safety by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IEEE Xplore-Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IEEE SysTol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Enevoldsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Galeazzi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Sampling-based collision and grounding avoidance for marine crafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Ocean Engineering, 261, 112078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='oceaneng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='112078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Hansen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Enevoldsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Papageorgiou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Autonomous Navigation in Confined Waters–A COLREGs Rule 9 Compliant Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='08227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Hansen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Papageorgiou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Galeazzi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', L¨utzen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Mogensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Bennedsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Hansen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' COLREGs-based Situation Awareness for Marine Vessels-a Discrete Event Systems Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IFAC-PapersOnLine, 53(2), 14501–14508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1016/ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='ifacol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Koschorrek, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Kosch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Nitsch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Abel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and J¨urgens, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Towards semi-autonomous opera- tion of an over-actuated river ferry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Automatisierung- stechnik, 70(5), 433–443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1515/auto-2021-0152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Papageorgiou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Hansen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Dittmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Blanke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Anticipation of ship behaviours in multi-vessel scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Ocean Engineering, 266, 112777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='oceaneng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='2022.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Nalpantidis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Vision-based object track- ing in marine environments using features from neural network detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IFAC-PapersOnLine, 53(2), 14517– 14523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Sch¨oller, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Enevoldsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Becktor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Hansen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Trajectory prediction for marine vessels using historical ais heatmaps and long short-term memory networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IFAC-PapersOnLine, 54(16), 83–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Søfartsstyrelsen (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Bekendtgørelse om sejlads p˚a Limfjorden mellem Egholm og Kattegat (BEK nr 953 af 18/12/1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Søfartsstyrelsen (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Bekendtgørelse om sejlads m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' i visse danske farvande (BEK nr 656 af 20/05/2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Thyri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' and Breivik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A domain-based and reactive colav method with a partially colregs-compliant domain for asvs operating in confined waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Field Robotics, 2, 637–677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Thyri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Breivik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Lekkas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' A path- velocity decomposition approach to collision avoidance for autonomous passenger ferries in confined waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IFAC-PapersOnLine, 53(2), 14628–14635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Shan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Leoni, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Fern´andez-Guti´errez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Meyers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', Ratti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=', and Rus, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Roboat II: A novel autonomous surface vessel for urban environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' In 2020 IEEE/RSJ Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' on Intelligent Robots and Systems (IROS), 1740–1747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E0T4oBgHgl3EQf2QLa/content/2301.02711v1.pdf'} diff --git 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College London,Gower Street, London and +Department of Physics, Faculty of Mathematics and Physics, +University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia +Marko ˇZnidariˇc +Department of Physics, Faculty of Mathematics and Physics, +University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia +The relaxation of observables to their non-equilibrium steady states in a disordered XX chain +subjected to dephasing at every site has been intensely studied in recent years. We comprehensively +analyse the relaxation of staggered magnetization, i.e., imbalance, in such a system, starting from +the N´eel initial state. We analytically predict emergence of several timescales in the system and +extract results which match with large-system numerics without any extra fitting parameter till a +universal timescale. An often reported stretched exponential decay is just one of the regimes which +holds in a finite window of time and is therefore in fact not a true stretched exponential decay. +Subsequently, the asymptotic decay of imbalance is governed by a power law irrespective of the +disorder. We show that this emerges from the continuum limit of the low magnitude eigenspectrum +of the Liouvillian. However, for finite systems, due to discreteness of the spectrum, the final phase +of relaxation is governed by the relevant smallest Liouvillian gap. +I. +INTRODUCTION +Non-interacting disordered systems in one-dimension, +isolated from external environment, are known to exhibit +Anderson localization.1. One expects that if a generic +initial state is allowed to evolve in such a system, at +long time scales, the wavefunction will not show sig- +nificant change. +Among the several quantifiers of this +phenomenon2, one of the most commonly used is imbal- +ance, a quantity easy to measure in experiments. Imbal- +ance I is the staggered magnetization and captures the +difference in orientation of spins on adjacent sites, +I = (1/L) +L +� +j=1 +(−1)j⟨σz +j ⟩, +(1) +where L is the length of the lattice, σz is the Pauli spin +z operator, and the expectation is taken with the state +we want to measure the quantity. It should be evident +that for computational states, the state with the largest +I, which equals 1, is the N´eel state. +One can also surmise that for an Anderson localized +system, if one starts from this state, one should see I ∼ 1 +at large timescales. However, if the localized system is +no longer isolated, then the external degrees of freedom +typically serve to break Anderson localization. The sys- +tem eventually forgets the memory of its initial state. +Accordingly, I would also evolve with time. In the pre- +vious works, the focus has mainly been to find the non +equilibrium steady state (NESS) in such systems3–10, or +perturbations around NESS11, while recently there has +been some work studying how different observables re- +lax to NESS in such systems12–18 or in Stark localized +systems19. Of particular focus was the decay of imbal- +ance. A slow stretched exponential (A exp(−tα)) decay +of I(t) has been reported15,20–23, though the value of α +obtained has some dispute. For example, a short theo- +retical analysis in Ref. 22 finds α ∼ 0.33 whereas a dif- +ferent analysis in Ref. 20 puts α ∼ 0.5. Numerical fits in +Refs. 20 puts α ∼ 0.38 and Refs. 15 and 23 put α ∼ 0.42. +Additionally, the analytical expressions suggested usu- +ally require at least one fitting parameter to be matched +with the numerical results. +In our work, we seek to understand and resolve this +inconsistency. Upon carefully analyzing the relaxation, +we see that several time scales emerge. Broadly speak- +ing one has a regime where off-diagonal matrix elements +of ρ(t) are large, then a regime in which ρ(t) becomes +increasingly diagonal due to dephasing and the scaling +variable is12 τ = 8γt/W 2, and finally a regime where the +system starts to feel its finite size L. In fact, we argue +that the ‘stretched exponential’ is not a true (asymp- +totic) description for the relaxation. It holds only in a +finite window, and is universal and independent of disor- +der type just in a linear-expansion regime, after which a +non-generic disorder-dependent behaviour follows. Addi- +tionally, beyond a timescale, τ = 8γt/W 2 ∼ 1, we show +that the relaxation smoothly changes to 1/(L +√ +t) from +the non-generic behaviour in open boundary conditions, +which is also the standard result for clean systems with +dephasing. Finally, for finite size systems, an exponen- +tial decay occurs at the longest timescale, governed by +the finite Liouvillian gap. +Using a toy model involving three lattice sites, we find +an expression of I(τ) analytically, using second order +perturbation theory, that matches with numerics very +accurately, till τ ∼ 1, without any fitting parameters. +From the model, we show that the universal behaviour +I(τ) = (1 − β√τ) occurs for very short times τ ≪ 1, be- +yond which the non universal behaviour follows till τ ∼ 1. +We first provide a brief description of our main results +in Sec. II before going into the details later. +arXiv:2301.05611v1 [cond-mat.dis-nn] 13 Jan 2023 + +2 +0.01 +0.10 +1 +10 +100 +0.1 +0.5 +1 +5 +10 +(b) +FIG. 1. (a) A schematic figure denoting the five timescales +of imbalance decay for an initial N´eel state. +(a) shows +schematic diagram with timescales t0 +∼ Min(1/γ, 1/W), +t1 ∼ 0.1W 2/(8γJ2), t2 ∼ W 2/(8γJ2) and t3 ∼ L2, whereas in +(b) disorder averaged exact numerical data for different sys- +tem sizes L is plotted with τ = 8γt/W 2 and J = 1, while β +is obtained from Taylor expansion of Eq. 12 with γ = 1 and +W = 8. The green vertical lines denote the timestamps of the +schematic diagram for L = 50 (open boundary conditions are +used). +II. +SUMMARY OF RESULTS +We begin by summarizing our findings. Starting from +the N´eel initial state, in the XX model with on-site dis- +order and dephasing, the behaviour of − ln I(t) can be +schematically represented by Fig. 1(a), where ln denotes +the natural logarithm, γ is the strength of dephasing, W +is the strength of disorder and J is the exchange inter- +action strength of XX model. +We use open boundary +conditions (obc) unless specified. +As evident from the figure, five timescales emerge dur- +ing the evolution of I(t). They are, +I. t < t0 ∼ Min(1/γ, 1/W): The shortest timescale +in the problem. +Here, I(t) ∼ 1 − t2 and hence +− ln I(t) ∼ t2. During this time period, off diago- +nal correlations first develop in the system and then +start decaying after reaching a maxima.24 When the +disorder strength W is small compared to the de- +phasing γ, t0 is given by ∼ 1/γ, akin to clean sys- +tems. However when W ≫ γ, then this behaviour +exists till t ∼ 1/W. +II. t0 < t < t1 ∼ 0.1W 2/(8γJ2): This is where lo- +cal relaxation of σz starts, and most (but not all) +off-diagonal correlations become negligible. In this +timescale one sees −lnI(t) ∼ −ln(1 − β +√ +t) ∼ β +√ +t +behaviour irrespective of the nature of disorder cho- +sen (β can depend on nature of disorder). This is +the short time, linear regime of the stretched expo- +nential behaviour noted in previous works15,20–23. +Unlike previous results, we find that this regime +is not asymptotic and is only valid till τ ∼ τ1 = +(8γJ2t1)/W 2 = 0.1. Therefore, the “stretched ex- +ponential” is not really a true stretched exponential +decay as it does not hold asymptotically, i.e., at ar- +bitrarily small values of I. +III. t1 < t < t2 ∼ W 2/(8γJ2): In this regime, which +holds till τ ∼ τ2 = (8γJ2t2)/W 2 = 1 a non-generic +decay dependent on the choice of disorder distribu- +tion is seen. t2 also marks the end of local relaxation +in the system. +IV. t2 < t < t3: In this regime, the decay smoothly +changes to a power law due to the bunching of low +magnitude eigenvalues of Liouvillian.21,25 This is the +asymptotic behaviour in the thermodynamic limit +for finite W. Here, I(t) ∝ 1/(L +√ +t). The exponent +– 1/2 in our case – can depend on boundary condi- +tions. The change of the nature of decay from Re- +gion III to IV occurs slowly at a timescale ∝ L, not +shown in the schematic diagram. For finite systems, +due to the discrete nature of the Liouvillian spec- +trum, this behaviour is not asymptotic and there is +another timescale. +V. t > t3 ∼ L2: The final timescale in finite sized sys- +tems is governed by the Liouvillian gap, i.e. +the +largest real non zero eigenvalue of the Liouvillian +(since it has an eigenvalue of zero). Here the decay is +given by I(t) ∼ exp(−|λ1|t) where λ1 is the Liouvil- +ian gap. This timescale begins around t3 ∼ 1/|λ1|. +As will be discussed in Sec. IV D, λ1 ∼ 1/L2, hence +this regime starts around t3 ∼ L2. +It is worth noting that till t ∼ t2, i.e. +till Region +III, there is no system size dependence of the results, +neither the time windows nor the behaviour depends on +L. +However, both the behaviour of I(t) and the time +span of Region IV is dependent on L. This feature is +clearly visible in Fig. 1(b) where we plot numerical results +obtained for evolution of −lnI(t) for a disorder strength +of W = 8 for different system sizes. We perform disorder +averaging over 102 realizations for L = 50, 500 and 10 for + +3 +L = 3000. We also mark the various timescales in this +plot, now with respect to τ. Clearly, for τ > 1, different +system sizes start showing different behaviour. +Since till τ ∼ 1, the rescaled data for the different +system sizes clearly overlap with one another and is al- +most linear in ln − ln scale, this prompted the stretched +exponential fits in previous works. In Fig. 1(b), we have +added a black dashed line showing the stretched exponen- +tial expression obtained from a theoretical computation +in Sec. IV B. As will be clear from the analysis in that +section, this behaviour is expected to hold best till small +τ ∼ 0.1, which is what is demonstrated in the plot.26 Fi- +nally for L = 50 one sees a fast growth of − ln I(t) (see a +slight bend upwards) at large times, which is due to the +exponential decay of I(t) from the finite size Liouvillian +gap. +In what follows, we shall discuss the above findings in +details and provide a theoretical understanding for the +same. In the next section, we describe the model in more +details and elaborate on the technique used to compute +the numerical results. Then in Sec. IV we elaborate on +the behaviour of I(t) in different timescales. Finally, in +Sec. V we provide some final remarks on our results and +possible extensions of this work. +III. +MODEL +We take the one dimensional disordered XX chain with +spin 1/2 particles, +H = −J +L−1 +� +j=1 +(σx +j σx +j+1 + σy +j σy +j+1) + +L +� +j=1 +hjσz +j +(2) +where hj are the on-site disorders and σx,y,z denotes the +Pauli matrices. In this work, unless otherwise mentioned, +hjs are chosen from a uniform distribution in (−W, W), +W being the strength of disorder. We set J = 1 for the +rest of our work. To simulate an open system, each spin +is exposed to a dephasing. The evolution of the system’s +density matrix is given by, +dρ +dt = i[ρ, H] + Ldeph(ρ) = L(ρ) +(3) +The non-unitary part can be written in terms of Lindblad +operators as, Ldeph(ρ) = � +k([Lkρ, L† +k] + [Lk, ρL† +k]). We +choose Lk = +� γk +2 σz +k to represent the on site dephasing +term. +For the rest of this work we shall put γk = γ, +i.e. have a spatially uniform dephasing. This choice of +dephasing causes an exponential decay of the off diagonal +elements of the density matrix with a strength γ, in the +diagonal basis of σz in absence of disorder. +If we want to solve Eq. 2 directly, numerically or oth- +erwise, we need to solve a set of 4L − 1 coupled differ- +ential equations. +Fortunately due to the choice of de- +phasing and the quadratic nature of the Hamiltonian, +the exponentially many equations can be decoupled into +blocks of polynomial complexity.3,11,27–32 In other words, +observables follow a hierarchy based on the number of +Fermionic operators they contain. For example, the block +of two point correlators decouples from the rest, and one +can write a closed set of equations for these observables. +Then this solution serves as a source term for the three +point correlations and so on. Since we are interested in +imbalance, which can be extracted from a two-point cor- +relator in the Fermionic language (σz ∼ c†c), we will just +consider the subspace of two point correlation functions. +Following the consideration of. Ref. 3 and 27, we define +the operators, +A(t) = +L +� +r=1 +L+1−r +� +j=1 +a(r) +j (t)A(r) +j +B(t) = +L +� +r=2 +L+1−r +� +j=1 +b(r) +j (t)B(r) +j , +(4) +where A(r+1) +j += +σj +xZ(r−1) +j+1 σj+r +x ++ σj +yZ(r−1) +j+1 σj+r +y +and +B(r+1) +j += σj +xZ(r−1) +j+1 σj+r +y +− σj +yZ(r−1) +j+1 σj+r +x +for r ≥ 2. +Z(r) +j += σj +z . . . σj+r−1 +z +are strings of σz operators, and +A(1) +j += −σj +z. Then the equation governing the time evo- +lution of the set of two point correlation functions can be +written compactly as, +dC(t) +dt ++2i(PC(t)−C(t)PT )+2(Γ ˜C(t)+ ˜C(t)Γ) = 0, (5) +where C, ˜C, P, Γ are L × L matrices. Their elements are +defined as, Cjk(t) = a(k−j+1) +j +(t) + ib(k−j+1) +j +(t) for k > j, +Cjj(t) = a(1) +j (t) and Cjk = C∗ +kj. +˜C = C − diag(C). +P = W − T where, Wjk = hkδjk, the on-site disorders +and for our model. Also for our model, Tjk = J(δj,k−1 + +δj,k+1), as we consider only nearest neighbour couplings33 +and Γk +j = γδjk. Clearly, I(t) = (1/L) � +j(−1)j+1Cjj(t). +Hence for the N´eel initial state Cjj(0) = (−1)j+1 and +I(0) = 1. +Eq. 5 can be recast into a linear differential equation +with L2 variables, in the form, +df +dt = Qf +(6) +where f = (C11, C12, . . . , C1L, C21, . . . , CLL) and Q is the +L2 × L2 matrix governing the evolution. In fact, due to +the hierarchical structure of observables, the eigenvalues +of Q are exactly the eigenvalues of the one particle sector +of the Liouvillian, L, while the eigenvectors of both are +connected via an appropriate rotation. This linearised +equation will be useful in understanding the behaviour +of I(t) in different regimes. +Unlike most of the previous results for this model +which are generated by either an effective Hamiltonian +or DMRG based techniques20,22, we use Eq. 5 or Eq. 6 +for the analysis that follows. This not only allows us to +obtain an exact description of the correlators but also + +4 +● +● +● +● +● +● +● +● +● +● +● +● +●●●●●●● +● +● +● +●●●●●●●●●●●●●●●●●●●●● +●● 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+▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ 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+▼ +▼ +▼ +▼ +▼ +▼ +▼ 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▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼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+○ +○ +○ +○ +○ +○ 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+● +■ +◆ +▲ +▼ +○ +0.1 +1 +10 +100 +0.01 +0.10 +1 +10 +FIG. 2. Figure showing the initial t2 behaviour of − ln I(t) in +region I for different disorder strengths W. +provides access to sizes and times an order of magni- +tude larger than usually accessed by DMRG techniques. +Thus, we can make better statements about system size +dependence of the results. +IV. +DIFFERENT TIMESCALES +Let us now discuss each timescale in more details. +A. +Region I +Doing a simple power series expansion of I(t) in t, the +first non-zero term turns out to be quadratic. Hence, we +expect I(t) ∼ 1−ηt2 behaviour at the smallest timescale. +As evident from Fig. 2, this is the case. This behaviour +continues till time t0 which is dependent on γ and W. +To be more precise, let us first consider the simpler case, +W → 0. +At t = 0, since we start from a pure state, +ρ(t = 0) has only one non zero element, which is in the +diagonal, and C(t = 0) is diagonal as well. Correlations +then spread rapidly throughout the system, reach a max- +ima and start decaying around t ∼ 1/ +� +|γ2 − 4| ∼ 1/γ for +γ ≫ 1. As we see from Fig. 2, indeed below t0 ∼ 0.5 for +γ = 1 and W = 0, one can approximate I(t) as 1 − ηt2, +and hence − ln I(t) ∼ ηt2. +From the numerical fit in +Fig. 2 shown by the black dashed line, we find η ∼ 16. +This behaviour can also be qualitatively extracted from +a simple two-site model (details in Appendix A), and we +get, for small t, I(t) ∼ 1 − 8t2 + O(t3). One can then +numerically check that on addition of a few more sites to +the system η approaches 16 rapidly. +However, on addition of disorder, t0 reduces with in- +creasing W, as can be seen Fig. 2. This can be quali- +tatively understood from Eq. 5, where we see that cor- +relations can develop in the system either due to P, i.e. +the terms involving disorder or due to Γ which involves +dephasing. +The dominant term would then determine +the smallest timescale. Also, it can be shown using the +two-state model again that, for W ≫ γ, the t0 now be- +comes 1/δ, δ being the difference of on-site disorders in +the two-site problem. Hence several timescales emerge +due to different δs at different lattice sites, and we will +get t0 ≪ 1/γ. +If we approximate the width of the +distribution of δs as W, we can say that t0 is around +Min(1/γ, 1/W). This is Region I. +B. +Regions II and III +Next, we shall understand the decay of imbalance in +Regions II and III. Local evolution is dominant in these +regimes and we can use the same analysis to describe +both. Hence we discuss them together. A timescale sep- +arating the two regimes will emerge naturally from the +computation that follows. +We consider W ≫ γ, a regime where localization ef- +fects would be prominent in the system. Then, to under- +stand the evolution of ⟨σz +j ⟩ for site j, we need to take into +account the influence of the neighbouring sites, j −1 and +j +1. Matrix C in Eq. 4 for this 3-site system, becomes a +3×3 matrix. We further ignore Cijs where |j −i| > 1, as +they are negligible compared to the rest in this timescale. +We can then use Eq. 6 with +f = +�C11, C12, C21, C22, C23, C32, C33,� +(7) +where we have taken j = 2 without loss of generality. +The form of Q for this system in Eq. 6 is given in Ap- +pendix B. We use second order perturbation theory to +find the eigenvalues as outlined in Appendix B, since ex- +act analytical diagonalization is not tractable. We also +just need to diagonalize the subspace where we have the +smaller modulus eigenvalues, as the larger modulus eigen- +values dictate the behaviour in Regime I. Doing so, we +obtain the relevant eigenvalues as given in Eq. B3 in Ap- +pendix. B. These in the regime W ≫ γ can be approxi- +mated as ∆(0) = 0 and, +∆(±) ∼ 8γ(δ2 +1 + δ2 +2 ± +� +δ4 +1 + δ4 +2 − δ2 +1δ2 +2) +δ2 +1δ2 +2 +, +(8) +where δ1 = |h1 − h2| and δ2 = |h2 − h3|. Hence the long +time behaviour of C22(t) = −⟨σz +2(t)⟩, can be generically +written as, +C22(t) = +� +m=±,0 +d(m) +2 +exp(−∆(m)t) +(9) +where d(m) +2 +is the corresponding element of the mth eigen- +vector, weighed by the factors arising from the initial con- +ditions. We shall drop the subscript 2 in what follows. +Further simplification of Eq. 9 can be made by observ- +ing that typically (see Appendix B) d(+) is the largest co- +efficient, hence the principal mode of relaxation is ∆(+). +The contribution from ∆(0) and ∆(−) modes weighted +by d(0) and d(−) respectively effectively act as pertur- +bations around the principal mode34 (see Appendix B). +In the random disorder case since ∆(+)s are different for + +5 +■ +■ +■ +■ ■ ■ ■■■■ +■ ■ ■ ■■■■■■■■■■■■■■■■■■■ ■ ■ ■■■■■■■■■■ ■ ■ ■■■■ ■ ■ ■■ +▲ +▲ +▲ +▲ ▲ ▲ ▲▲▲▲ +▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲▲▲▲ ▲ ▲ ▲▲ +● +● +● +● ● ●●●●● +● ● ●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●● ● ●●●●● ● ●●● +■ +▲ +● +0.001 +0.010 +0.100 +1 +10 +0.1 +0.5 +1 +5 +● +● +● +● +● +● ● ● ●● +● +● +● +● ● ●●●●●●●●●●●●●●●●● +● +● ● ● ●●●●●●●● +● +● ● ● ●● +● ● ● ● +■ +■ +■ +■ +■ ■ ■ ■ ■ ■ +■ +■ +■ ■ ■ ■ ■■■■■■■■■■■■■■■■ +■ ■ ■ ■ ■ ■■■■■■■ +■ ■ ■ ■ ■ ■ +■ ■ ■ ■ +● +■ +0.01 +0.1 +1. +10. +0.01 +0.05 +0.10 +0.50 +1 +5 +10 +FIG. 3. +Comparison of analytical predictions with exact numerics in Regions II and III. (a) Plot showing how the rescaled +time causes complete collapse of data with different L, γ, W, and agreement of the said numerical results for the box distributed +random on site disorder, with the theoretical predictions. f(τ) is defined in Eq. 10 and g(τ) is defined in Eq. 12. (b) Plot +showing agreement of our theoretical prediction with numerical results derived from for two other disorder distributions, viz. +gaussian random and alternating disorder, see Eq. 13 and Eq. 14 respectively. fG(τ) is obtained from Eq. 10 where the random +numbers are picked from a Gaussian distribution. +different sites, we reintroduce the subscript j, and write +for the N´eel initial state, I(t) = (1/L) �L +j=1 |Cj(t)| ∼ +(1/L) �L +j=1 |d(+) +j +|e−∆(+) +j +t, where the perturbative correc- +tions due to the other modes can be neglected due to +averaging. +Additionally, since d(+) +j +s are O(1)(see Ap- +pendix B), it is not essential to keep track of them. +Hence, given a disorder distribution one can calculate +the evolution of imbalance for the N´eel initial state in +the thermodynamic limit as, +I(t) = f(t, W, γ) = lim +L→∞(1/L) +L +� +j=1 +e−∆(+) +j +t, +(10) +where different ∆(+) +j +s are obtained from the distribution +of hjs, using Eq. 8. From Eq. 10 we can see that in this +timescale, I(t) is the average over the decay of magne- +tization at individual sites with effective decay rates at +each site controlled by the disorders in its nearest neigh- +bours. From Eq. 8, we also see an emergent energy scale +8γ/W 2 (recall J = 1), which gives us the rescaled time +τ = 8γt/W 2. +In Fig. 3(a) we show comparison between exact nu- +merical results with Eq. 10 and other approximations +described later in the section. +The plots for different +system sizes L, disorder strengths W and dephasing γ +obtained via exact numerics collapse on each other when +we use rescaled τ = 8γt/W 2, confirming the presence of +the universal timescale12. As shown in Fig. 3(a) with the +black dashed line, computing Eq. 10 by sampling a large +number of random numbers from the box disorder distri- +bution, we get a result that is a very accurate match with +exact numerics till τ ∼ 1. Note how a numerical result +obtained via solving many coupled differential equations +can be replicated by correctly sampling the underlying +disorder distribution, due to locality of the evolution. +However, it is difficult to perform the summation an- +alytically to obtain a closed form expression for dif- +ferent disorder distributions. +To proceed further we +need to go to the continuum limit and write Eq. 10 as +I(t) = +� +d∆(+)p(∆(+))e−∆(+)t, p(∆(+)) being the proba- +bility distribution of ∆(+). Even so, computing p(∆(+)) +analytically is a very difficult task. Hence, we need to +make further simplifying assumptions. If we approximate +δ4 +1+δ2 +2−δ2 +1δ2 +2 ∼ (δ2 +1+δ2 +2)2 and hence ∆(+) ∼ 16γ( 1 +δ2 +1 + 1 +δ2 +2 ), +the integral becomes tractable. +This approximation is +valid in the limit of |δ2 +1 − δ2 +2| ≫ 0, where we have +δ2 +1δ2 +2 ≪ δ4 +1 + δ4 +2 and hence works in the large ∆(+) tail of +p(∆(+)) very well, see Fig. 7 in Appendix. B.35. Then, +we consider that δ1 and δ2 are independently drawn from +different sets of random numbers to avoid calculating the +complicated convolution term. This allows us to write, +I(t) = +�� +dδp(δ)e−16γt/δ2�2 +(11) +where p(δ) is the probability distribution of δs. Eq. 11 +allows us to compute the evolution of I(t) analytically +for any given disorder distribution. For example, for a +disorder distributed uniformly in (−W, W), i.e. the box +distribution, Eq. 11 gives the result in terms of known +mathematical functions as, +I(τ = 8γt/W 2) = g(τ) = [− +√ +2π√τerfc +��τ +2 +� ++e− τ +2 + 1 +2τΓ +� +0, τ +2 +� +]2 (12) +where erfc(z) = 1 − erf(z), erf denotes the Error func- +tion and Γ is the incomplete gamma function defined by +Γ(0, z) = +� ∞ +z +e−t/tdt. The green dashed line in Fig. 3(a), +obtained from Eq. 12 shows good agreement till τ ∼ 0.1, + +6 +which is expected since our approximation works best in +the large ∆(+) tail. +1 − β +√ +t scaling regime: +Actually, in the regime of +τ ≪ 1 or t ≪ W 2, Taylor expanding Eq. 12 in τ, we get +I(τ) = 1 − β√τ + O(τ) with β = +√ +8π, which is exactly +the linear term in the stretched exponential21 e−β√τ. In +Fig. 3(a), we plot − ln I(τ) ∼ β√τ as the brown dashed +line, and it also agrees well with exact numerics till τ ∼ +0.1. +Generally, if p(δ) is an analytic function of δ, then +it can be expanded as p(δ) = +1 +N (c + O(δ)), where +N = +� +dδ(c+O(δ)). Since +� +dδe−16t/δ2 ∼ t1/2+O(t), the +lowest order term obtained from p(δ) in this form would +be +√ +t (for c = 0 the result would be different). For many +different disorder distributions this is a good approxima- +tion of results till higher order terms become important +with larger t. +To highlight this special scaling, we label the regime +where the linear approximation of β√τ is valid as Region +II which is approximately till τ = 0.1, and denote the +non-generic regime during 0.1 ≲ τ ≲ 1 as Region III. +Other disorder distributions: +To further consol- +idate our claims, we repeat the above calculations for +other disorder distributions, viz. the Gaussian random +distribution and staggered on-site or alternating poten- +tial, shown in Fig. 3(b). +We have also checked the +quasiperiodic Aubre Andre distribution, data not shown +to avoid cluttering. In Fig. 3(b), results from Eq. 10 is +represented by the green dashed line for the Gaussian +disorder case, and the black dashed line for the alter- +nating potential case. In both cases (and the qausiperi- +odic one not shown), we see that Eq. 10, obtained from +the second order perturbation result of the 3-site model, +captures the nuances in the evolution very well till τ = 1 +(longer for alternating potential see Appendix B). In gen- +eral, our analysis is valid for any disorder distribution +where typical |hj − hj+1| ≫ 0, to make the second order +perturbation theory hold.36 +For the Gaussian distribution it is also possible to com- +pute Eq. 11 analytically and the result is, +I(τ) =e− +√ +8τ, +(13) +a different result to Eq. 12 and exactly the stretched +exponential37. This is shown as the purple dot-dashed +line in Fig. 3(b) which we again see to be valid till τ ∼ 0.1, +thus showing a separation of timescales similar to box dis- +tribution. Other standard disorder distributions such as +the exponential and student’s T distribution also show +similar timescale separation. +Finally, for the staggered potential, i.e when hj = +(−1)jW, Eq. 10 has a simple form. +Since we have +δ1 = −δ2 = δ/2 = W, ∆(+) = 6γ/W 2 for all sites. In +this case the weight of the ∆(−) mode, d(−) = 0, hence +I(t) ∼ d(0) + d(+)e−∆(+)t. For large W, this can be ap- +proximated as falt = e−∆efft, where ∆eff = +8γ +W 2 and hence +I(τ) ∼ e−τ. +(14) +See Appendix B for more details. In Fig. 3(b), we see ex- +actly the expected behaviour. The almost perfect agree- +ment between our theoretical predictions with the numer- +ical data provides a good benchmark about the generality +of our theory. +Effective Hamiltonian approach: +Before we end +this section we make a digression to briefly discuss an al- +ternative approach. In Ref. 12 it was shown that, for the +interacting XX chain with dephasing and strong disor- +der, beyond a timescale given by 1/γ, off-diagonal terms +of the density matrix are negligible and the evolution of +the diagonal terms of the density matrix is given by the +differential equation, dρD +dt = −HeffρD, where ρD denotes +the diagonal part of ρ and +Heff = +L +� +j=1 +2 +(γj + γj+1) +(hj − hj+1)2 + (γj + γj+1)2 (1 − σj.σj+1), +(15) +where σ = (σx, σy, σz). +An emergent rescaled time, +4γt/(W 2 + 4γ2) is seen the overall prefactor of Heff, if +we consider γj = γ and δj = hj −hj+1 ∼ W, reminiscent +of the one we extracted from the three site model, since +this is also a second order perturbative description. In +our non-interacting XX model, this effective description +still holds as there is no term involving any interaction +in Eq. 15. Beyond a cutoff timescale, which for L = 24 +and W = 30 is given by τ ∼ 0.01 (check Appendix C) +the agreement of exact numerics and evolution of I(t) +with Heff is excellent. This means the effective Hamilto- +nian describes the system approximately for τ ≫ τ0 in +our model (τ0 ∼ 0.001 for W = 30), while before that +timescale the off diagonal elements of the density matrix +still has significant contribution to the evolution. While, +a direct use of Heff to repeat the analysis of this section to +understand Region II and III are a bit involved, we shall +use this effective Hamiltonian to explain the behaviour +in Region IV and V in the following sections. +Finally, it is worth noting that For W = 0, Regions +II and III no longer exist and there is an oscillatory be- +haviour in −lnI(t) as can be seen in Fig. 2, with an en- +velope growing as 4γt. For large γ one can again apply +the effective Heisenberg model to obtain these results for +this system as in Ref. 21. Upon analysis via our three +site model, we observe that the magnitude of the real +part of eigenvalues involved in the evolution ∼ 4γ as the +corresponding eigenvectors carry almost all the weight, +a distinct shift from what happens in the disordered +case, where lower magnitude eigenvalues carry most of +the weight. Near the end of this time-scale the system +begins to realize its non-local nature, and the evolution +slowly changes to what is seen in Region IV, discussed +below, irrespective of the presence of disorder. +C. +Region IV +In this section, we shall discuss the behaviour of I(t) +in the fourth time-window, where we observe an asymp- + +7 +Case +disorder α +β +I(t) +− ln I(t) +pbc, even L +0 +2 no overlap +e−4γt +4γ +pbc, odd L +0 +2 +0 +1 +L +√ +8πt +1 +2 ln(tL2) + 1 +2 ln(8π) +obc, any L +0 +2 +0 +1 +L +√ +8πt +1 +2 ln(tL2) + 1 +2 ln(8π) +obc, any L +W +2 +0 +√π +2L +√ +at +1 +2 ln(tL2) + 1 +2 ln 4a +π +pbc, any L +W +2 +2 +√π +2(at)3/2 +3 +2 ln t + 1 +2 ln 4a3 +π +TABLE I. Table showing the decay of I(t) in Regime IV for different cases, α and β are exponents defined in Eq. 17. For obc, +the scaling of I(t) with t do not qualitatively change on addition of disorder, whereas for pbc they are remarkably different. +See text for details +totic power law decay irrespective of the nature of dis- +order distribution. This is the first regime where we see +L dependent behaviour, as evident from Fig. 1(b), where +the evolution of − ln I(t) is plotted for different Ls with +open boundary conditions. Additionally, unlike the pre- +vious regimes which did not depend on the boundary +conditions, behaviour of I(t) in Region IV is strongly +dependent on such factors. As shown in Fig. 4(c), for +open boundary conditions, we see a I(t) ∝ 1/(L +√ +t) +or − ln I(t) ∼ +1 +2 ln(tL2) behaviour(dependence on γ is +more complicated, discussed later in the section). How- +ever for periodic boundary conditions (pbc), as plotted in +Fig. 4(d), − ln I(t) ∼ 3 +2 ln t, show a completely different +behaviour with a different exponent and no system size +dependence! +To explain this behaviour, we first realize that in this +regime charge is transported on longer scales and the +system can no longer be described by three site models +of the previous section. +Hence, we need to study the +eigenspectrum of the Liouvillian. Additionally, we need +to focus on the low magnitude eigenvalues as this regime +is asymptotic. However, since we have the hierarchical +structure of observables, we do not need to study the full +4L ×4L Liouvillian, but the eigenspectrum of Q in Eq. 6, +which, as mentioned earlier, is related to the one-particle +Liouvillian of the problem. We can write the solution of +Eq. 6 generically as, +f(t) = fNESS + +� +j +δf(j)e(λj+iΩj)t, +(16) +where f is defined under Eq. 6, fNESS is the steady state +value of the observables, λj and Ωj are the real and imag- +inary parts of the eigenvalues of Q and δf(j) contains the +corresponding eigenvectors weighted by the initial condi- +tions. +Typically, the weights are similar to the corre- +sponding elements of the eigenvector, i.e. δf(j) ∼ m2 +j +where mj is the jth eigenvector (See Appendix D for +more details). +From Eq. 16, it might naively seem that any observable +should show exponential decay with different rates at dif- +ferent timescales, depending on λjs. However, in reality, +in the thermodynamic limit the eigenspectrum becomes +continuous, and this leads to a power law approach of +observables towards NESS. To demonstrate this, assume +without loss of generality that +λj ∼ −jα, +δf(j) ∼ jβ +(17) +and38 Ωj ≪ λj. +Then Eq. 16 can be written in the +continuum limit as21,22,25, +f(t) − fNESS ∼ +� +djjβe−jαt ∼ t− β+1 +α . +(18) +In what follows, we shall understand the behaviour of +I(t) for different systems using Eq. 16. +In Table I we first summarize the results of various +cases which shall be discussed in this section. +Since f contains all the two particle Fermionic opera- +tors in the system, while I(t) = 1 +L +�L +k=1(−1)k+1Ckk(t), +we can rewrite Eq. 16 in the continuum for I(t) as, +I(t) = INESS + +� +djI(j)eλjt +(19) +where +I(j) = 1 +L +L +� +k=1 +(−1)k+1[δf(j)](k−1)L+k, +(20) +sums over the relevant operator subspace for the finite L +system. Eq. 18 gets appropriately modified. +Let us first look at the simpler case of systems without +disorder. It is known that the low lying eigenenergies in +the one-particle Liouvillian, in open boundary conditions +are approximately39, λj = 2 +�� +4 cos +� πj +L +� +− 3 − 1 +� +, j = +0 . . . L − 1. For periodic boundary conditions the expres- +sion becomes, 2 +�� +4 cos +� 2πj +L +� +− 3 − 1 +� +, j = 0 . . . L − 1, +i.e. there is a degeneracy of 2. Since we are interested in +the regime j/L ≪ 1, we can approximate obc eigenvalues +as 2π2j2 +L2γ . For pbc this gets multiplied by a factor of 4, +and hence in both cases α = 2. +I(j) can be computed from the relevant elements of +the eigenvectors of the one-particle Liouvillian. It turns +out that the relevant elements of the jth eigenvector can +be approximated to be exactly the free fermion wave- +function with the corresponding boundary condition, +i.e. +for pbc, they are +1 +√ +L +�L +k=1 exp(2iπjk/L). +Hence +I(j) ∼ 1 +L[�L +k=1 +1 +√ +L cos(πk) exp(2iπjk/L)]2 is identically + +8 +● +● +● +● +● +● +● +● +● ● ● ● ● ● ●●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■ +■ +■ +■ ■ +■ ■ +■ ■ ■ ■ ■■■■■■■■■■■■■ +● +■ +1 +2 +5 +10 +20 +1.×10-6 +5.×10-6 +1.×10-5 +5.×10-5 +1.×10-4 +5.×10-4 +10-3 +● +● +● +● +● +● +● +● +● ● ● ● ● ● ● +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ ◆ ◆ +◆ ◆ +◆ ◆ +◆ +◆◆◆ +◆ +◆◆ +◆◆◆◆◆◆◆◆ +● +■ +◆ +1 +2 +5 +10 +20 +1.×10-9 +5.×10-9 +1.×10-8 +5.×10-8 +1.×10-7 +5.×10-7 +1.×10-6 +● +● +● ● ● ● +● +● +● ● ● ● ●●●●●●●●●●●●●●●● +● +● ● ● ● ●●●●●●● +● +● ● ● ● ● +● +● ● ● +■ +■ +■ ■ ■ ■ +■ +■ +■ +■ ■ ■ ■ ■ ■■■■■■■■■■■■■■ +■ +■ ■ ■ ■ ■ ■■■■■■ +■ +■ ■ ■ ■ ■ +■ +■ ■ ■ +◆ +◆ +◆ +◆ ◆◆ +◆ +◆ +◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆ +◆ ◆ ◆ ◆ ◆◆ +◆ ◆ ◆ ◆ +▲ +▲ +▲ +▲ ▲ ▲ +▲ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲ +▲ +▲ ▲ ▲ ▲ ▲ +▲ +▲ ▲ ▲ +● +■ +◆ +▲ +5 +10 +50 +100 +500 1000 +4 +6 +8 +10 +12 +● +● +● +● +● ● ● ●●● +● +● +● +● ● ●●●●●●●●●●●●●●●●● +● +● ● ●●●●●●●●● +● ● ● ●●● +● ● ● ●●●●● +■ +■ +■ +■ +■ ■ ■ ■ ■ ■ +■ +■ +■ +■ ■ ■ ■■■■■■■■■■■■■■■■ +■ ■ ■ ■ ■ ■■■■■■■ +■ ■ ■ ■ ■ ■ +■ +■ ■ ■ ■ ■■■ +● +■ +1 +10 +100 +1000 +0 +2 +4 +6 +8 +10 +12 +FIG. 4. Power law decay of I(t) in Region IV. (a) Plot showing j2 scaling of low magnitude Liouvillian eigenvalues averaged over +100 disorder realizations compared with approximate theoretical prediction in Eq. 21. (b) Plot showing scaling of I(j) defined +in Eq. 20 averaged over 100 disorder realizations. obc and pbc denote open and periodic boundary conditions respectively. See +text for details. (c) Plots showing the scaling of − ln I(t) with t for obc at different system sizes L. (d) Same as (c) but for +periodic boundary conditions. +0 for even L due to the symmetry of the wavefunction.40 +Since I has no support on the low magnitude eigenspec- +trum of the Liouvillian, it does not show a power law de- +cay. Instead it decays exponentially with the the largest +λj = 4γ. +The situation for odd L in pbc is different as +the one site is unpaired. +In this case I(j) +∼ +1 +L[�L +k=1 +1 +√ +L cos(πk) exp(2iπjk/L)]2 +∼ +1 +L2 +for j +≪ +L. +This +means +β += +0, +and +hence +I(t) +∼ +2 +� +dj 1 +L2 e−8π2j2/L2 = +1 +L +√ +8πt, where the factor 2 in front +comes from the degeneracy of the eigenvalues. +For +obc +the +eigenvectors +are +given +by, +� +2 +L +�L +k=1 cos(πjk/L). +Hence +I(j) += +1 +L[�L +k=1 +� +2 +L cos(πk) cos(πjk/L)]2. +This +equals +0 +when both L and j are even or odd, and +2 +L2 otherwise. +This gives I(t) ∼ +� +dj 2 +L2 e−2π2(2j)2/L2 = +1 +L +√ +8πt. These +results for the clean systems have been matched with +exact numerics, data not shown. +Now we focus our attention to disordered systems. In- +tuitively, we can predict that since the behaviour in this +regime involves the low energy eigenspectrum of the Liou- +villian, it should not drastically change on addition of dis- +order. If we look into the obc case, we indeed see ∝ 1/ +√ +t +behaviour for both clean and disordered systems, but not +so for the pbc case where we see a t−3/2 behaviour. Let +us investigate why. +Unfortunately it is very difficult to find an analyti- +cal expression for the eigenspectrum when we add the +disorder in the system, so we have to resort to exact +diagonalization. However, we can make some crude ap- +proximations about the low lying eigenvalues and their +scaling with L, W and γ since the evolution is described +by the effective Heisenberg Hamiltonian, Eq. 15 in this +time regime. Firstly, since the effective hamiltonian is +a Heisenberg model, we conclude that the low energy +distribution would be ∝ 1/L2. We make a further edu- +cated guess by borrowing the result λj = 2π2j2 +L2γ +for clean +systems from Ref. 39 and then introducing the relevant +prefactor in Heff for disordered systems. Thus, we have +disorder averaged +λj ∼ −a(γ, W)j2/L2, +(21) + +9 +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 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+▲ +▲ +▲ +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +● +■ +◆ +▲ +0 +500 +1000 +1500 +2000 +0 +5 +10 +15 +FIG. 5. Plot showing the linear growth of − ln I(t) in Region +V for finite size systems of different lengths L and at disorder +strengths W. +The black dashed lines denote the expected +growth of − ln I(t) with the rate given by corresponding λ1 +dependent on L, W and γ. +for j ≪ L, where +a(γ, W) = +8π2γ +( 2W 2 +3 ++ 4γ2) +(22) +and we have used ⟨(hj−hj+1)2⟩ ∼ 2W 2/3 for the box dis- +tribution (−W, W). Henceforth we shall write a(γ, W) as +a. +In Fig. 4(a) we see a very good agreement between our +approximate prediction in Eq. 21 and numerical results +for the low magnitude eigenvalues of the Liouvillian. Now +we turn our attention to the eigenvectors. As shown be- +fore, imbalance has a constant support which was signif- +icant for odd/even j in clean obc systems. Small pertur- +bations around this value due to disorder, does not affect +the overall scheme of things upon averaging over many +disorder realizations, as can be verified from Fig. 4(b). +As seen in Fig. 4(b), for odd j, Ij is almost the same for +clean and disordered obc systems with even L. For even +j on the other had this quantity is no longer identically 0 +as the symmetry of the system is broken due to disorder, +and shows a ( +√ +2j)2/L2 scaling. However, this does not +seriously affect the evolution as their magnitude is much +smaller than for odd j. Consequently, for disordered obc +systems, +I(t) ∼ 1 +L +� +dj 2 +Le−a(2j)2/L2 += +√π +2L +√ +at. +(23) +This is the behaviour seen in Fig. 4(c), where disorder, +W = 4 , is chosen to be smaller to highlight Region IV +(data for W = 10 is similar at larger times, not shown). +However, for the periodic case, we have a different sce- +nario. Previously I(j) had no support for any j in this +regime for clean systems, but now, due to breaking of +symmetry, I(j) ∼ ( +√ +2j)2/L2 as evident from Fig. 4(b). +The eigenvalues as seen in Fig. 4(a) follow the same scal- +ing law as obc (the degeneracy in the smallest magnitude +eigenvalue is resolved as we increase j). Hence, we get, +I(t) ∼ 1 +L +� +dj 2j2 +L2 e−a(j)2/L2 += +√π +2(at)3/2 . +(24) +Indeed this is what we see in Fig. 4(d), including the lack +of L dependence in the result. +D. +Region V +As mentioned in Sec. II, Region IV becomes asymp- +totic in the thermodynamic limit, when we have a con- +tinuum in the eigenspectrum. +For finite size systems +though, the spectrum eventually gets resolved and hence +for larger t we observe a different behaviour, which we +classify as region V. In this region, I(t) shows an expo- +nential decay, with the rate governed by the Liouvillian +gap λ1.41 λ1 is thelargest real part of non-zero Liouvil- +lian eigenvalue for even L systems. For odd L systems, +it is the next largest as the imbalance operator does not +have support for the largest non-zero eigenvalue in this +case. In Fig. 5, we see that − ln I(t) grows linearly with +time at large times, with the slope |λ1(L, γ, W)| verifying +our expectation. As before, analytical results for clean +systems are well known39, but upon addition of disorder +one can only resort to numerics for exact results. Nev- +ertheless, the description given by Eq. 15 is still valid in +this time scale, and explains the scaling behaviour of λ1 +with L, W and γ as discussed below. +In Fig. 6(a) we show the change of λ1 with L for con- +stant W and γ. It is known that the 1/L2 scaling of λ1 +works for sufficiently large L for clean systems. In fact, it +has already been shown in Ref. 39 that the critical length +required to observe this scaling is Lc ∼ π +√ +2 +γ . For γ = 0.1 +this is at L ∼ 45 and is shown by the blue gridline in the +Fig. 6(a). This behaviour also continues when we intro- +duce disorder in the system. It is evident from the figure +that upon increasing W, Lc decreases. This can be ex- +plained as in the previous section using Eq 15 and Eq. 21, +which tells us that on addition of disorder of strength W, +γ can be effectively replaced by (4γ2 + 2W 2 +3 )/(4γ). This +gives +Lc(W) ∼ +4 +√ +2πγ +4γ2 + 2W 2 +3 +, +(25) +shown as different coloured gridlines in the plot, and cap- +ture the transition point between the two regimes quite +well. +Finally in Fig. 6(b) we show the change of λ1 on chang- +ing W and γ for constant L = 3000 via a contour plot. +We expect |λ1L2| ∼ a and hence the equation of constant +contours can be obtained from Eq. 22 as +ln a = constant. +(26) + +10 +● +● +● +● +● +● +● +● ● ● ● ● ● ●●●●●●●● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■ +■ +■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ +▲ +▲ ▲ +▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ +● +■ +◆ +▲ +5 +10 +50 +100 +0.005 +0.010 +0.050 +0.100 +0.500 +0 +2 +4 +6 +8 +10 +0.5 +1.0 +1.5 +2.0 +-1.0 +-0.5 +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +FIG. 6. +Scaling of λ1, the smallest Liouvillian gap, with different parameters. (a) Plot showing 1/L2 scaling of λ1, for different +disorder strengths, W. Lc from Eq. 25 is marked by appropriate coloured gridlines in the plot. Averaging is done over 100 +realizations. (b) Plot showing dependence of ln |λ1L2| with W and γ for L = 3000 averaged over 100 disorder realizations. The +red dashed lines indicate three constant contours at 0, 2.5 and 3.5 for ln a obtained from Eq. 22. +Three such contours are plotted as red dashed lines in +Fig. 6(b) and they match very well with the contours +obtained from exact numerics. +V. +DISCUSSION +In this work we have studied the evolution of imbal- +ance, I(t) for the disordered XX chain with on-site de- +phasing, starting from the N´eel initial state. Using the +hierarchical nature of equations for the observables, we +have computed I(t) for large system sizes (L ∼ 103) and +long times (t ∼ 103). Our analysis showed the emergence +of five timescales in disordered systems. +The shortest time scale, t < t0 ∼ Min(1/γ, 1/W) where +I(t) ∼ (1 − ηt2) constitutes the linear response regime +of the system. Then, due to the localization via disor- +der, two timescales denoted by Region II and III emerge, +which are absent in clean systems. Region II is the linear +regime of what has been called a stretched exponential +decay in previous works15,20–23, where I(t) ∼ (1 − β +√ +t), +and is universal irrespective of the nature of disorder cho- +sen, and continues till t1 ∼ 0.1W 2/8γ, before smoothly +transitioning to a disorder dependent behaviour in region +III which continues tillt2 ∼ W 2/(8γJ2). We thus con- +clude that the stretched exponential fits are neither uni- +versal nor asymptotic for the system under study. Sim- +ple local three-site models describe the behaviour in these +two regimes, and a rescaled time emerges during the anal- +ysis τ = 8γt/W 2. An effective Heisenberg model given +by Eq. 15 also provides the correct description of the +evolution from Region II. +For t > t2, where Region IV begins, dephasing breaks +localization and the system shows size and boundary con- +ditions dependent power law decay of I(t). In fact for +obc it shows a 1/(L +√ +t) decay and for pbc a 1/ +√ +t3 decay. +These were explained from the continuum limit of the low +magnitude eigenspectrum of the Liouvillian. Finally, we +discussed that for finite sized systems due to resolution +of the eigenspectrum, the decay of I(t) in final Region +V at t > t3 is exponential with the rate governed by the +relevant Liouvillian gap λ1. We also provided analysis of +how λ1 typically scales with W, L, γ using the effective +Heisenberg Hamiltonian approach. +Our analysis gives insight to the mechanism behind +the evolution of imbalance in such a system. It demys- +tifies the behaviour of the system in Region II and III +clearly showing its local origin and gives us the ability to +make predictions about the evolution for many different +disorder distributions. Additionally, the proper analysis +of Liouvillian eigenspectrum provides us with the correct +asymptotic description of evolution for different models +and boundary conditions. +We would also like to briefly mention the nuances of +crossover from one regime to another. Since the tran- +sition is smooth, there is a always a finite time taken +by the system to go from one regime to another. +For +example, from Region III to IV, where the finite size +effect first appears, the system typically takes t ∼ L +to transition to a power law for open boundary condi- +tions. Hence, if the thermodynamic limit is taken first, +we would see that even if region III ends at L independent +τ2 = 8γt2/W 2 ∼ 1, the power law in Region IV is only +reached asymptotically. However, this does not seem to +be the case for periodic boundary conditions. This is also +consistent with the system size dependence of the results +in Region IV. +Additionally, it is worthy to remark that the effec- +tive Hamiltonian description12 of Eq. 15 is also valid +for weakly interacting systems where the interaction + +11 +strength is much smaller than disorder strength. This +means weak interaction does not play much role in evo- +lution in such timescales, so we expect most of our results +to hold for weakly interacting systems as well. +Furthermore, while we have discussed the N´eel ini- +tial state in this work, decay of I(t) from other generic +computational initial states also share some similar fea- +tures. The behaviour in regions I, IV and V seen for the +N´eel state is still observed for other typical states. The +differences occur in t2 and the behaviour in Regions II +and III. The first difference in other initial states is that +I(t = 0) < 1 i.e. − ln I(t = 0) > 0. Since t2 is defined by +a finite non-zero value of I universal for a given disorder +distribution, relaxation from different I(t = 0) typically +reaches this value at different times. Secondly, all the +sites of the initial state are not locally equivalent unlike +the N´eel state. Hence, the scaling in Regions II and III +show a difference due to variation in local behaviours. A +detailed analysis of this aspect is beyond the scope of the +present work. +There are still a few open questions left in the context +of this work. One natural extension would be to study the +evolution of other observables such as current in such a +model. The question of whether the scaling laws depend +on the type of dephasing is also worth studying. Indeed, +the hierarchical structure of observables or the effective +Heisenberg model might break down if we choose a dif- +ferent model of dephasing. +Additionally, while Region +I-III has been studied under different disorder settings, +we have provided a general explanation for the power +law decay in Region IV, discussing the two cases where +it shows different exponents. Specific potentials may give +rise to unique behaviour in this time scale, and an anal- +ysis of that is left for a future work. +ACKNOWLEDGEMENT +We would like to acknowledge support by Grants +No. J1-1698, J1-4385 and No. P1-0402 from the Slove- +nian Research Agency. RG would also like to acknowl- +edge support from UKRI grant EP/R029075/1. +1 P. W. Anderson, Phys. Rev. 109, 1492 (1958). +2 F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 +(2008). +3 M. ˇZnidariˇc, Journal of Statistical Mechanics: Theory and +Experiment 2010, L05002 (2010). +4 S. R. Taylor and A. Scardicchio, Phys. Rev. B 103, 184202 +(2021). +5 C. Guo and D. Poletti, Phys. Rev. A 95, 052107 (2017). +6 I. Vakulchyk, I. Yusipov, M. Ivanchenko, S. Flach, and +S. Denisov, Phys. Rev. B 98, 020202 (2018). +7 L.-N. Wu, A. Schnell, G. D. Tomasi, M. Heyl, and +A. Eckardt, New Journal of Physics 21, 063026 (2019). +8 C. Monthus, Journal of Statistical Mechanics: Theory and +Experiment 2017, 043302 (2017). +9 I. Yusipov, T. Laptyeva, S. Denisov, and M. Ivanchenko, +Phys. Rev. Lett. 118, 070402 (2017). +10 O. S. Vershinina, +I. I. Yusipov, +S. Denisov, +M. V. +Ivanchenko, and T. V. Laptyeva, Europhysics Letters 119, +56001 (2017). +11 X. Turkeshi and M. Schir´o, Phys. Rev. B 104, 144301 +(2021). +12 M. V. Medvedyeva, T. Prosen, and M. ˇZnidariˇc, Phys. Rev. +B 93, 094205 (2016). +13 S. Gopalakrishnan, K. R. Islam, and M. Knap, Phys. Rev. +Lett. 119, 046601 (2017). +14 B. Sciolla, D. Poletti, and C. Kollath, Phys. Rev. Lett. +114, 170401 (2015). +15 E. Levi, M. Heyl, I. Lesanovsky, and J. P. Garrahan, Phys. +Rev. Lett. 116, 237203 (2016). +16 T. L. M. Lezama and Y. B. Lev, SciPost Phys. 12, 174 +(2022). +17 B. Everest, I. Lesanovsky, J. P. Garrahan, and E. Levi, +Phys. Rev. B 95, 024310 (2017). +18 S. Wolff, J.-S. Bernier, D. Poletti, A. Sheikhan, and C. Kol- +lath, Phys. Rev. B 100, 165144 (2019). +19 L. N. Wu and A. Eckardt, Phys. Rev. Lett. 123, 030602 +(2019). +20 M. H. Fischer, M. Maksymenko, and E. Altman, Phys. +Rev. Lett. 116, 160401 (2016). +21 Z. Cai and T. Barthel, Phys. Rev. Lett. 111, 150403 +(2013). +22 J. Ren, Q. Li, W. Li, Z. Cai, and X. Wang, Phys. Rev. +Lett. 124, 130602 (2020). +23 S. Marcantoni, F. Carollo, F. M. Gambetta, I. Lesanovsky, +U. Schneider, and J. P. Garrahan, Phys. Rev. B 106, +134211 (2022). +24 Recall that since we start from the N´eel state, a computa- +tional state, and hence at t = 0 the system is completely +uncorrelated. +25 M. V. Medvedyeva and S. Kehrein, Phys. Rev. B 90, +205410 (2014). +26 Hence numerical fits of the form exp(−tα) in Refs. 20–23 +found different αs depending on fit windows as the be- +haviour is not a true stretched exponential. +27 M. ˇZnidariˇc and M. Horvat, The European Physical Jour- +nal B 86, 67 (2013). +28 C. Guo and D. Poletti, Phys. Rev. A 98, 052126 (2018). +29 V. Eisler, Journal of Statistical Mechanics: Theory and +Experiment 2011, P06007 (2011). +30 K. Temme, M. M. Wolf, and F. Verstraete, New Journal +of Physics 14, 075004 (2012). +31 B. Horstmann, J. I. Cirac, and G. Giedke, Phys. Rev. A +87, 012108 (2013). +32 A. Nava, G. Campagnano, P. Sodano, and D. Giuliano, +Arxiv:2210.10856 (2022). +33 For Fermionic Hamiltonians, if longer range couplings are +present then in the most general case, Tjk = Jjk where Jjk +denotes the hopping strength between two different sites +at positions j and k. + +12 +34 This is difficult to rigorously prove analytically, but can be +seen numerically. +35 This choice of approximation is used to obtain a tractable +form for p(∆(+)). Other choices are also possible in the +said limit, but they do not simplify the calculation. We +show a comparison of the probability distribution of the +approximated and exact ∆(+) in Appendix B. +36 One can still numerically solve the small size Hamiltonians +to obtain the eigenenergies non-perturbatively and con- +tinue with the analysis. +37 Ref. 20 arrived at this result using a different approach ef- +fectively leading to the same integral. However, under their +approximations one arrives at this result irrespective of the +disorder chosen and then needs to fix a free parameter via +numerical fitting to fit to different disorders. +38 Ωj gives an oscillatory term which is usually ∼ 0 for the +relevant low magnitude eigenvalues of the Liouvillian spec- +trum. If they are included the power law reduces by 125. +39 M. ˇZnidariˇc, Phys. Rev. E 92, 042143 (2015). +40 The square comes from the weightage of the initial condi- +tions, whose weights are of the same order as the element +of the eigenfunction. Hence we use ∼ instead of =. +41 The exception being clean pbc systems with even L where +imbalance has no support on the eigenfunction as shown +before. +Appendix A: Two site model +In this Appendix we shall derive the eigenspectrum ex- +pressions for the two site model using Eq. 6 as the starting +point of our computation. For Regime I this simple model +is enough to qualitatively explain the features seen. For +the two site model, f = (C11, C12, C21, C22) and, +Q = +� +� +� +0 +2i +−2i +0 +2i +2iδ + 4γ +0 +−2i +−2i +0 +−2iδ + 4γ +2i +0 +−2i +2i +0 +� +� +� +(A1) +where we have taken J = 1 and δ = h1 − h2. For δ = 0 +(clean system), one can compute the eigenvalues exactly +as , +� +0, 4γ, 2 +� +γ − +� +γ2 − 4 +� +, 2 +�� +γ2 − 4 + γ +�� +, +(A2) +and the time dependent solutions to the C11(22) can be +written as, +C11(22) = e2γt[(−) cosh +� +2t +� +γ2 − 4 +� +−(+) +γ sinh +� +2t +� +γ2 − 4 +� +� +γ2 − 4 +]. +(A3) +Hence, +I(t) = 1 +2(C11(t) − C22(t)) ∼ 1 − 8t2 + O(t3). +(A4) +This is the demonstration of the short time quadratic +behaviour when t < 1/ +� +γ2 − 4. +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +FIG. 7. Comparison of distribution of p(∆(+)), between ∆(+) +given by Eq. B6 and approximated by Eq. B7. +However when δ > 0 the exact solution is given from a +cubic equation since one of the eigenvalues of Q is always +0. The expressions are complicated and hence not shown. +But we can formulate a simple perturbative result for +δ ≫ 1, γ. To do so we first rearrange Q to separate the +degenerate block and non-degenerate blocks as, +Q = +� +� +� +0 +0 +2i +−2i +0 +0 +−2i +2i +2i +−2i 4γ + 2iδ +0 +−2i +2i +0 +4γ − 2iδ +� +� +� = +� +O2×2 B2×2 +C2×2 D2×2 +� +. +(A5) +Then applying second order degenerate perturbation the- +ory to O and second order non degenerate perturbation +to D we have the eigenvalues as, +� +0, +16γ +4γ2 + δ2 , 4γ + 2iδ + +8 +4γ + 2iδ , 4γ − 2iδ − +8 +−4γ + 2iδ +� +(A6) +and the solution for δ ≫ γ is given by +I(t) = −4 cos +� +2 +√ +δ2 + 4t +� ++ δ2 +δ2 + 4 += 1−8t2 +O +� +t4� +(A7) +whence the quadratic decay with t remains but it is valid +till t ∼ 1/ +√ +δ2 + 4. +Appendix B: Three site model +Now we repeat the above calculation for a three site +model which describes the behaviour in Regime II and +III. In this case f is given by Eq. 7, while Q is given by, +� +� +� +� +� +� +� +� +� +0 +2i +−2i +0 +0 +0 +0 +2i +4γ + 2iδ1 +0 +−2i +0 +0 +0 +−2i +0 +4γ − 2iδ1 +2i +0 +0 +0 +0 +−2i +2i +0 +2iτ +−2i +0 +0 +0 +0 +2i +4γ + 2iδ2 +0 +−2i +0 +0 +0 +−2i +0 +4γ − 2iδ2 +2i +0 +0 +0 +0 +−2i +2i +0 +� +� +� +� +� +� +� +� +� +(B1) + +13 +where, h1 − h2 = δ1 and h2 − h3 = δ2. Rearranging in +the form of Eq. A5, we get the general form in this case +as, +Q = +� +O3×3 B3×4 +C4×3 D4×4 +� +. +(B2) +Then using second order degenerate perturbation theory +in the subspace of O, we can compute the eigenvalues as +∆(0) = 0 and +∆± = +8γ +� +8γ2 ± +� +16γ4 + 4γ2 (δ2 +1 + δ2 +2) + δ4 +1 − δ2 +1δ2 +2 + δ4 +2 + δ2 +1 + δ2 +2 +� +(4γ2 + δ2 +1) (4γ2 + δ2 +2) +. +(B3) +Since we are in the regime where W ≫ γ, we have δ1, +δ2 ≫ γ, and hence we can approximate Eq. B3 as Eq. 8. +One can also compute the eigenvectors in this subspace +and plug in the initial N´eel state to find the coefficients, +d(0) = −1 +3 +d(+) = (2δ2 +1 − δ2 +2 + κ)(δ2 +2 + κ) +3δ2 +1κ +d(−) = (2δ2 +1 − δ2 +2 − κ)(−δ2 +2 + κ) +3δ2 +1κ +(B4) +where κ = +� +δ4 +1 − δ2 +1δ2 +2 + δ4 +2. We can see that when |δ1| = +|δ2|, d(+) = 4 +3, d(−) = 0. In the opposite regime, i.e. when +|δ2 +1 −δ2 +2| ≫ 0, d(+) → 1 and d(−) → −d(0). Consequently, +it can be shown 1 ≤ d(+) ≤ 4/3, 0 ≤ d(−) ≤ 1/3 and +hence d(+) always provides the largest contribution to +the evolution. +We use the result for |δ1| = |δ2| when we compute I(t) +for alternating potential plotted in Fig. 3(b) as, +I(t) = −1 +3 + 4 +3e−∆(+)t +∼ e−4/3∆(+)t = e−τ, +(B5) +where we have used ∆(+) ∼ 6γ/W 2 for W ≫ γ. One +surprising aspect is that Eq. B5 is a good fit to exact +numerics in Fig. 3(b) beyond τ ∼ 1. This can be ex- +plained by observing that if we take local models of higher +sizes, then we will find numerically that d(0) decreases +and the weight gets shifted to a mode which evolves with +∼ 4/3∆(+). Thus Eq. B5 remains valid even when the +transport occurs in a more non-local region (but not in +the full lattice) and hence to longer times, before the +system shows the asymptotic power law behaviour. Also +notice that because there is only one mode of relaxation, +the decay is exponential in this timescale. +For the random disorder case we will have a distribu- +tion of d and ∆(±) based on the distribution of disorder. +However, the principal mode of decay is via ∆(+). To +the leading order, the effect of d0 on this mode becomes +progressively smaller as we move away from the limiting +alternating potential case |δ1| = |δ2|, as it is countered by +the d(−) term. Furthermore ∆(−) is typically too small +to have a significant effect on evolution in Regimes II +and III. Finally, the shifting of weight towards a mode +∼ ∆(+) when we take slightly bigger but still local mod- +els, is valid for the disordered case as well. Hence, simply +averaging over ∆(+) modes is usually enough to get ac- +curate results as presented in Fig. 3(a). +The final point we need to address is the validity of +the approximation we have used to arrive at Eq. 11 from +Eq. 10. Let us recall that +∆(+) ∼ 8γ(δ2 +1 + δ2 +2 + +� +δ4 +1 + δ4 +2 − δ2 +1δ2 +2) +δ2 +1δ2 +2 +. +(B6) +For very large W, we can expect terms with |δ1| ∼ |δ2| +would be statistically insignificant. +Hence to get a +tractable expression, we take the approximation |δ2 +1 − +δ2 +2| ≫ 0. +This would constitute the long tails of the +distribution of ∆(+). Finally, to be able to perform the +analytical computation simply, we choose to approximate +∆(+) by, +∆(+) ∼ 16γ(δ2 +1 + δ2 +2) +δ2 +1δ2 +2 +(B7) +and show its agreement with full ∆(+) in Fig. 7. The +approximation qualitatively represents the exact curve +reasonably well. Also, since it becomes exact in the tail, +i.e. for large ∆(+), consequently the expression computed +using these result represent the exact result for I(t) for +smaller t more accurately than the rest, a feature seen in +Fig. 3(a) and Fig. 3(b). The exact stretched exponential +result which arises out of this approximation for Gaussian +disorders is also valid only for small t. +Appendix C: Agreement of exact numerics with +evolution by Heff +In the end of Sec. IV B we mentioned that we could +use an effective Hamiltonian to replicate the behaviour +of I(t) from a bit later than τ = τ0. Let us discuss this +aspect in a bit of detail. + +14 +● +● +● +● +● +● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +● +■ +0.001 +0.010 +0.100 +1 +0.05 +0.10 +0.50 +1 +5 +FIG. 8. +Comparison between exact numerical results for evo- +lution of I(t) with those obtained by solving Eq. C1 averaged +over 20 realizations for L = 24 and W = 30. +Following Ref. 12, at a sufficiently large timescale the +density matrix becomes effectively diagonal and the evo- +lution of the diagonal elements ρi is given by, +dρi(t) +dt += +� +j +[Heff]ijρj(t) +(C1) +where Heff is given in Eq. 15. I(t) can then be computed +from ρi(t) as +I(t) = +2L +� +i +ρi(t) +L +� +j +⟨αi| (−1)jσz +j |αi⟩ +(C2) +where |αi⟩ can be taken as the computational basis states. +We plot the comparison of results obtained from exact +numerics using Eq. 11 and evolution of the density ma- +trix using Eq. C1 and Eq. C2 in Fig. 8. We see that from +τ ∼ 0.01 the two results are almost equal. This effective +description reproduces the low energy eigenspectrum of +the Liouvillian quite accurately and hence correctly ap- +proximates the evolution from τ ≫ τ0, thus is effective in +capturing the evolution of I(t) from approximately Re- +gion II, as expected. +Appendix D: Computation of low magnitude +eigenspectrum of Liouvillian +Following the arguments of Ref. 39, we have extracted +the low magnitude eigenspectrum of the system by diag- +onalizing a matrix of size ∼ L × L instead of ∼ L2 × L2 +one-particle Liouvillian. If we denote the computational +basis as |j⟩, then the low magnitude eigenspectrum of the +one-particle Liouvillian can be approximated by eigen- +spectrum the following matrix +� +� +0 +RT +0 +R −4Iγ +iIX +0 −iIX −4Iγ +� +� +(D1) +● +● +● +● +● +● +● +● +● ● ● +● ● ●● +●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■ +■ +■ +■ ■ ■ +■ ■ ■ +■ +■■■■■ +■■ +■■■■■■■ +● +■ +1 +2 +5 +10 +20 +0.001 +0.005 +0.010 +0.050 +0.100 +0.500 +FIG. 9. Comparison of exactly computed Ij via solving L×L +linear equations for the initial conditions with our approxima- +tion using the squares of the eigenvector elements for a system +with L = 80, W = 30, γ = 1 and periodic boundary condi- +tions, averaged over 103 disorder realizations. +The dashed +line is the approximate scaling we have used in Sec. IV C. +written in the basis |j⟩⟨j|, +|j⟩⟨j + 1| + |j + 1⟩⟨j| +and |j⟩⟨j + 1| − |j + 1⟩⟨j|. +Here γ is the dephasing, +Rjk = −2i +√ +2(δj,k − δk,j+1) andXjk = (hj − hj+1)δjk, hi +being the on site disorders. Note that unlike the clean +Hamiltonian case treated in Ref. 39, |j⟩⟨j +1|+|j +1⟩⟨j| +are not eigenvectors of L as disorder breaks transla- +tional invariance. +Thus we need to effectively solve a +3L−2×3L−2 [3L×3L] problem for obc [pbc], to obtain +the eigenspectrum, which is the ‘tridiagonal’ approxima- +tion. This allows us to compute the eigenspectrum for +large systems and results are shown in Fig. 4(a) and (b). +Furthermore, for small system sizes one can numeri- +cally show that the δf(j)s are proportional to the cor- +responding eigenvector elements and that approximating +I(j) with the square of corresponding eigenvector terms +is justified. In Fig. 9 we show the comparison of our ap- +proximation with exact I(j) found by numerically com- +puting the correct weights due to the initial N´eel state. +We do exact diagonalization for a system size of L = 80 +at large W = 30, averaged over 103 realizations under +periodic boundary conditions to obtain the exact data. +As one can see the qualitative nature of both the plots +are the same, they vary by a factor of approximately 1.5 +(computed numerically). Hence in Fig. 4(b) our approx- +imation was able to qualitatively replicate the behaviour +of I(t) with insignificant deviation from exact numerics. + diff --git a/YdE5T4oBgHgl3EQfdg8f/content/tmp_files/load_file.txt b/YdE5T4oBgHgl3EQfdg8f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5392960cf54aa224aca03a59dbc4dad3e6c9e56e --- /dev/null +++ b/YdE5T4oBgHgl3EQfdg8f/content/tmp_files/load_file.txt @@ -0,0 +1,1173 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf,len=1172 +page_content='Relaxation of imbalance in disordered XX model with on site dephasing Roopayan Ghosh Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='Gower Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' London and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Faculty of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' University of Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Jadranska 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' SI-1000 Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Slovenia Marko ˇZnidariˇc Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Faculty of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' University of Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Jadranska 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' SI-1000 Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Slovenia The relaxation of observables to their non-equilibrium steady states in a disordered XX chain subjected to dephasing at every site has been intensely studied in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We comprehensively analyse the relaxation of staggered magnetization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=', imbalance, in such a system, starting from the N´eel initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We analytically predict emergence of several timescales in the system and extract results which match with large-system numerics without any extra fitting parameter till a universal timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' An often reported stretched exponential decay is just one of the regimes which holds in a finite window of time and is therefore in fact not a true stretched exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Subsequently, the asymptotic decay of imbalance is governed by a power law irrespective of the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We show that this emerges from the continuum limit of the low magnitude eigenspectrum of the Liouvillian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, for finite systems, due to discreteness of the spectrum, the final phase of relaxation is governed by the relevant smallest Liouvillian gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' INTRODUCTION Non-interacting disordered systems in one-dimension, isolated from external environment, are known to exhibit Anderson localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' One expects that if a generic initial state is allowed to evolve in such a system, at long time scales, the wavefunction will not show sig- nificant change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Among the several quantifiers of this phenomenon2, one of the most commonly used is imbal- ance, a quantity easy to measure in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Imbal- ance I is the staggered magnetization and captures the difference in orientation of spins on adjacent sites, I = (1/L) L � j=1 (−1)j⟨σz j ⟩, (1) where L is the length of the lattice, σz is the Pauli spin z operator, and the expectation is taken with the state we want to measure the quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It should be evident that for computational states, the state with the largest I, which equals 1, is the N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' One can also surmise that for an Anderson localized system, if one starts from this state, one should see I ∼ 1 at large timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, if the localized system is no longer isolated, then the external degrees of freedom typically serve to break Anderson localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The sys- tem eventually forgets the memory of its initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Accordingly, I would also evolve with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In the pre- vious works, the focus has mainly been to find the non equilibrium steady state (NESS) in such systems3–10, or perturbations around NESS11, while recently there has been some work studying how different observables re- lax to NESS in such systems12–18 or in Stark localized systems19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Of particular focus was the decay of imbal- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A slow stretched exponential (A exp(−tα)) decay of I(t) has been reported15,20–23, though the value of α obtained has some dispute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For example, a short theo- retical analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 22 finds α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='33 whereas a dif- ferent analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 20 puts α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Numerical fits in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 20 puts α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='38 and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15 and 23 put α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Additionally, the analytical expressions suggested usu- ally require at least one fitting parameter to be matched with the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In our work, we seek to understand and resolve this inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Upon carefully analyzing the relaxation, we see that several time scales emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Broadly speak- ing one has a regime where off-diagonal matrix elements of ρ(t) are large, then a regime in which ρ(t) becomes increasingly diagonal due to dephasing and the scaling variable is12 τ = 8γt/W 2, and finally a regime where the system starts to feel its finite size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In fact, we argue that the ‘stretched exponential’ is not a true (asymp- totic) description for the relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It holds only in a finite window, and is universal and independent of disor- der type just in a linear-expansion regime, after which a non-generic disorder-dependent behaviour follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Addi- tionally, beyond a timescale, τ = 8γt/W 2 ∼ 1, we show that the relaxation smoothly changes to 1/(L √ t) from the non-generic behaviour in open boundary conditions, which is also the standard result for clean systems with dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally, for finite size systems, an exponen- tial decay occurs at the longest timescale, governed by the finite Liouvillian gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Using a toy model involving three lattice sites, we find an expression of I(τ) analytically, using second order perturbation theory, that matches with numerics very accurately, till τ ∼ 1, without any fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' From the model, we show that the universal behaviour I(τ) = (1 − β√τ) occurs for very short times τ ≪ 1, be- yond which the non universal behaviour follows till τ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We first provide a brief description of our main results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' II before going into the details later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='05611v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='dis-nn] 13 Jan 2023 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='10 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 1 5 10 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (a) A schematic figure denoting the five timescales of imbalance decay for an initial N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (a) shows schematic diagram with timescales t0 ∼ Min(1/γ, 1/W), t1 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1W 2/(8γJ2), t2 ∼ W 2/(8γJ2) and t3 ∼ L2, whereas in (b) disorder averaged exact numerical data for different sys- tem sizes L is plotted with τ = 8γt/W 2 and J = 1, while β is obtained from Taylor expansion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12 with γ = 1 and W = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The green vertical lines denote the timestamps of the schematic diagram for L = 50 (open boundary conditions are used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' SUMMARY OF RESULTS We begin by summarizing our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Starting from the N´eel initial state, in the XX model with on-site dis- order and dephasing, the behaviour of − ln I(t) can be schematically represented by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 1(a), where ln denotes the natural logarithm, γ is the strength of dephasing, W is the strength of disorder and J is the exchange inter- action strength of XX model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We use open boundary conditions (obc) unless specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As evident from the figure, five timescales emerge dur- ing the evolution of I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' They are, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' t < t0 ∼ Min(1/γ, 1/W): The shortest timescale in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Here, I(t) ∼ 1 − t2 and hence − ln I(t) ∼ t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' During this time period, off diago- nal correlations first develop in the system and then start decaying after reaching a maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='24 When the disorder strength W is small compared to the de- phasing γ, t0 is given by ∼ 1/γ, akin to clean sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However when W ≫ γ, then this behaviour exists till t ∼ 1/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' t0 < t < t1 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1W 2/(8γJ2): This is where lo- cal relaxation of σz starts, and most (but not all) off-diagonal correlations become negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In this timescale one sees −lnI(t) ∼ −ln(1 − β √ t) ∼ β √ t behaviour irrespective of the nature of disorder cho- sen (β can depend on nature of disorder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This is the short time, linear regime of the stretched expo- nential behaviour noted in previous works15,20–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Unlike previous results, we find that this regime is not asymptotic and is only valid till τ ∼ τ1 = (8γJ2t1)/W 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Therefore, the “stretched ex- ponential” is not really a true stretched exponential decay as it does not hold asymptotically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=', at ar- bitrarily small values of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' t1 < t < t2 ∼ W 2/(8γJ2): In this regime, which holds till τ ∼ τ2 = (8γJ2t2)/W 2 = 1 a non-generic decay dependent on the choice of disorder distribu- tion is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' t2 also marks the end of local relaxation in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' t2 < t < t3: In this regime, the decay smoothly changes to a power law due to the bunching of low magnitude eigenvalues of Liouvillian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='21,25 This is the asymptotic behaviour in the thermodynamic limit for finite W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Here, I(t) ∝ 1/(L √ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The exponent – 1/2 in our case – can depend on boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The change of the nature of decay from Re- gion III to IV occurs slowly at a timescale ∝ L, not shown in the schematic diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For finite systems, due to the discrete nature of the Liouvillian spec- trum, this behaviour is not asymptotic and there is another timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' t > t3 ∼ L2: The final timescale in finite sized sys- tems is governed by the Liouvillian gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' the largest real non zero eigenvalue of the Liouvillian (since it has an eigenvalue of zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Here the decay is given by I(t) ∼ exp(−|λ1|t) where λ1 is the Liouvil- ian gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This timescale begins around t3 ∼ 1/|λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' IV D, λ1 ∼ 1/L2, hence this regime starts around t3 ∼ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It is worth noting that till t ∼ t2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' till Region III, there is no system size dependence of the results, neither the time windows nor the behaviour depends on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, both the behaviour of I(t) and the time span of Region IV is dependent on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This feature is clearly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 1(b) where we plot numerical results obtained for evolution of −lnI(t) for a disorder strength of W = 8 for different system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We perform disorder averaging over 102 realizations for L = 50, 500 and 10 for 3 L = 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We also mark the various timescales in this plot, now with respect to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Clearly, for τ > 1, different system sizes start showing different behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since till τ ∼ 1, the rescaled data for the different system sizes clearly overlap with one another and is al- most linear in ln − ln scale, this prompted the stretched exponential fits in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 1(b), we have added a black dashed line showing the stretched exponen- tial expression obtained from a theoretical computation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As will be clear from the analysis in that section, this behaviour is expected to hold best till small τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1, which is what is demonstrated in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='26 Fi- nally for L = 50 one sees a fast growth of − ln I(t) (see a slight bend upwards) at large times, which is due to the exponential decay of I(t) from the finite size Liouvillian gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In what follows, we shall discuss the above findings in details and provide a theoretical understanding for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In the next section, we describe the model in more details and elaborate on the technique used to compute the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Then in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' IV we elaborate on the behaviour of I(t) in different timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' V we provide some final remarks on our results and possible extensions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' MODEL We take the one dimensional disordered XX chain with spin 1/2 particles, H = −J L−1 � j=1 (σx j σx j+1 + σy j σy j+1) + L � j=1 hjσz j (2) where hj are the on-site disorders and σx,y,z denotes the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In this work, unless otherwise mentioned, hjs are chosen from a uniform distribution in (−W, W), W being the strength of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We set J = 1 for the rest of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To simulate an open system, each spin is exposed to a dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The evolution of the system’s density matrix is given by, dρ dt = i[ρ, H] + Ldeph(ρ) = L(ρ) (3) The non-unitary part can be written in terms of Lindblad operators as, Ldeph(ρ) = � k([Lkρ, L† k] + [Lk, ρL† k]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We choose Lk = � γk 2 σz k to represent the on site dephasing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For the rest of this work we shall put γk = γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' have a spatially uniform dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This choice of dephasing causes an exponential decay of the off diagonal elements of the density matrix with a strength γ, in the diagonal basis of σz in absence of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' If we want to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2 directly, numerically or oth- erwise, we need to solve a set of 4L − 1 coupled differ- ential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Fortunately due to the choice of de- phasing and the quadratic nature of the Hamiltonian, the exponentially many equations can be decoupled into blocks of polynomial complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='3,11,27–32 In other words, observables follow a hierarchy based on the number of Fermionic operators they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For example, the block of two point correlators decouples from the rest, and one can write a closed set of equations for these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Then this solution serves as a source term for the three point correlations and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since we are interested in imbalance, which can be extracted from a two-point cor- relator in the Fermionic language (σz ∼ c†c), we will just consider the subspace of two point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Following the consideration of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3 and 27, we define the operators, A(t) = L � r=1 L+1−r � j=1 a(r) j (t)A(r) j B(t) = L � r=2 L+1−r � j=1 b(r) j (t)B(r) j , (4) where A(r+1) j = σj xZ(r−1) j+1 σj+r x + σj yZ(r−1) j+1 σj+r y and B(r+1) j = σj xZ(r−1) j+1 σj+r y − σj yZ(r−1) j+1 σj+r x for r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Z(r) j = σj z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' σj+r−1 z are strings of σz operators, and A(1) j = −σj z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Then the equation governing the time evo- lution of the set of two point correlation functions can be written compactly as, dC(t) dt +2i(PC(t)−C(t)PT )+2(Γ ˜C(t)+ ˜C(t)Γ) = 0, (5) where C, ˜C, P, Γ are L × L matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Their elements are defined as, Cjk(t) = a(k−j+1) j (t) + ib(k−j+1) j (t) for k > j, Cjj(t) = a(1) j (t) and Cjk = C∗ kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ˜C = C − diag(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' P = W − T where, Wjk = hkδjk, the on-site disorders and for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Also for our model, Tjk = J(δj,k−1 + δj,k+1), as we consider only nearest neighbour couplings33 and Γk j = γδjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Clearly, I(t) = (1/L) � j(−1)j+1Cjj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence for the N´eel initial state Cjj(0) = (−1)j+1 and I(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 5 can be recast into a linear differential equation with L2 variables, in the form, df dt = Qf (6) where f = (C11, C12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' , C1L, C21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' , CLL) and Q is the L2 × L2 matrix governing the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In fact, due to the hierarchical structure of observables, the eigenvalues of Q are exactly the eigenvalues of the one particle sector of the Liouvillian, L, while the eigenvectors of both are connected via an appropriate rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This linearised equation will be useful in understanding the behaviour of I(t) in different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Unlike most of the previous results for this model which are generated by either an effective Hamiltonian or DMRG based techniques20,22, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 5 or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6 for the analysis that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This not only allows us to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='obtain an exact description of the correlators but also ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='4 ●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='◆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='▲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='▼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='10 1 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Figure showing the initial t2 behaviour of − ln I(t) in region I for different disorder strengths W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' provides access to sizes and times an order of magni- tude larger than usually accessed by DMRG techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Thus, we can make better statements about system size dependence of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' DIFFERENT TIMESCALES Let us now discuss each timescale in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Region I Doing a simple power series expansion of I(t) in t, the first non-zero term turns out to be quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, we expect I(t) ∼ 1−ηt2 behaviour at the smallest timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2, this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This behaviour continues till time t0 which is dependent on γ and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To be more precise, let us first consider the simpler case, W → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' At t = 0, since we start from a pure state, ρ(t = 0) has only one non zero element, which is in the diagonal, and C(t = 0) is diagonal as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Correlations then spread rapidly throughout the system, reach a max- ima and start decaying around t ∼ 1/ � |γ2 − 4| ∼ 1/γ for γ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As we see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2, indeed below t0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 for γ = 1 and W = 0, one can approximate I(t) as 1 − ηt2, and hence − ln I(t) ∼ ηt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' From the numerical fit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2 shown by the black dashed line, we find η ∼ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This behaviour can also be qualitatively extracted from a simple two-site model (details in Appendix A), and we get, for small t, I(t) ∼ 1 − 8t2 + O(t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' One can then numerically check that on addition of a few more sites to the system η approaches 16 rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, on addition of disorder, t0 reduces with in- creasing W, as can be seen Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This can be quali- tatively understood from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 5, where we see that cor- relations can develop in the system either due to P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' the terms involving disorder or due to Γ which involves dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The dominant term would then determine the smallest timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Also, it can be shown using the two-state model again that, for W ≫ γ, the t0 now be- comes 1/δ, δ being the difference of on-site disorders in the two-site problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence several timescales emerge due to different δs at different lattice sites, and we will get t0 ≪ 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' If we approximate the width of the distribution of δs as W, we can say that t0 is around Min(1/γ, 1/W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This is Region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Regions II and III Next, we shall understand the decay of imbalance in Regions II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Local evolution is dominant in these regimes and we can use the same analysis to describe both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence we discuss them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A timescale sep- arating the two regimes will emerge naturally from the computation that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We consider W ≫ γ, a regime where localization ef- fects would be prominent in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Then, to under- stand the evolution of ⟨σz j ⟩ for site j, we need to take into account the influence of the neighbouring sites, j −1 and j +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Matrix C in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4 for this 3-site system, becomes a 3×3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We further ignore Cijs where |j −i| > 1, as they are negligible compared to the rest in this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We can then use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6 with f = �C11, C12, C21, C22, C23, C32, C33,� (7) where we have taken j = 2 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The form of Q for this system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6 is given in Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We use second order perturbation theory to find the eigenvalues as outlined in Appendix B, since ex- act analytical diagonalization is not tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We also just need to diagonalize the subspace where we have the smaller modulus eigenvalues, as the larger modulus eigen- values dictate the behaviour in Regime I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Doing so, we obtain the relevant eigenvalues as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B3 in Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' These in the regime W ≫ γ can be approxi- mated as ∆(0) = 0 and, ∆(±) ∼ 8γ(δ2 1 + δ2 2 ± � δ4 1 + δ4 2 − δ2 1δ2 2) δ2 1δ2 2 , (8) where δ1 = |h1 − h2| and δ2 = |h2 − h3|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence the long time behaviour of C22(t) = −⟨σz 2(t)⟩, can be generically written as, C22(t) = � m=±,0 d(m) 2 exp(−∆(m)t) (9) where d(m) 2 is the corresponding element of the mth eigen- vector, weighed by the factors arising from the initial con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We shall drop the subscript 2 in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Further simplification of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 9 can be made by observ- ing that typically (see Appendix B) d(+) is the largest co- efficient, hence the principal mode of relaxation is ∆(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The contribution from ∆(0) and ∆(−) modes weighted by d(0) and d(−) respectively effectively act as pertur- bations around the principal mode34 (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In the random disorder case since ∆(+)s are different for 5 ■ ■ ■ ■ ■ ■ ■■■■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■ ■ ■ ■■■■■■■■■■ ■ ■ ■■■■ ■ ■ ■■ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲▲▲▲ ▲ ▲ ▲▲ ● ●●●●● ● ●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●● ● ●●●●● ● ●●● ■ ▲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='100 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 1 5 ● ● ●● ● ●●●●●●●●●●●●●●●●● ● ● ●●●●●●●● ● ● ●● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='50 1 5 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Comparison of analytical predictions with exact numerics in Regions II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (a) Plot showing how the rescaled time causes complete collapse of data with different L, γ, W, and agreement of the said numerical results for the box distributed random on site disorder, with the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' f(τ) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 and g(τ) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (b) Plot showing agreement of our theoretical prediction with numerical results derived from for two other disorder distributions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' gaussian random and alternating disorder, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 13 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 14 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' fG(τ) is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 where the random numbers are picked from a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' different sites, we reintroduce the subscript j, and write for the N´eel initial state, I(t) = (1/L) �L j=1 |Cj(t)| ∼ (1/L) �L j=1 |d(+) j |e−∆(+) j t, where the perturbative correc- tions due to the other modes can be neglected due to averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Additionally, since d(+) j s are O(1)(see Ap- pendix B), it is not essential to keep track of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, given a disorder distribution one can calculate the evolution of imbalance for the N´eel initial state in the thermodynamic limit as, I(t) = f(t, W, γ) = lim L→∞(1/L) L � j=1 e−∆(+) j t, (10) where different ∆(+) j s are obtained from the distribution of hjs, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 we can see that in this timescale, I(t) is the average over the decay of magne- tization at individual sites with effective decay rates at each site controlled by the disorders in its nearest neigh- bours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 8, we also see an emergent energy scale 8γ/W 2 (recall J = 1), which gives us the rescaled time τ = 8γt/W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(a) we show comparison between exact nu- merical results with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 and other approximations described later in the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The plots for different system sizes L, disorder strengths W and dephasing γ obtained via exact numerics collapse on each other when we use rescaled τ = 8γt/W 2, confirming the presence of the universal timescale12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(a) with the black dashed line, computing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 by sampling a large number of random numbers from the box disorder distri- bution, we get a result that is a very accurate match with exact numerics till τ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Note how a numerical result obtained via solving many coupled differential equations can be replicated by correctly sampling the underlying disorder distribution, due to locality of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, it is difficult to perform the summation an- alytically to obtain a closed form expression for dif- ferent disorder distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To proceed further we need to go to the continuum limit and write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 as I(t) = � d∆(+)p(∆(+))e−∆(+)t, p(∆(+)) being the proba- bility distribution of ∆(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Even so, computing p(∆(+)) analytically is a very difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, we need to make further simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' If we approximate δ4 1+δ2 2−δ2 1δ2 2 ∼ (δ2 1+δ2 2)2 and hence ∆(+) ∼ 16γ( 1 δ2 1 + 1 δ2 2 ), the integral becomes tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This approximation is valid in the limit of |δ2 1 − δ2 2| ≫ 0, where we have δ2 1δ2 2 ≪ δ4 1 + δ4 2 and hence works in the large ∆(+) tail of p(∆(+)) very well, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 7 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Then, we consider that δ1 and δ2 are independently drawn from different sets of random numbers to avoid calculating the complicated convolution term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This allows us to write, I(t) = �� dδp(δ)e−16γt/δ2�2 (11) where p(δ) is the probability distribution of δs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 11 allows us to compute the evolution of I(t) analytically for any given disorder distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For example, for a disorder distributed uniformly in (−W, W), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' the box distribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 11 gives the result in terms of known mathematical functions as, I(τ = 8γt/W 2) = g(τ) = [− √ 2π√τerfc ��τ 2 � +e− τ 2 + 1 2τΓ � 0, τ 2 � ]2 (12) where erfc(z) = 1 − erf(z), erf denotes the Error func- tion and Γ is the incomplete gamma function defined by Γ(0, z) = � ∞ z e−t/tdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The green dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(a), obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12 shows good agreement till τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1, 6 which is expected since our approximation works best in the large ∆(+) tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 1 − β √ t scaling regime: Actually, in the regime of τ ≪ 1 or t ≪ W 2, Taylor expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12 in τ, we get I(τ) = 1 − β√τ + O(τ) with β = √ 8π, which is exactly the linear term in the stretched exponential21 e−β√τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(a), we plot − ln I(τ) ∼ β√τ as the brown dashed line, and it also agrees well with exact numerics till τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Generally, if p(δ) is an analytic function of δ, then it can be expanded as p(δ) = 1 N (c + O(δ)), where N = � dδ(c+O(δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since � dδe−16t/δ2 ∼ t1/2+O(t), the lowest order term obtained from p(δ) in this form would be √ t (for c = 0 the result would be different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For many different disorder distributions this is a good approxima- tion of results till higher order terms become important with larger t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To highlight this special scaling, we label the regime where the linear approximation of β√τ is valid as Region II which is approximately till τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1, and denote the non-generic regime during 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1 ≲ τ ≲ 1 as Region III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Other disorder distributions: To further consol- idate our claims, we repeat the above calculations for other disorder distributions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' the Gaussian random distribution and staggered on-site or alternating poten- tial, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We have also checked the quasiperiodic Aubre Andre distribution, data not shown to avoid cluttering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(b), results from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 is represented by the green dashed line for the Gaussian disorder case, and the black dashed line for the alter- nating potential case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In both cases (and the qausiperi- odic one not shown), we see that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10, obtained from the second order perturbation result of the 3-site model, captures the nuances in the evolution very well till τ = 1 (longer for alternating potential see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In gen- eral, our analysis is valid for any disorder distribution where typical |hj − hj+1| ≫ 0, to make the second order perturbation theory hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='36 For the Gaussian distribution it is also possible to com- pute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 11 analytically and the result is, I(τ) =e− √ 8τ, (13) a different result to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12 and exactly the stretched exponential37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This is shown as the purple dot-dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(b) which we again see to be valid till τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1, thus showing a separation of timescales similar to box dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Other standard disorder distributions such as the exponential and student’s T distribution also show similar timescale separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally, for the staggered potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e when hj = (−1)jW, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 has a simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since we have δ1 = −δ2 = δ/2 = W, ∆(+) = 6γ/W 2 for all sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In this case the weight of the ∆(−) mode, d(−) = 0, hence I(t) ∼ d(0) + d(+)e−∆(+)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For large W, this can be ap- proximated as falt = e−∆efft, where ∆eff = 8γ W 2 and hence I(τ) ∼ e−τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (14) See Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(b), we see ex- actly the expected behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The almost perfect agree- ment between our theoretical predictions with the numer- ical data provides a good benchmark about the generality of our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Effective Hamiltonian approach: Before we end this section we make a digression to briefly discuss an al- ternative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12 it was shown that, for the interacting XX chain with dephasing and strong disor- der, beyond a timescale given by 1/γ, off-diagonal terms of the density matrix are negligible and the evolution of the diagonal terms of the density matrix is given by the differential equation, dρD dt = −HeffρD, where ρD denotes the diagonal part of ρ and Heff = L � j=1 2 (γj + γj+1) (hj − hj+1)2 + (γj + γj+1)2 (1 − σj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='σj+1), (15) where σ = (σx, σy, σz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' An emergent rescaled time, 4γt/(W 2 + 4γ2) is seen the overall prefactor of Heff, if we consider γj = γ and δj = hj −hj+1 ∼ W, reminiscent of the one we extracted from the three site model, since this is also a second order perturbative description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In our non-interacting XX model, this effective description still holds as there is no term involving any interaction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Beyond a cutoff timescale, which for L = 24 and W = 30 is given by τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='01 (check Appendix C) the agreement of exact numerics and evolution of I(t) with Heff is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This means the effective Hamilto- nian describes the system approximately for τ ≫ τ0 in our model (τ0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='001 for W = 30), while before that timescale the off diagonal elements of the density matrix still has significant contribution to the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' While, a direct use of Heff to repeat the analysis of this section to understand Region II and III are a bit involved, we shall use this effective Hamiltonian to explain the behaviour in Region IV and V in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally, it is worth noting that For W = 0, Regions II and III no longer exist and there is an oscillatory be- haviour in −lnI(t) as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2, with an en- velope growing as 4γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For large γ one can again apply the effective Heisenberg model to obtain these results for this system as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Upon analysis via our three site model, we observe that the magnitude of the real part of eigenvalues involved in the evolution ∼ 4γ as the corresponding eigenvectors carry almost all the weight, a distinct shift from what happens in the disordered case, where lower magnitude eigenvalues carry most of the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Near the end of this time-scale the system begins to realize its non-local nature, and the evolution slowly changes to what is seen in Region IV, discussed below, irrespective of the presence of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Region IV In this section, we shall discuss the behaviour of I(t) in the fourth time-window, where we observe an asymp- 7 Case disorder α β I(t) − ln I(t) pbc, even L 0 2 no overlap e−4γt 4γ pbc, odd L 0 2 0 1 L √ 8πt 1 2 ln(tL2) + 1 2 ln(8π) obc, any L 0 2 0 1 L √ 8πt 1 2 ln(tL2) + 1 2 ln(8π) obc, any L W 2 0 √π 2L √ at 1 2 ln(tL2) + 1 2 ln 4a π pbc, any L W 2 2 √π 2(at)3/2 3 2 ln t + 1 2 ln 4a3 π TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Table showing the decay of I(t) in Regime IV for different cases, α and β are exponents defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For obc, the scaling of I(t) with t do not qualitatively change on addition of disorder, whereas for pbc they are remarkably different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' See text for details totic power law decay irrespective of the nature of dis- order distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This is the first regime where we see L dependent behaviour, as evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 1(b), where the evolution of − ln I(t) is plotted for different Ls with open boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Additionally, unlike the pre- vious regimes which did not depend on the boundary conditions, behaviour of I(t) in Region IV is strongly dependent on such factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(c), for open boundary conditions, we see a I(t) ∝ 1/(L √ t) or − ln I(t) ∼ 1 2 ln(tL2) behaviour(dependence on γ is more complicated, discussed later in the section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' How- ever for periodic boundary conditions (pbc), as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(d), − ln I(t) ∼ 3 2 ln t, show a completely different behaviour with a different exponent and no system size dependence!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To explain this behaviour, we first realize that in this regime charge is transported on longer scales and the system can no longer be described by three site models of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, we need to study the eigenspectrum of the Liouvillian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Additionally, we need to focus on the low magnitude eigenvalues as this regime is asymptotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, since we have the hierarchical structure of observables, we do not need to study the full 4L ×4L Liouvillian, but the eigenspectrum of Q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6, which, as mentioned earlier, is related to the one-particle Liouvillian of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We can write the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6 generically as, f(t) = fNESS + � j δf(j)e(λj+iΩj)t, (16) where f is defined under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6, fNESS is the steady state value of the observables, λj and Ωj are the real and imag- inary parts of the eigenvalues of Q and δf(j) contains the corresponding eigenvectors weighted by the initial condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Typically, the weights are similar to the corre- sponding elements of the eigenvector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' δf(j) ∼ m2 j where mj is the jth eigenvector (See Appendix D for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 16, it might naively seem that any observable should show exponential decay with different rates at dif- ferent timescales, depending on λjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, in reality, in the thermodynamic limit the eigenspectrum becomes continuous, and this leads to a power law approach of observables towards NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To demonstrate this, assume without loss of generality that λj ∼ −jα, δf(j) ∼ jβ (17) and38 Ωj ≪ λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 16 can be written in the continuum limit as21,22,25, f(t) − fNESS ∼ � djjβe−jαt ∼ t− β+1 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (18) In what follows, we shall understand the behaviour of I(t) for different systems using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Table I we first summarize the results of various cases which shall be discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since f contains all the two particle Fermionic opera- tors in the system, while I(t) = 1 L �L k=1(−1)k+1Ckk(t), we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 16 in the continuum for I(t) as, I(t) = INESS + � djI(j)eλjt (19) where I(j) = 1 L L � k=1 (−1)k+1[δf(j)](k−1)L+k, (20) sums over the relevant operator subspace for the finite L system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 18 gets appropriately modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Let us first look at the simpler case of systems without disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It is known that the low lying eigenenergies in the one-particle Liouvillian, in open boundary conditions are approximately39, λj = 2 �� 4 cos � πj L � − 3 − 1 � , j = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' L − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For periodic boundary conditions the expres- sion becomes, 2 �� 4 cos � 2πj L � − 3 − 1 � , j = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' L − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' there is a degeneracy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since we are interested in the regime j/L ≪ 1, we can approximate obc eigenvalues as 2π2j2 L2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For pbc this gets multiplied by a factor of 4, and hence in both cases α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' I(j) can be computed from the relevant elements of the eigenvectors of the one-particle Liouvillian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It turns out that the relevant elements of the jth eigenvector can be approximated to be exactly the free fermion wave- function with the corresponding boundary condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' for pbc, they are 1 √ L �L k=1 exp(2iπjk/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence I(j) ∼ 1 L[�L k=1 1 √ L cos(πk) exp(2iπjk/L)]2 is identically 8 ● ● ● ● ● ●●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■ ■ 1 2 5 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-4 10-3 ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆ ◆ ◆◆ ◆◆◆◆◆◆◆◆ ■ ◆ 1 2 5 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='×10-6 ● ● ● ● ● ● ●●●●●●●●●●●●●●●● ● ● ● ●●●●●●● ● ● ● ● ● ● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ■ ■ ■ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Power law decay of I(t) in Region IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (a) Plot showing j2 scaling of low magnitude Liouvillian eigenvalues averaged over 100 disorder realizations compared with approximate theoretical prediction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (b) Plot showing scaling of I(j) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 20 averaged over 100 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' obc and pbc denote open and periodic boundary conditions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (c) Plots showing the scaling of − ln I(t) with t for obc at different system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (d) Same as (c) but for periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 0 for even L due to the symmetry of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='40 Since I has no support on the low magnitude eigenspec- trum of the Liouvillian, it does not show a power law de- cay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Instead it decays exponentially with the the largest λj = 4γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The situation for odd L in pbc is different as the one site is unpaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In this case I(j) ∼ 1 L[�L k=1 1 √ L cos(πk) exp(2iπjk/L)]2 ∼ 1 L2 for j ≪ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This means β = 0, and hence I(t) ∼ 2 � dj 1 L2 e−8π2j2/L2 = 1 L √ 8πt, where the factor 2 in front comes from the degeneracy of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For obc the eigenvectors are given by, � 2 L �L k=1 cos(πjk/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence I(j) = 1 L[�L k=1 � 2 L cos(πk) cos(πjk/L)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This equals 0 when both L and j are even or odd, and 2 L2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This gives I(t) ∼ � dj 2 L2 e−2π2(2j)2/L2 = 1 L √ 8πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' These results for the clean systems have been matched with exact numerics, data not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Now we focus our attention to disordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In- tuitively, we can predict that since the behaviour in this regime involves the low energy eigenspectrum of the Liou- villian, it should not drastically change on addition of dis- order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' If we look into the obc case, we indeed see ∝ 1/ √ t behaviour for both clean and disordered systems, but not so for the pbc case where we see a t−3/2 behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Let us investigate why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Unfortunately it is very difficult to find an analyti- cal expression for the eigenspectrum when we add the disorder in the system, so we have to resort to exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, we can make some crude ap- proximations about the low lying eigenvalues and their scaling with L, W and γ since the evolution is described by the effective Heisenberg Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15 in this time regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Firstly, since the effective hamiltonian is a Heisenberg model, we conclude that the low energy distribution would be ∝ 1/L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We make a further edu- cated guess by borrowing the result λj = 2π2j2 L2γ for clean systems from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 39 and then introducing the relevant prefactor in Heff for disordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' we have disorder averaged λj ∼ −a(γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' W)j2/L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='(21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='◆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='▲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Plot showing the linear growth of − ln I(t) in Region V for finite size systems of different lengths L and at disorder strengths W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The black dashed lines denote the expected growth of − ln I(t) with the rate given by corresponding λ1 dependent on L, W and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' for j ≪ L, where a(γ, W) = 8π2γ ( 2W 2 3 + 4γ2) (22) and we have used ⟨(hj−hj+1)2⟩ ∼ 2W 2/3 for the box dis- tribution (−W, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Henceforth we shall write a(γ, W) as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(a) we see a very good agreement between our approximate prediction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 21 and numerical results for the low magnitude eigenvalues of the Liouvillian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Now we turn our attention to the eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As shown be- fore, imbalance has a constant support which was signif- icant for odd/even j in clean obc systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Small pertur- bations around this value due to disorder, does not affect the overall scheme of things upon averaging over many disorder realizations, as can be verified from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(b), for odd j, Ij is almost the same for clean and disordered obc systems with even L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For even j on the other had this quantity is no longer identically 0 as the symmetry of the system is broken due to disorder, and shows a ( √ 2j)2/L2 scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, this does not seriously affect the evolution as their magnitude is much smaller than for odd j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Consequently, for disordered obc systems, I(t) ∼ 1 L � dj 2 Le−a(2j)2/L2 = √π 2L √ at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (23) This is the behaviour seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(c), where disorder, W = 4 , is chosen to be smaller to highlight Region IV (data for W = 10 is similar at larger times, not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, for the periodic case, we have a different sce- nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Previously I(j) had no support for any j in this regime for clean systems, but now, due to breaking of symmetry, I(j) ∼ ( √ 2j)2/L2 as evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The eigenvalues as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(a) follow the same scal- ing law as obc (the degeneracy in the smallest magnitude eigenvalue is resolved as we increase j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, we get, I(t) ∼ 1 L � dj 2j2 L2 e−a(j)2/L2 = √π 2(at)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (24) Indeed this is what we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(d), including the lack of L dependence in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Region V As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' II, Region IV becomes asymp- totic in the thermodynamic limit, when we have a con- tinuum in the eigenspectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For finite size systems though, the spectrum eventually gets resolved and hence for larger t we observe a different behaviour, which we classify as region V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In this region, I(t) shows an expo- nential decay, with the rate governed by the Liouvillian gap λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='41 λ1 is thelargest real part of non-zero Liouvil- lian eigenvalue for even L systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For odd L systems, it is the next largest as the imbalance operator does not have support for the largest non-zero eigenvalue in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 5, we see that − ln I(t) grows linearly with time at large times, with the slope |λ1(L, γ, W)| verifying our expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As before, analytical results for clean systems are well known39, but upon addition of disorder one can only resort to numerics for exact results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Nev- ertheless, the description given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15 is still valid in this time scale, and explains the scaling behaviour of λ1 with L, W and γ as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6(a) we show the change of λ1 with L for con- stant W and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It is known that the 1/L2 scaling of λ1 works for sufficiently large L for clean systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In fact, it has already been shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 39 that the critical length required to observe this scaling is Lc ∼ π √ 2 γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1 this is at L ∼ 45 and is shown by the blue gridline in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This behaviour also continues when we intro- duce disorder in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It is evident from the figure that upon increasing W, Lc decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This can be ex- plained as in the previous section using Eq 15 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 21, which tells us that on addition of disorder of strength W, γ can be effectively replaced by (4γ2 + 2W 2 3 )/(4γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This gives Lc(W) ∼ 4 √ 2πγ 4γ2 + 2W 2 3 , (25) shown as different coloured gridlines in the plot, and cap- ture the transition point between the two regimes quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6(b) we show the change of λ1 on chang- ing W and γ for constant L = 3000 via a contour plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We expect |λ1L2| ∼ a and hence the equation of constant contours can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 22 as ln a = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (26) 10 ● ● ● ● ● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ■ ◆ ▲ 5 10 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='500 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Scaling of λ1, the smallest Liouvillian gap, with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (a) Plot showing 1/L2 scaling of λ1, for different disorder strengths, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lc from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 25 is marked by appropriate coloured gridlines in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Averaging is done over 100 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (b) Plot showing dependence of ln |λ1L2| with W and γ for L = 3000 averaged over 100 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The red dashed lines indicate three constant contours at 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 for ln a obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Three such contours are plotted as red dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6(b) and they match very well with the contours obtained from exact numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' DISCUSSION In this work we have studied the evolution of imbal- ance, I(t) for the disordered XX chain with on-site de- phasing, starting from the N´eel initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Using the hierarchical nature of equations for the observables, we have computed I(t) for large system sizes (L ∼ 103) and long times (t ∼ 103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Our analysis showed the emergence of five timescales in disordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The shortest time scale, t < t0 ∼ Min(1/γ, 1/W) where I(t) ∼ (1 − ηt2) constitutes the linear response regime of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Then, due to the localization via disor- der, two timescales denoted by Region II and III emerge, which are absent in clean systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Region II is the linear regime of what has been called a stretched exponential decay in previous works15,20–23, where I(t) ∼ (1 − β √ t), and is universal irrespective of the nature of disorder cho- sen, and continues till t1 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1W 2/8γ, before smoothly transitioning to a disorder dependent behaviour in region III which continues tillt2 ∼ W 2/(8γJ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We thus con- clude that the stretched exponential fits are neither uni- versal nor asymptotic for the system under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Sim- ple local three-site models describe the behaviour in these two regimes, and a rescaled time emerges during the anal- ysis τ = 8γt/W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' An effective Heisenberg model given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15 also provides the correct description of the evolution from Region II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For t > t2, where Region IV begins, dephasing breaks localization and the system shows size and boundary con- ditions dependent power law decay of I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In fact for obc it shows a 1/(L √ t) decay and for pbc a 1/ √ t3 decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' These were explained from the continuum limit of the low magnitude eigenspectrum of the Liouvillian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally, we discussed that for finite sized systems due to resolution of the eigenspectrum, the decay of I(t) in final Region V at t > t3 is exponential with the rate governed by the relevant Liouvillian gap λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We also provided analysis of how λ1 typically scales with W, L, γ using the effective Heisenberg Hamiltonian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Our analysis gives insight to the mechanism behind the evolution of imbalance in such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' It demys- tifies the behaviour of the system in Region II and III clearly showing its local origin and gives us the ability to make predictions about the evolution for many different disorder distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Additionally, the proper analysis of Liouvillian eigenspectrum provides us with the correct asymptotic description of evolution for different models and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We would also like to briefly mention the nuances of crossover from one regime to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since the tran- sition is smooth, there is a always a finite time taken by the system to go from one regime to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For example, from Region III to IV, where the finite size effect first appears, the system typically takes t ∼ L to transition to a power law for open boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, if the thermodynamic limit is taken first, we would see that even if region III ends at L independent τ2 = 8γt2/W 2 ∼ 1, the power law in Region IV is only reached asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, this does not seem to be the case for periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This is also consistent with the system size dependence of the results in Region IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Additionally, it is worthy to remark that the effec- tive Hamiltonian description12 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15 is also valid for weakly interacting systems where the interaction 11 strength is much smaller than disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This means weak interaction does not play much role in evo- lution in such timescales, so we expect most of our results to hold for weakly interacting systems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Furthermore, while we have discussed the N´eel ini- tial state in this work, decay of I(t) from other generic computational initial states also share some similar fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The behaviour in regions I, IV and V seen for the N´eel state is still observed for other typical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The differences occur in t2 and the behaviour in Regions II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The first difference in other initial states is that I(t = 0) < 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' − ln I(t = 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Since t2 is defined by a finite non-zero value of I universal for a given disorder distribution, relaxation from different I(t = 0) typically reaches this value at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Secondly, all the sites of the initial state are not locally equivalent unlike the N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, the scaling in Regions II and III show a difference due to variation in local behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A detailed analysis of this aspect is beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' There are still a few open questions left in the context of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' One natural extension would be to study the evolution of other observables such as current in such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The question of whether the scaling laws depend on the type of dephasing is also worth studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Indeed, the hierarchical structure of observables or the effective Heisenberg model might break down if we choose a dif- ferent model of dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Additionally, while Region I-III has been studied under different disorder settings, we have provided a general explanation for the power law decay in Region IV, discussing the two cases where it shows different exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Specific potentials may give rise to unique behaviour in this time scale, and an anal- ysis of that is left for a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ACKNOWLEDGEMENT We would like to acknowledge support by Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' J1-1698, J1-4385 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' P1-0402 from the Slove- nian Research Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' RG would also like to acknowl- edge support from UKRI grant EP/R029075/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Anderson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 109, 1492 (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Evers and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Mirlin, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 80, 1355 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ˇZnidariˇc, Journal of Statistical Mechanics: Theory and Experiment 2010, L05002 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Taylor and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Scardicchio, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 103, 184202 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Guo and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Poletti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A 95, 052107 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Vakulchyk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Yusipov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Ivanchenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Flach, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Denisov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 98, 020202 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 7 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Wu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Schnell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Tomasi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Heyl, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Eckardt, New Journal of Physics 21, 063026 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Monthus, Journal of Statistical Mechanics: Theory and Experiment 2017, 043302 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 9 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Yusipov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Laptyeva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Denisov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Ivanchenko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 118, 070402 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Vershinina, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Yusipov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Denisov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Ivanchenko, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Laptyeva, Europhysics Letters 119, 56001 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 11 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Turkeshi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Schir´o, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 104, 144301 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Medvedyeva, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Prosen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ˇZnidariˇc, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 93, 094205 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 13 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Gopalakrishnan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Islam, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Knap, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 119, 046601 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Sciolla, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Poletti, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Kollath, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 114, 170401 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Levi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Heyl, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lesanovsky, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Garrahan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 116, 237203 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 16 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lezama and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lev, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12, 174 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Everest, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lesanovsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Garrahan, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Levi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 95, 024310 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 18 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Wolff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Bernier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Poletti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Sheikhan, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Kol- lath, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 100, 165144 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 19 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Wu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Eckardt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 123, 030602 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Fischer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Maksymenko, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Altman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 116, 160401 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 21 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Cai and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Barthel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 111, 150403 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Ren, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Cai, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 124, 130602 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 23 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Marcantoni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Carollo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Gambetta, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Lesanovsky, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Schneider, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Garrahan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 106, 134211 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 24 Recall that since we start from the N´eel state, a computa- tional state, and hence at t = 0 the system is completely uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Medvedyeva and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Kehrein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B 90, 205410 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 26 Hence numerical fits of the form exp(−tα) in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 20–23 found different αs depending on fit windows as the be- haviour is not a true stretched exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 27 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ˇZnidariˇc and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Horvat, The European Physical Jour- nal B 86, 67 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 28 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Guo and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Poletti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A 98, 052126 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 29 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Eisler, Journal of Statistical Mechanics: Theory and Experiment 2011, P06007 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 30 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Temme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Wolf, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Verstraete, New Journal of Physics 14, 075004 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 31 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Horstmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Cirac, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Giedke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A 87, 012108 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Nava, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Campagnano, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Sodano, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Giuliano, Arxiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='10856 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 33 For Fermionic Hamiltonians, if longer range couplings are present then in the most general case, Tjk = Jjk where Jjk denotes the hopping strength between two different sites at positions j and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12 34 This is difficult to rigorously prove analytically, but can be seen numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 35 This choice of approximation is used to obtain a tractable form for p(∆(+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Other choices are also possible in the said limit, but they do not simplify the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We show a comparison of the probability distribution of the approximated and exact ∆(+) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 36 One can still numerically solve the small size Hamiltonians to obtain the eigenenergies non-perturbatively and con- tinue with the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 37 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 20 arrived at this result using a different approach ef- fectively leading to the same integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, under their approximations one arrives at this result irrespective of the disorder chosen and then needs to fix a free parameter via numerical fitting to fit to different disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 38 Ωj gives an oscillatory term which is usually ∼ 0 for the relevant low magnitude eigenvalues of the Liouvillian spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' If they are included the power law reduces by 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 39 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ˇZnidariˇc, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' E 92, 042143 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 40 The square comes from the weightage of the initial condi- tions, whose weights are of the same order as the element of the eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence we use ∼ instead of =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 41 The exception being clean pbc systems with even L where imbalance has no support on the eigenfunction as shown before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Appendix A: Two site model In this Appendix we shall derive the eigenspectrum ex- pressions for the two site model using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 6 as the starting point of our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For Regime I this simple model is enough to qualitatively explain the features seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For the two site model, f = (C11, C12, C21, C22) and, Q = � � � 0 2i −2i 0 2i 2iδ + 4γ 0 −2i −2i 0 −2iδ + 4γ 2i 0 −2i 2i 0 � � � (A1) where we have taken J = 1 and δ = h1 − h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For δ = 0 (clean system), one can compute the eigenvalues exactly as , � 0, 4γ, 2 � γ − � γ2 − 4 � , 2 �� γ2 − 4 + γ �� , (A2) and the time dependent solutions to the C11(22) can be written as, C11(22) = e2γt[(−) cosh � 2t � γ2 − 4 � −(+) γ sinh � 2t � γ2 − 4 � � γ2 − 4 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (A3) Hence, I(t) = 1 2(C11(t) − C22(t)) ∼ 1 − 8t2 + O(t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (A4) This is the demonstration of the short time quadratic behaviour when t < 1/ � γ2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Comparison of distribution of p(∆(+)), between ∆(+) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B6 and approximated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However when δ > 0 the exact solution is given from a cubic equation since one of the eigenvalues of Q is always 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The expressions are complicated and hence not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' But we can formulate a simple perturbative result for δ ≫ 1, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To do so we first rearrange Q to separate the degenerate block and non-degenerate blocks as, Q = � � � 0 0 2i −2i 0 0 −2i 2i 2i −2i 4γ + 2iδ 0 −2i 2i 0 4γ − 2iδ � � � = � O2×2 B2×2 C2×2 D2×2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (A5) Then applying second order degenerate perturbation the- ory to O and second order non degenerate perturbation to D we have the eigenvalues as, � 0, 16γ 4γ2 + δ2 , 4γ + 2iδ + 8 4γ + 2iδ , 4γ − 2iδ − 8 −4γ + 2iδ � (A6) and the solution for δ ≫ γ is given by I(t) = −4 cos � 2 √ δ2 + 4t � + δ2 δ2 + 4 = 1−8t2 +O � t4� (A7) whence the quadratic decay with t remains but it is valid till t ∼ 1/ √ δ2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Appendix B: Three site model Now we repeat the above calculation for a three site model which describes the behaviour in Regime II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In this case f is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 7, while Q is given by, � � � � � � � � � 0 2i −2i 0 0 0 0 2i 4γ + 2iδ1 0 −2i 0 0 0 −2i 0 4γ − 2iδ1 2i 0 0 0 0 −2i 2i 0 2iτ −2i 0 0 0 0 2i 4γ + 2iδ2 0 −2i 0 0 0 −2i 0 4γ − 2iδ2 2i 0 0 0 0 −2i 2i 0 � � � � � � � � � (B1) 13 where, h1 − h2 = δ1 and h2 − h3 = δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Rearranging in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' A5, we get the general form in this case as, Q = � O3×3 B3×4 C4×3 D4×4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (B2) Then using second order degenerate perturbation theory in the subspace of O, we can compute the eigenvalues as ∆(0) = 0 and ∆± = 8γ � 8γ2 ± � 16γ4 + 4γ2 (δ2 1 + δ2 2) + δ4 1 − δ2 1δ2 2 + δ4 2 + δ2 1 + δ2 2 � (4γ2 + δ2 1) (4γ2 + δ2 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (B3) Since we are in the regime where W ≫ γ, we have δ1, δ2 ≫ γ, and hence we can approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B3 as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' One can also compute the eigenvectors in this subspace and plug in the initial N´eel state to find the coefficients, d(0) = −1 3 d(+) = (2δ2 1 − δ2 2 + κ)(δ2 2 + κ) 3δ2 1κ d(−) = (2δ2 1 − δ2 2 − κ)(−δ2 2 + κ) 3δ2 1κ (B4) where κ = � δ4 1 − δ2 1δ2 2 + δ4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We can see that when |δ1| = |δ2|, d(+) = 4 3, d(−) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In the opposite regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' when |δ2 1 −δ2 2| ≫ 0, d(+) → 1 and d(−) → −d(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Consequently, it can be shown 1 ≤ d(+) ≤ 4/3, 0 ≤ d(−) ≤ 1/3 and hence d(+) always provides the largest contribution to the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We use the result for |δ1| = |δ2| when we compute I(t) for alternating potential plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(b) as, I(t) = −1 3 + 4 3e−∆(+)t ∼ e−4/3∆(+)t = e−τ, (B5) where we have used ∆(+) ∼ 6γ/W 2 for W ≫ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' One surprising aspect is that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B5 is a good fit to exact numerics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(b) beyond τ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This can be ex- plained by observing that if we take local models of higher sizes, then we will find numerically that d(0) decreases and the weight gets shifted to a mode which evolves with ∼ 4/3∆(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' B5 remains valid even when the transport occurs in a more non-local region (but not in the full lattice) and hence to longer times, before the system shows the asymptotic power law behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Also notice that because there is only one mode of relaxation, the decay is exponential in this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' For the random disorder case we will have a distribu- tion of d and ∆(±) based on the distribution of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' However, the principal mode of decay is via ∆(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' To the leading order, the effect of d0 on this mode becomes progressively smaller as we move away from the limiting alternating potential case |δ1| = |δ2|, as it is countered by the d(−) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Furthermore ∆(−) is typically too small to have a significant effect on evolution in Regimes II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally, the shifting of weight towards a mode ∼ ∆(+) when we take slightly bigger but still local mod- els, is valid for the disordered case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence, simply averaging over ∆(+) modes is usually enough to get ac- curate results as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The final point we need to address is the validity of the approximation we have used to arrive at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 11 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Let us recall that ∆(+) ∼ 8γ(δ2 1 + δ2 2 + � δ4 1 + δ4 2 − δ2 1δ2 2) δ2 1δ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' (B6) For very large W, we can expect terms with |δ1| ∼ |δ2| would be statistically insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence to get a tractable expression, we take the approximation |δ2 1 − δ2 2| ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This would constitute the long tails of the distribution of ∆(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Finally, to be able to perform the analytical computation simply, we choose to approximate ∆(+) by, ∆(+) ∼ 16γ(δ2 1 + δ2 2) δ2 1δ2 2 (B7) and show its agreement with full ∆(+) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The approximation qualitatively represents the exact curve reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Also, since it becomes exact in the tail, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' for large ∆(+), consequently the expression computed using these result represent the exact result for I(t) for smaller t more accurately than the rest, a feature seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The exact stretched exponential result which arises out of this approximation for Gaussian disorders is also valid only for small t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Appendix C: Agreement of exact numerics with evolution by Heff In the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' IV B we mentioned that we could use an effective Hamiltonian to replicate the behaviour of I(t) from a bit later than τ = τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Let us discuss this aspect in a bit of detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='14 ● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ■ ■ ■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='100 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='50 1 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Comparison between exact numerical results for evo- lution of I(t) with those obtained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' C1 averaged over 20 realizations for L = 24 and W = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 12, at a sufficiently large timescale the density matrix becomes effectively diagonal and the evo- lution of the diagonal elements ρi is given by, dρi(t) dt = � j [Heff]ijρj(t) (C1) where Heff is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' I(t) can then be computed from ρi(t) as I(t) = 2L � i ρi(t) L � j ⟨αi| (−1)jσz j |αi⟩ (C2) where |αi⟩ can be taken as the computational basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We plot the comparison of results obtained from exact numerics using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 11 and evolution of the density ma- trix using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' C1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' C2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We see that from τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='01 the two results are almost equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This effective description reproduces the low energy eigenspectrum of the Liouvillian quite accurately and hence correctly ap- proximates the evolution from τ ≫ τ0, thus is effective in capturing the evolution of I(t) from approximately Re- gion II, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Appendix D: Computation of low magnitude eigenspectrum of Liouvillian Following the arguments of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 39, we have extracted the low magnitude eigenspectrum of the system by diag- onalizing a matrix of size ∼ L × L instead of ∼ L2 × L2 one-particle Liouvillian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' If we denote the computational basis as |j⟩, then the low magnitude eigenspectrum of the one-particle Liouvillian can be approximated by eigen- spectrum the following matrix � � 0 RT 0 R −4Iγ iIX 0 −iIX −4Iγ � � (D1) ● ● ● ●● ●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■ ■■ ■■■■■■■ ■ 1 2 5 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='500 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Comparison of exactly computed Ij via solving L×L linear equations for the initial conditions with our approxima- tion using the squares of the eigenvector elements for a system with L = 80, W = 30, γ = 1 and periodic boundary condi- tions, averaged over 103 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' The dashed line is the approximate scaling we have used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' written in the basis |j⟩⟨j|, |j⟩⟨j + 1| + |j + 1⟩⟨j| and |j⟩⟨j + 1| − |j + 1⟩⟨j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Here γ is the dephasing, Rjk = −2i √ 2(δj,k − δk,j+1) andXjk = (hj − hj+1)δjk, hi being the on site disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Note that unlike the clean Hamiltonian case treated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 39, |j⟩⟨j +1|+|j +1⟩⟨j| are not eigenvectors of L as disorder breaks transla- tional invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Thus we need to effectively solve a 3L−2×3L−2 [3L×3L] problem for obc [pbc], to obtain the eigenspectrum, which is the ‘tridiagonal’ approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' This allows us to compute the eigenspectrum for large systems and results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Furthermore, for small system sizes one can numeri- cally show that the δf(j)s are proportional to the cor- responding eigenvector elements and that approximating I(j) with the square of corresponding eigenvector terms is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 9 we show the comparison of our ap- proximation with exact I(j) found by numerically com- puting the correct weights due to the initial N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' We do exact diagonalization for a system size of L = 80 at large W = 30, averaged over 103 realizations under periodic boundary conditions to obtain the exact data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' As one can see the qualitative nature of both the plots are the same, they vary by a factor of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content='5 (computed numerically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' Hence in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} +page_content=' 4(b) our approx- imation was able to qualitatively replicate the behaviour of I(t) with insignificant deviation from exact numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfdg8f/content/2301.05611v1.pdf'} diff --git a/_NFLT4oBgHgl3EQfvy-O/content/tmp_files/2301.12161v1.pdf.txt b/_NFLT4oBgHgl3EQfvy-O/content/tmp_files/2301.12161v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19957d92cef8c7af2567e2907939c8edb3960932 --- /dev/null +++ b/_NFLT4oBgHgl3EQfvy-O/content/tmp_files/2301.12161v1.pdf.txt @@ -0,0 +1,842 @@ +Knowledge-Aware Semantic Communication +System Design +Sachin Kadam and Dong In Kim +Department of Electrical and Computer Engineering +Sungkyunkwan University (SKKU), Suwon 16419, Republic of Korea +Email: sachinkadam@skku.edu, dikim@skku.ac.kr +Abstract—The recent emergence of 6G raises the challenge +of increasing the transmission data rate even further in order +to break the barrier set by the Shannon limit. Traditional +communication methods fall short of the 6G goals, paving the way +for Semantic Communication (SemCom) systems. These systems +find applications in wide range of fields such as economics, +metaverse, autonomous transportation systems, healthcare, smart +factories, etc. In SemCom systems, only the relevant information +from the data, known as semantic data, is extracted to eliminate +unwanted overheads in the raw data and then transmitted after +encoding. In this paper, we first use the shared knowledge base to +extract the keywords from the dataset. Then, we design an auto- +encoder and auto-decoder that only transmit these keywords and, +respectively, recover the data using the received keywords and the +shared knowledge. We show analytically that the overall semantic +distortion function has an upper bound, which is shown in the +literature to converge. We numerically compute the accuracy +of the reconstructed sentences at the receiver. Using simulations, +we show that the proposed methods outperform a state-of-the-art +method in terms of the average number of words per sentence. +Index Terms—Semantic Communications, Knowledge Base, +6G, Data Compression, Wireless Communications +I. INTRODUCTION +As per the prediction in [1], semantic communication (Sem- +Com) technology is identified as one of the key ingredients +in 6G due to the requirement of low latency and high data +rate transmissions. The recent emergence of SemCom tech- +nologies finds applications in wide range of fields such as +economics [2], metaverse [3], autonomous transportation sys- +tems [4], smart factories [5], and so on. In SemCom, we only +transmit useful and necessary information to the recipients. +The semantic extraction (SE) is a process wherein the useful +and necessary features are extracted from the original raw data. +For example, the essential speech features are extracted using +an attention-based mechanism in [6]–[8]. +During critical applications such as military operations, +search operations by forest personnel in a dense forest, medical +emergencies in remote areas, fire incidents in a remote agri- +cultural land, the release of water from a nearby dam, etc., +only the essential information needs to be communicated on +an urgent basis. The messages could be in the form of text or +audio and they come from a limited dataset. In a non-critical +application, such as broadcasting a text/audio summary of +This research was supported in part by the Korean Government (MSIT) un- +der the ICT Creative Consilience program (IITP-2020-0-01821) supervised by +the IITP (Institute for Information & Communications Technology Planning +& Evaluation). +commentary provided by live football commentators. Among +all the words spoken by them, only a limited set of useful +or important words are relevant to the game. These words +are drawn from a limited dataset such as football vocabu- +lary [9] which includes words such as goal, player names, +red card, football, score, assist, half-time, etc. This limited +dataset provides an opportunity, in the context of SemCom +design, for a significant overhead reduction by extracting and +processing only the relevant keywords. For example, an uttered +commentary sentence is: ‘Ronaldo shoots the ball into the +right-bottom of the net and it’s a goal!’ The extracted keywords +in this example are Ronaldo, shoots, ball, right-bottom, net, +goal. Only these keywords are transmitted in place of the entire +sentence, and the receiver reconstructs a meaningful sentence. +The reconstructed sentence in this case is: ‘Ronaldo shoots +the ball into the right-bottom of the net to score a goal.’ This +sentence is not exactly the same as the original sentence, but +it conveys the same meaning. +The main goal of this paper is to use SemCom technology to +reduce communication overhead, in the context of natural lan- +guage processing (NLP) problems, while maintaining a certain +minimum accuracy in wireless communication systems. The +overhead reduction is performed with high accuracy in the +literature [10], [11]. However, in some applications, high data +rates are preferred over high accuracy. As a result, we present +the results of the trade-off between overhead reduction and +accuracy. Model parameters are chosen based on the context. +Instead of transmitting raw data, the transmitter is designed to +transmit semantic data, which significantly reduces network +data traffic. A knowledge base (KB) is a technology that +collects, stores, and manages data. A knowledge graph (KG) +is a KB that integrates data using a graph-structured topology. +They are used to store interconnected event descriptions. These +are used to predict the missing words in the received data +(keywords) to construct a meaningful sentence. +The organization of the paper is as follows: A brief literature +review on SemCom technologies is provided in Section II. +We introduce our proposed system model in Section III and +provide a few useful simulation results in Section IV. Finally, +we conclude the paper in Section V. +II. RELATED WORK +The following state-of-the-art survey papers provide in- +depth discussions on various SemCom technologies and their +arXiv:2301.12161v1 [cs.NI] 28 Jan 2023 + +applications [12]–[14]. Deep learning based SemCom tech- +nologies are proposed in +[10], [11]. A brief tutorial on +the framework of SemCom and a method to calculate a +bound on semantic data compression is provided in [15]. The +SemCom technology wherein both transmitter and receiver +are empowered with the capability of contextual reasoning is +proposed in [16]. The SemCom technology for a system where +transmitter and receiver speak different languages is designed +in [17]. In [18], a SemCom framework for textual data trans- +mission is proposed. In this framework, semantic information +is represented by a KG made up of a set of semantic triples +and the receiver recovers the original text using a graph-to-text +generation model. All of these works focused on achieving an +overhead reduction without compromising the accuracy of the +received data. None of these works investigated the possibility +of further overhead reduction, thereby improving transmission +data rates, while sacrificing a little accuracy. This issue is +addressed in this paper using a shared knowledge base. +A significant research on the usage of KBs and KGs is +carried out in the field of natural language processing (NLP). +A survey paper based on KG is presented in [19]. Similarly, +another survey paper on KB text generation is presented +in [20]. A method to generate a summary of sentences by +using a given set of keywords is proposed in [21]. Similarly, +a method to generate a summary of sentences by using a +knowledge base is shown in [22]. Recently, KGs are utlized +in the context of SemCom design [18], [23], [24]. But these +works do not focus on the issue presented in this paper, which +is to design a SemCom system with a significant overhead +reduction with a little compromise on accuracy. +III. SYSTEM MODEL +The system model of the proposed SemCom system is +shown in Fig. 1. Let X be the input text dataset with N +sentences, Xi be the ith, i ∈ {1, . . . , N}, sentence of X, and +K be the shared knowledge base (KB). First, we extract the +keywords from X using K. Let the total set of keywords be +Ω = �N +i=1 Ωi, where Ωi denotes the set of keywords present +in Xi. The keyword extraction process at every sentence +Xi, i ∈ {1, . . . , N}, is executed by multiplying it with a binary +vector bi = [bi(ℓ), ℓ = {1, . . . , |Xi|}],1 which is defined as +follows: +bi(ℓ) ≜ +� +1, +if ℓth word of Xi, Xi(ℓ), is a keyword in K +0, +else. +(1) +Hence, Ωi, i ∈ {1, . . . , N}, is obtained by collecting the +non-zero elements from Xi ⊙ bi, where ⊙ is a word-wise +multiplication operator. Here Xi ⊙ bi ≜ [Xi(ℓ)bi(ℓ), ∀ℓ = +{1, . . . , |Xi|}].2 +1|A| denotes the cardinality of set A. +2For ease of understanding, let us consider the example discussed in +Section I. Let Xi be ‘Ronaldo shoots the ball into the right-bottom of the net +and it’s a goal!’. If the set of keywords present in Xi is {Ronaldo, shoots, +ball, right-bottom, net, goal} then bi = [11010010010001]. Now, Xi ⊙ bi +gives +[Ronaldo, shoots, 0, ball, 0, 0, right-bottom, 0, 0, net, 0, 0, 0, goal]. +Next, +Ωi +is +obtained +by +collecting +the +non-zero +elements, +i.e., +Ωi = {Ronaldo, shoots, ball, right-bottom, net, goal}. +Now, let us define the quantity BLEU score (bilingual eval- +uation understudy [25]) to compare the similarities between +two sentences quantitatively. The BLEU(s, ˆs) ∈ [0, 1] score +between transmitted sentence s and reconstructed sentence ˆs +is computed as follows: +BLEU(s, ˆs) = BP(s, ˆs) exp +� W +� +n=1 +wn ln pn(s, ˆs) +� +, +(2) +where pn denotes the modified n-gram precision function up +to length W, wn denotes the weights, and brevity penalty (BP) +is given by the following expression: +BP(s, ˆs) = +� +1 +ℓc > ℓr +e1−ℓr/ℓc +ℓc ≤ ℓr, +(3) +where ℓc is the length of the candidate translation and ℓr is +the effective reference corpus length [25]. +Let ξ be a function which generates a set of M (say) +sentences from a given set of keywords with the help of a +given knowledge base. Using the keywords in Ωi, bi, i ∈ +{1, . . . , N}, and the knowledge K, for a given sentence Xi, +the sentence generator at the transmitter generates a set of M +sentences using the function ξλ, where λ is a parameter. Let +that set of sentences be � +Xij, j = {1, . . . , M}. So, +� +Xij = ξλ(Ωi, K), j = {1, . . . , M}. +(4) +Next, out of these M sentences we choose the most seman- +tically equivalent sentence based on the BLEU scores [25] +compared with input sentence Xi, i.e., +� +Xi = +arg max +� +Xij,j=1,...,M +BLEU( � +Xij; Xi), i ∈ {1, . . . , N}. +(5) +Note that the set of sentences � +X = { � +Xi, i ∈ {1, . . . , N}}, is +generated at the transmitter during the training process only +and it is shared with the receiver a priori. Next, ith keyword +set Ωi is encoded using the auto-encoder which consists of +semantic and channel encoders. The auto-encoder uses the +binary vector bi, i ∈ {1, . . . , N}, to assign a common symbol +to all non-keywords of Xi and a unique symbol to keywords +of Xi, respectively. This enables the receiver to construct +appropriate sentences from the received symbol sets. Let us +denote Sθe and Cφe as the semantic and channel encoders with +θe and φe as the parameters sets, respectively. After encoding +Ωi, we get the following set of symbols: +�Ωi = Cφe(Sθe(Ωi)), i ∈ {1, . . . , N}. +(6) +The encoded set of symbols �Ωi is transmitted via the AWGN +(additive white Gaussion noise) channel. Let h be the channel +gain and η be the noise which gets added to �Ωi during +transmission. So, the set of received symbols at the receiver +is Ωi = h�Ωi + η. After receiving, this set of symbols is +decoded using the auto-decoder which consists of channel and +semantic decoders. Let us denote Cφd and Sθd as the channel +and semantic decoders with φd and θd as the parameters sets, + +Auto-Decoder +Auto-Encoder +Semantic +Encoder +Channel +Encoder +Channel +Decoder +Channel Noise +Semantic +Decoder +Destination +Sentence +Generator +Source +Shared Knowledge +(K) +Context, +keywords, +events, etc. +Input +Text +Output +Text +Keyword +Extraction +𝑋𝑖 +Ω𝑖 +෩Ω𝑖 +ഥΩ𝑖 +෡Ω𝑖 +෠𝑌𝑖 +෠𝑌𝑖𝑗 +෠𝑋𝑖 +Sentence +Generator +෠𝑋𝑖𝑗 +argmax +෠𝑋𝑖𝑗,𝑗=1,…,𝑀 +BLEU( ෠𝑋𝑖𝑗; 𝑋𝑖) +argmax +෠𝑌𝑖𝑗,𝑗=1,…,𝑀 +BLEU(෠𝑌𝑖𝑗; ෠𝑋𝑖) +𝑋𝑖 +Auto-Decoder +Auto-Encoder +Semantic +Encoder +Channel +Encoder +Channel +Decoder +Channel Noise +Semantic +Decoder +Sentence +Generator +Source +Input +Text +Output +Text +Keyword +Extraction +෩Ω𝑖 +ഥΩ𝑖 +෡Ω𝑖 +෠𝑌𝑖 +Shared Knowledge +Context, +keywords, +events, etc. +(a) +(b) +Destination +𝑖 ∈ {1, … , 𝑁} +𝑋𝑖 +𝑖 ∈ {1, … , 𝑁} +Ω𝑖 +Fig. 1: The block diagram of our proposed SemCom system model. The model in Fig. (a) is used for training the system parameters and +the model in Fig. (b) is used for evaluating the system model. +respectively. After decoding Ωi, we get the following set of +keywords: +�Ωi = Sθd(Cφd(Ωi)), i ∈ {1, . . . , N}. +(7) +From the decoded set of keywords and the shared knowl- +edge K, the sentence generator at the receiver generates a set +of M sentences using the function ξµ, where µ is a parameter, +and let that set of sentences be �Yij, j = {1, . . . , M}, i.e., +�Yij = ξµ(�Ωi, K), j = {1, . . . , M}. +(8) +To select the most desired sentence among these sentences, we +compute the BLEU scores between �Yij, j = {1, . . . , M} and +the sentence at the transmitter � +Xi, and choose the one which +maximizes the BLEU score. That is: +�Yi = +arg max +�Yij,j=1,...,M +BLEU(�Yij; � +Xi), i = {1, . . . , N}, +(9) +where �Y = {�Yi, i ∈ {1, . . . , N}} denotes the desired set of +sentences generated at the receiver. +A. Training the System Model +Let X be the set of all possible sentences and pX(x), pλ(�x), +and pµ(�y) denote the probability distributions of sentences +X ∈ X, �X ∈ � +X, and �Y ∈ �Y , respectively, for all x, �x, �y ∈ X.3 +The sentences �X and �Y are generated by the sentence genera- +tors in transmitter and receiver, respectively, parameterized by +λ and µ, respectively, with the help of shared knowledge K +3Note that �x, �y ∈ X ⊆ X. To obtain the closed-form expressions for +certain quantities, we relax the condition and assume X = X. +(see Fig. 1(a)). Note that, since �X and �Y are generated with the +help of knowledge K, throughout the paper, the conditional +distributions pλ(·|·) and pµ(·|·) are conditioned on the event +K = k, ∀k ∈ K. +Let H(X) and DKL(p||q) represent, respectively, the en- +tropy of a random variable X whose probability distribution +is p and the Kullback Leibler (KL) divergence between the +probability distributions p and q. These quantities are defined +as follows [26]: +H(X) ≜ − +� +x∈X +p(x) log p(x), +(10) +DKL(p||q) ≜ +� +x∈X +p(x) log p(x) +q(x). +(11) +The overall cross entropy (CE) loss measures the difference +between the actual probability distribution at the input and +the estimated probability distribution at the output and it can +be minimized using the stochastic gradient descent (SGD) +methods [27]. So the overall cross entropy (CE) loss is defined +as follows [28]: +LCE(µ) ≜ H(X) + DKL(pX(x)||pµ(�y|k)). +(12) +By using the expressions of (10) and (11), we simplify the +expression for overall CE loss as follows: +LCE(µ) = − +� +x∈X +pX(x) log pµ(�y|k). +(13) +From Fig. 1, we observe that there are information losses +at auto-encoder, auto-decoder, and sentence generation blocks +both in transmitter and receiver. We aim to minimize the + +summation of all these losses. The characterization of these +losses are as follows. +• The loss of information between sentences X and �X at +the transmitter is measured using the CE loss, i.e., +LCE +1 +(λ) = H(X) + DKL(pX(x)||pλ(�x|k)) +(14) += − +� +x∈X +pX(x) log pλ(�x|k). +(15) +• Similarly, the loss of information between sentences �X +and �Y at the receiver is also measured using the CE loss, +i.e., +LCE +2 +(µ, λ) = H(�X|K) + DKL(pλ(�x|k)||pµ(�y|k)) (16) += − +� +�x∈X +pλ(�x|k) log pµ(�y|k). +(17) +• Lastly, the loss of information in the channel is measured +in terms of mutual information (MI) between transmitted +symbols and received symbols, i.e., +LMI +3 +(θe, φe, θd, φd) = I(�Ω; Ω). +(18) +Now, we define the overall semantic distortion function as +follows:4 +L(θe, φe, θd, φd, λ, µ) ≜ LCE +1 +(·) + LCE +2 +(·) − γLMI +3 +(·), (19) +where γ ≥ 0 is a hyper-parameter. In Theorem 1, we show +that the overall semantic distortion L(·) attains an upper bound +which can be optimized. We aim to compute the optimal +parameters of auto-encoder, auto-decoder, and sentence gener- +ation blocks. These blocks are characterized by the parameters +θe, φe, θd, φd, λ, µ and are obtained by training the system +model, as shown in Fig. 1(a), by minimizing the overall loss +function defined in (19). +Theorem 1. The overall semantic distortion function attains +the following upper-bound: +L(θe, φe, θd, φd, λ, µ) ≤ LCE(·) − γLMI +3 +(·) = B. +(20) +Proof. The proof is given in Appendix. +Remark 1. The upper-bound B, provided in Theorem 1, is +proved to be optimized using the SGD algorithms [27]. Also, +in the published works [10], [17], the semantic distortion of +the overall system is minimized using a similar expression as +that of B. +Hence, to minimize the overall semantic distortion defined +in (19), we seek to minimize the upper-bound B provided in +Theorem 1. The loss due to mutual information LMI +3 +(·) can +be estimated using state-of-the-art mutual information neural +estimator (MINE) [29]. +4We use (·) in place of parameters, wherever convenient, for ease of +representation. +B. Accuracy versus Overhead Reduction Trade-off +Given the limited size of knowledge base, though the +accuracy of the reconstructed sentences in �Y may not be +sufficiently high, the useful content in those sentences is +summarized and conveyed to the receiver. This novel approach +saves a significant amount of overhead. +There exists a trade-off between overhead reduction and +the accuracy that depends on the size of the knowledge base +K. For example, if the set K is small, only a few keywords +are extracted, encoded, and transmitted from the given input +sentences in X, implying a higher amount of average overhead +reduction. On average, this results in a large amount of missing +information, so the accuracy of the reconstructed sentences in +�Y is expected to be low. On the other hand, if the set K is +large, a significant number of keywords are extracted from +the given sentences in X, encoded, and transmitted, implying +a lower average overhead reduction. On average, this results +in a small amount of missing information, so the accuracy of +the reconstructed sentences in �Y is expected to be high. This +phenomenon is numerically shown in Section IV. +So, we aim at minimizing the transmission of average +number of words per sentence (equivalent to maximizing the +average overhead reduction) by keeping a certain minimum +accuracy information τ in the received sentence, i.e., +min 1 +N +N +� +i=1 +|Ωi| +(21) +BLEU(�Yi; Xi) ≥ τ, i = {1, . . . , N}, +(22) +where |Ωi| denotes the number of keywords in Ωi that corre- +sponds to sentence Xi. +C. Shared Knowledge Base +We generate the shared knowledge base K by using the +keywords from a limited dataset Ω which consists of only the +relevant words of a particular event, like that of a football +game in our case. We assume that both the transmitter and +receiver have access to K. It is shown in [30] that the +capacity of the channel can be increased beyond Shannon’s +limit by using a semantic encoder with low semantic ambiguity +and a semantic decoder with strong inference ability and a +large shared knowledge base. From Section III, recall that in +every sentence, only the words w ∈ Ω are uniquely encoded +and transmitted to the receiver in their corresponding time +slots. At other time slots, a common symbol is transmitted. +By utilizing K, the receiver reconstructs the sentence based +on the received symbols. To improve the accuracy of the +reconstructed sentences, we can increase the size of K by +adding more keywords from the vocabulary generated using +X. This result is shown using simulations in Section IV. +IV. SIMULATION RESULTS +First, we evaluate the performance of the text data trans- +mission in terms of accuracy using BLEU score [25].5 In +our work, we use the dataset provided in [31]. We parse the +5We defined the BLEU score in Section III. + +TABLE I: Simulation parameters +Number of matches used in training +1580 +Number of matches used in evaluation +340 +Number of epochs during training +10 +SNR +6 dB +Learning rate +0.001 +Batch Size +64 +Channel +AWGN +football commentary data of 1920 matches from the website +goal.com. The considered football matches are from Union of +European Football Associations (UEFA) Champions League, +UEFA Europa League, and Premier League between 2016 and +2020. The simulation parameters used for plots in this section +are shown in Table I. The simulations are performed in a +computer with NVIDIA GeForce RTX 3090 GPU and Intel +Core i9-10980XE CPU with 256GB RAM. +Let ρ be the fraction of the total vocabulary V , which +contains all the dataset words, to be added to K. ρ = 0 +indicates that no additional vocabulary is added and the system +is evaluated only with the initial keyword set Ω0. Based on the +way of adding the vocabulary words to Ω0, we propose two +types of schemes. In the first type, ρ|V | vocabulary words are +uniformly chosen at random from V and added to K. In the +second type, the words in V are first arranged in the decreasing +order of the frequency of appearances in the dataset, and then +the first ρ|V | vocabulary words are added to K. We call these +schemes as ‘RANDOM’ and ‘ORDERED’, respectively. +The accuracy performances of both the schemes and a deep +learning based SemCom system method named DeepSC [10], +in terms of BLEU score vs. ρ, are shown in Fig. 2. From the +plot we can infer that even with ρ = 0, the initial keyword +set can produce a BLEU score of 0.55 (for 1-gram). This +shows that the context-related keywords produce good results. +Also, we see that as we add more vocabulary words to Ω0, +the BLEU score increases. For the same value of ρ and n, +the ORDERED scheme performs better than the RANDOM +scheme because of the addition of high frequency words. +And, in terms of different n-grams, BLEU score decreases +as n increases, which is an expected result. In comparison to +the DeepSC scheme, the proposed schemes perform poorly +in terms of accuracy but outperform it in terms of overhead +reduction, as shown below. +Next, we evaluate the performance of the proposed schemes, +in terms of the transmission of average number of words per +sentence, with respect to DeepSC [10] and the results are +shown in Fig. 3a. Let W denote the average number of words +per sentence. From the plot we observe that both the schemes +outperform DeepSC. Among the proposed schemes, for a +given ρ the RANDOM scheme outperforms the ORDERED +scheme. This is because, in the ORDERED scheme high +frequency words are added which increases the number of +words to be encoded in the input data as compared with the +RANDOM scheme. +Now, we solve the optimization problem presented in (21) +and (22) using both the proposed schemes. For this purpose, +we evaluate W vs. τ and the results are shown in Fig. 3b. From +the plot we observe that both the schemes outperform DeepSC. +Fig. 2: This plot shows the BLEU score vs. ρ for different values of +n-grams, where n = {1, 2, 3, 4}, for the proposed schemes and the +DeepSC scheme [10]. +(a) +(b) +Fig. 3: These plots show the average number of words per sentence +vs. ρ in the left plot and vs. τ in the right plot, respectively, for the +proposed schemes and the DeepSC scheme [10]. +Also, we see that the performance of both the schemes is same +for a given accuracy threshold τ. This is because, as shown in +Fig. 2, for a given value of ρ ∈ (0, 1), the ORDERED scheme +outperforms the RANDOM scheme in terms of accuracy, +whereas in Fig. 3a, the RANDOM scheme outperforms the +ORDERED scheme in terms of overhead reduction. Hence, +we can choose any one of the proposed methods to solve the +optimization problem. +V. CONCLUSIONS +In this paper, we first extracted relevant keywords from +the dataset using the shared knowledge base. Then, using the +received keywords and the shared knowledge, we designed +an auto-encoder and auto-decoder that only transmit these +keywords and, respectively, recover the data. We proved that +the overall semantic distortion function has an upper bound, +which is shown to be optimized using the SGD algorithms in +the literature. We computed the accuracy of the reconstructed +sentences at the receiver quantitatively. We demonstrated +through simulations that the proposed methods outperform a +state-of-the-art method in terms of average number of words +per sentence. Furthermore, the proposed approach makes no +new hardware modifications to the existing infrastructure. We +focused solely on the text dataset; however, similar approaches +can be used in the future for other types of datasets such as +image, audio, and video. + +0.8 +BLEU Score +0.6 +0.4 +1-gram(RANDOM) +3-gram(ORDERED) +0.2 +2-gram (RANDOM) +4-gram(ORDERED) +3-gram(RANDOM) +1-gram (DeepSC) +4-gram(RANDOM) +2-gram(DeepSC) +1-gram(ORDERED) +3-gram(DeepSC) +2-gram(ORDERED +4-gram(DeepSC) +0 +0 +0.2 +0.4 +0.6 +0.8 +120 +15 +M +10 +RANDOM +-ORDERED +-DeepsC +5 +0 +0.2 +0.4 +0.6 +0.8 +1 +p20 +RANDOM +15 +ORDERED +M +DeepsC +10 +5 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +TAPPENDIX +Now we provide the proof of Theorem 1. Let us define the +following term, parameterized by λ, µ, and k ∈ K: +δ(λ, µ, k) ≜ log +�pµ(�y|k) +pλ(�x|k) +� +, ∀�x, �y ∈ X. +(23) +From (19), we know that +L(·) = LCE +1 +(·) + LCE +2 +(·) − γLMI +3 +(·) +(24a) += − +� +x∈X +pX(x) log pλ(�x|k) +− +� +�x∈X +pλ(�x|k) log pµ(�y|k) − γI(�Ω; Ω) +(24b) += − +� +x∈X +pX(x) log +� +pλ(�x|k)pµ(�y|k) +pµ(�y|k) +� +− +� +�x∈X +pλ(�x|k) log +� +pµ(�y|k)pλ(�x|k) +pλ(�x|k) +� +− γI(�Ω; Ω) +(24c) += − +� +x∈X +pX(x) log pµ(�y|k) + δ(λ, µ, k) +− +� +�x∈X +pλ(�x|k) log pλ(�x|k) − δ(λ, µ, k) − γI(�Ω; Ω) +(24d) += LCE(·) + H(�X|K) − γI(�Ω; Ω) +(24e) +≤ LCE(·) − γI(�Ω; Ω). +(24f) +In (24b), we expand the loss function expressions using +their respective definitions provided in (15), (17), and (18), +respectively. In (24c), we multiply and divide pµ(�y|k) and +pλ(�x|k) in the first and second terms, respectively. Using (23) +and algebraic simplifications, we get (24d). By using (12) +and the definition of entropy, we write (24e). And, finally the +inequality in (24f) is due to H(�X|K) ≥ 0 [26]. +REFERENCES +[1] N. Rajatheva, I. Atzeni, E. Bj¨ornson, A. Bourdoux, S. Buzzi, J.-B. +Dor´e, S. Erkucuk, M. Fuentes, K. Guan, Y. Hu, et al., “White paper +on broadband connectivity in 6G,” 6G Research Visions, vol. 10, 2020. +[2] Z. Q. Liew, H. Du, W. Y. B. Lim, Z. Xiong, D. Niyato, C. Miao, and +D. I. Kim, “Economics of Semantic Communication System: An Auction +Approach,” arXiv preprint arXiv:2208.05040, 2022. +[3] L. Ismail, D. Niyato, S. Sun, D. I. Kim, M. Erol-Kantarci, and C. Miao, +“Semantic Information Market For The Metaverse: An Auction Based +Approach,” arXiv preprint arXiv:2204.04878, 2022. +[4] W. Yang, Z. Q. Liew, W. Y. B. Lim, Z. Xiong, D. Niyato, X. Chi, X. Cao, +and K. B. Letaief, “Semantic communication meets edge intelligence,” +arXiv preprint arXiv:2202.06471, 2022. +[5] X. Luo, H.-H. Chen, and Q. Guo, “Semantic communications: Overview, +open issues, and future research directions,” IEEE Wireless Communi- +cations, 2022. +[6] Z. Weng and Z. Qin, “Semantic communication systems for speech +transmission,” IEEE Journal on Selected Areas in Communications, +vol. 39, no. 8, pp. 2434–2444, 2021. +[7] Z. Weng, Z. Qin, and G. Y. Li, “Semantic communications for speech +signals,” in ICC 2021-IEEE International Conference on Communica- +tions, pp. 1–6, IEEE, 2021. +[8] H. Tong, Z. Yang, S. Wang, Y. Hu, W. Saad, and C. Yin, “Federated +learning based audio semantic communication over wireless networks,” +in 2021 IEEE Global Communications Conference (GLOBECOM), +pp. 1–6, IEEE, 2021. +[9] “Football/ +soccer +english +vocabulary, +https://www.vocabulary.cl/english/football-soccer.htm.” +[10] H. Xie, Z. Qin, G. Y. Li, and B.-H. Juang, “Deep learning enabled +semantic communication systems,” IEEE Transactions on Signal Pro- +cessing, vol. 69, pp. 2663–2675, 2021. +[11] H. Xie and Z. Qin, “A lite distributed semantic communication system +for internet of things,” IEEE Journal on Selected Areas in Communica- +tions, vol. 39, no. 1, pp. 142–153, 2020. +[12] W. Yang, H. Du, Z. Liew, W. Y. B. Lim, Z. Xiong, D. Niyato, X. Chi, +X. S. Shen, and C. Miao, “Semantic communications for 6G future +internet: Fundamentals, applications, and challenges,” arXiv preprint +arXiv:2207.00427, 2022. +[13] Z. Qin, X. Tao, J. Lu, and G. Y. Li, “Semantic communications: +Principles and challenges,” arXiv preprint arXiv:2201.01389, 2021. +[14] Q. Lan, D. Wen, Z. Zhang, Q. Zeng, X. Chen, P. Popovski, and +K. Huang, “What is semantic communication? A view on conveying +meaning in the era of machine intelligence,” Journal of Communications +and Information Networks, vol. 6, no. 4, pp. 336–371, 2021. +[15] K. Niu, J. Dai, S. Yao, S. Wang, Z. Si, X. Qin, and P. Zhang, +“Towards Semantic Communications: A Paradigm Shift,” arXiv preprint +arXiv:2203.06692, 2022. +[16] H. Seo, J. Park, M. Bennis, and M. Debbah, “Semantics-native commu- +nication with contextual reasoning,” arXiv preprint arXiv:2108.05681, +2021. +[17] M. Sana and E. C. Strinati, “Learning semantics: An opportunity for +effective 6G communications,” in 2022 IEEE 19th Annual Consumer +Communications & Networking Conference (CCNC), pp. 631–636, +IEEE, 2022. +[18] Y. Wang, M. Chen, T. Luo, W. Saad, D. Niyato, H. V. Poor, and S. Cui, +“Performance optimization for semantic communications: An attention- +based reinforcement learning approach,” IEEE Journal on Selected Areas +in Communications, vol. 40, no. 9, pp. 2598–2613, 2022. +[19] S. Ji, S. Pan, E. Cambria, P. Marttinen, and S. Y. Philip, “A survey on +knowledge graphs: Representation, acquisition, and applications,” IEEE +Transactions on Neural Networks and Learning Systems, vol. 33, no. 2, +pp. 494–514, 2021. +[20] W. Yu, C. Zhu, Z. Li, Z. Hu, Q. Wang, H. Ji, and M. Jiang, “A survey of +knowledge-enhanced text generation,” ACM Computing Surveys (CSUR), +2022. +[21] H. Li, J. Zhu, J. Zhang, C. Zong, and X. He, “Keywords-guided ab- +stractive sentence summarization,” Proceedings of the AAAI conference +on artificial intelligence, vol. 34, no. 05, pp. 8196–8203, 2020. +[22] L. Huang, L. Wu, and L. Wang, “Knowledge Graph-Augmented Abstrac- +tive Summarization with Semantic-Driven Cloze Reward,” in Proceed- +ings of the 58th Annual Meeting of the Association for Computational +Linguistics, pp. 5094–5107, 2020. +[23] F. Zhou, Y. Li, X. Zhang, Q. Wu, X. Lei, and R. Q. Hu, “Cognitive +semantic communication systems driven by knowledge graph,” arXiv +preprint arXiv:2202.11958, 2022. +[24] J. Liang, Y. Xiao, Y. Li, G. Shi, and M. Bennis, “Life-long learn- +ing for reasoning-based semantic communication,” arXiv preprint +arXiv:2202.01952, 2022. +[25] K. Papineni, S. Roukos, T. Ward, and W.-J. Zhu, “BLEU: A method +for automatic evaluation of machine translation,” in Proceedings of the +40th annual meeting of the Association for Computational Linguistics, +pp. 311–318, 2002. +[26] T. M. Cover, Elements of information theory. John Wiley & Sons, 1999. +[27] H. Yao, D.-l. Zhu, B. Jiang, and P. Yu, “Negative log likelihood ratio +loss for deep neural network classification,” in Proceedings of the Future +Technologies Conference (FTC) 2019: Volume 1, pp. 276–282, Springer, +2020. +[28] I. Goodfellow, Y. Bengio, and A. Courville, Deep learning. MIT press, +2016. +[29] M. I. Belghazi, A. Baratin, S. Rajeshwar, S. Ozair, Y. Bengio, +A. Courville, and D. Hjelm, “Mutual Information Neural Estimation,” +in International Conference on Machine Learning, pp. 531–540, PMLR, +2018. +[30] J. Bao, P. Basu, M. Dean, C. Partridge, A. Swami, W. Leland, and J. A. +Hendler, “Towards a theory of semantic communication,” in 2011 IEEE +Network Science Workshop, pp. 110–117, IEEE, 2011. +[31] R. Zhang and C. Eickhoff, “SOCCER: An Information-Sparse Discourse +State Tracking Collection in the Sports Commentary Domain,” in +Proceedings of the 2021 Conference of the North American Chapter +of the Association for Computational Linguistics: Human Language +Technologies, pp. 4325–4333, 2021. + diff --git a/_NFLT4oBgHgl3EQfvy-O/content/tmp_files/load_file.txt b/_NFLT4oBgHgl3EQfvy-O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9697add12d95d2582f8ce4acb35093144677b2cd --- /dev/null +++ b/_NFLT4oBgHgl3EQfvy-O/content/tmp_files/load_file.txt @@ -0,0 +1,594 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf,len=593 +page_content='Knowledge-Aware Semantic Communication System Design Sachin Kadam and Dong In Kim Department of Electrical and Computer Engineering Sungkyunkwan University (SKKU), Suwon 16419, Republic of Korea Email: sachinkadam@skku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='edu, dikim@skku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='kr Abstract—The recent emergence of 6G raises the challenge of increasing the transmission data rate even further in order to break the barrier set by the Shannon limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Traditional communication methods fall short of the 6G goals, paving the way for Semantic Communication (SemCom) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' These systems find applications in wide range of fields such as economics, metaverse, autonomous transportation systems, healthcare, smart factories, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In SemCom systems, only the relevant information from the data, known as semantic data, is extracted to eliminate unwanted overheads in the raw data and then transmitted after encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In this paper, we first use the shared knowledge base to extract the keywords from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Then, we design an auto- encoder and auto-decoder that only transmit these keywords and, respectively, recover the data using the received keywords and the shared knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We show analytically that the overall semantic distortion function has an upper bound, which is shown in the literature to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We numerically compute the accuracy of the reconstructed sentences at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Using simulations, we show that the proposed methods outperform a state-of-the-art method in terms of the average number of words per sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Index Terms—Semantic Communications, Knowledge Base, 6G, Data Compression, Wireless Communications I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' INTRODUCTION As per the prediction in [1], semantic communication (Sem- Com) technology is identified as one of the key ingredients in 6G due to the requirement of low latency and high data rate transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The recent emergence of SemCom tech- nologies finds applications in wide range of fields such as economics [2], metaverse [3], autonomous transportation sys- tems [4], smart factories [5], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In SemCom, we only transmit useful and necessary information to the recipients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The semantic extraction (SE) is a process wherein the useful and necessary features are extracted from the original raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' For example, the essential speech features are extracted using an attention-based mechanism in [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' During critical applications such as military operations, search operations by forest personnel in a dense forest, medical emergencies in remote areas, fire incidents in a remote agri- cultural land, the release of water from a nearby dam, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', only the essential information needs to be communicated on an urgent basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The messages could be in the form of text or audio and they come from a limited dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In a non-critical application, such as broadcasting a text/audio summary of This research was supported in part by the Korean Government (MSIT) un- der the ICT Creative Consilience program (IITP-2020-0-01821) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' commentary provided by live football commentators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Among all the words spoken by them, only a limited set of useful or important words are relevant to the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' These words are drawn from a limited dataset such as football vocabu- lary [9] which includes words such as goal, player names, red card, football, score, assist, half-time, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This limited dataset provides an opportunity, in the context of SemCom design, for a significant overhead reduction by extracting and processing only the relevant keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' For example, an uttered commentary sentence is: ‘Ronaldo shoots the ball into the right-bottom of the net and it’s a goal!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The extracted keywords in this example are Ronaldo, shoots, ball, right-bottom, net, goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Only these keywords are transmitted in place of the entire sentence, and the receiver reconstructs a meaningful sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The reconstructed sentence in this case is: ‘Ronaldo shoots the ball into the right-bottom of the net to score a goal.’ This sentence is not exactly the same as the original sentence, but it conveys the same meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The main goal of this paper is to use SemCom technology to reduce communication overhead, in the context of natural lan- guage processing (NLP) problems, while maintaining a certain minimum accuracy in wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The overhead reduction is performed with high accuracy in the literature [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' However, in some applications, high data rates are preferred over high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' As a result, we present the results of the trade-off between overhead reduction and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Model parameters are chosen based on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Instead of transmitting raw data, the transmitter is designed to transmit semantic data, which significantly reduces network data traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A knowledge base (KB) is a technology that collects, stores, and manages data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A knowledge graph (KG) is a KB that integrates data using a graph-structured topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' They are used to store interconnected event descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' These are used to predict the missing words in the received data (keywords) to construct a meaningful sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The organization of the paper is as follows: A brief literature review on SemCom technologies is provided in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We introduce our proposed system model in Section III and provide a few useful simulation results in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Finally, we conclude the paper in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' RELATED WORK The following state-of-the-art survey papers provide in- depth discussions on various SemCom technologies and their arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='12161v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='NI] 28 Jan 2023 applications [12]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Deep learning based SemCom tech- nologies are proposed in [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A brief tutorial on the framework of SemCom and a method to calculate a bound on semantic data compression is provided in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The SemCom technology wherein both transmitter and receiver are empowered with the capability of contextual reasoning is proposed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The SemCom technology for a system where transmitter and receiver speak different languages is designed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In [18], a SemCom framework for textual data trans- mission is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In this framework, semantic information is represented by a KG made up of a set of semantic triples and the receiver recovers the original text using a graph-to-text generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' All of these works focused on achieving an overhead reduction without compromising the accuracy of the received data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' None of these works investigated the possibility of further overhead reduction, thereby improving transmission data rates, while sacrificing a little accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This issue is addressed in this paper using a shared knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A significant research on the usage of KBs and KGs is carried out in the field of natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A survey paper based on KG is presented in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Similarly, another survey paper on KB text generation is presented in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A method to generate a summary of sentences by using a given set of keywords is proposed in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Similarly, a method to generate a summary of sentences by using a knowledge base is shown in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Recently, KGs are utlized in the context of SemCom design [18], [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' But these works do not focus on the issue presented in this paper, which is to design a SemCom system with a significant overhead reduction with a little compromise on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' SYSTEM MODEL The system model of the proposed SemCom system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let X be the input text dataset with N sentences, Xi be the ith, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}, sentence of X, and K be the shared knowledge base (KB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' First, we extract the keywords from X using K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let the total set of keywords be Ω = �N i=1 Ωi, where Ωi denotes the set of keywords present in Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The keyword extraction process at every sentence Xi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}, is executed by multiplying it with a binary vector bi = [bi(ℓ), ℓ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , |Xi|}],1 which is defined as follows: bi(ℓ) ≜ � 1, if ℓth word of Xi, Xi(ℓ), is a keyword in K 0, else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (1) Hence, Ωi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}, is obtained by collecting the non-zero elements from Xi ⊙ bi, where ⊙ is a word-wise multiplication operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Here Xi ⊙ bi ≜ [Xi(ℓ)bi(ℓ), ∀ℓ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , |Xi|}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='2 1|A| denotes the cardinality of set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2For ease of understanding, let us consider the example discussed in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let Xi be ‘Ronaldo shoots the ball into the right-bottom of the net and it’s a goal!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' If the set of keywords present in Xi is {Ronaldo, shoots, ball, right-bottom, net, goal} then bi = [11010010010001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Now, Xi ⊙ bi gives [Ronaldo, shoots, 0, ball, 0, 0, right-bottom, 0, 0, net, 0, 0, 0, goal].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Next, Ωi is obtained by collecting the non-zero elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', Ωi = {Ronaldo, shoots, ball, right-bottom, net, goal}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Now, let us define the quantity BLEU score (bilingual eval- uation understudy [25]) to compare the similarities between two sentences quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The BLEU(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ˆs) ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1] score between transmitted sentence s and reconstructed sentence ˆs is computed as follows: BLEU(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ˆs) = BP(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ˆs) exp � W � n=1 wn ln pn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ˆs) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (2) where pn denotes the modified n-gram precision function up to length W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' wn denotes the weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' and brevity penalty (BP) is given by the following expression: BP(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ˆs) = � 1 ℓc > ℓr e1−ℓr/ℓc ℓc ≤ ℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (3) where ℓc is the length of the candidate translation and ℓr is the effective reference corpus length [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let ξ be a function which generates a set of M (say) sentences from a given set of keywords with the help of a given knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Using the keywords in Ωi, bi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}, and the knowledge K, for a given sentence Xi, the sentence generator at the transmitter generates a set of M sentences using the function ξλ, where λ is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let that set of sentences be � Xij, j = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' So, � Xij = ξλ(Ωi, K), j = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (4) Next, out of these M sentences we choose the most seman- tically equivalent sentence based on the BLEU scores [25] compared with input sentence Xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', � Xi = arg max � Xij,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=',M BLEU( � Xij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xi), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (5) Note that the set of sentences � X = { � Xi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}}, is generated at the transmitter during the training process only and it is shared with the receiver a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Next, ith keyword set Ωi is encoded using the auto-encoder which consists of semantic and channel encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The auto-encoder uses the binary vector bi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}, to assign a common symbol to all non-keywords of Xi and a unique symbol to keywords of Xi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This enables the receiver to construct appropriate sentences from the received symbol sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let us denote Sθe and Cφe as the semantic and channel encoders with θe and φe as the parameters sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' After encoding Ωi, we get the following set of symbols: �Ωi = Cφe(Sθe(Ωi)), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (6) The encoded set of symbols �Ωi is transmitted via the AWGN (additive white Gaussion noise) channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let h be the channel gain and η be the noise which gets added to �Ωi during transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' So, the set of received symbols at the receiver is Ωi = h�Ωi + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' After receiving, this set of symbols is decoded using the auto-decoder which consists of channel and semantic decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let us denote Cφd and Sθd as the channel and semantic decoders with φd and θd as the parameters sets, Auto-Decoder Auto-Encoder Semantic Encoder Channel Encoder Channel Decoder Channel Noise Semantic Decoder Destination Sentence Generator Source Shared Knowledge (K) Context, keywords, events, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Input Text Output Text Keyword Extraction 𝑋𝑖 Ω𝑖 ෩Ω𝑖 ഥΩ𝑖 \u0de1Ω𝑖 \u0de0𝑌𝑖 \u0de0𝑌𝑖𝑗 \u0de0𝑋𝑖 Sentence Generator \u0de0𝑋𝑖𝑗 argmax \u0de0𝑋𝑖𝑗,𝑗=1,…,𝑀 BLEU( \u0de0𝑋𝑖𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 𝑋𝑖) argmax \u0de0𝑌𝑖𝑗,𝑗=1,…,𝑀 BLEU(\u0de0𝑌𝑖𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' \u0de0𝑋𝑖) 𝑋𝑖 Auto-Decoder Auto-Encoder Semantic Encoder Channel Encoder Channel Decoder Channel Noise Semantic Decoder Sentence Generator Source Input Text Output Text Keyword Extraction ෩Ω𝑖 ഥΩ𝑖 \u0de1Ω𝑖 \u0de0𝑌𝑖 Shared Knowledge Context, keywords, events, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (a) (b) Destination 𝑖 ∈ {1, … , 𝑁} 𝑋𝑖 𝑖 ∈ {1, … , 𝑁} Ω𝑖 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1: The block diagram of our proposed SemCom system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (a) is used for training the system parameters and the model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (b) is used for evaluating the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' After decoding Ωi, we get the following set of keywords: �Ωi = Sθd(Cφd(Ωi)), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (7) From the decoded set of keywords and the shared knowl- edge K, the sentence generator at the receiver generates a set of M sentences using the function ξµ, where µ is a parameter, and let that set of sentences be �Yij, j = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , M}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', �Yij = ξµ(�Ωi, K), j = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (8) To select the most desired sentence among these sentences, we compute the BLEU scores between �Yij, j = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , M} and the sentence at the transmitter � Xi, and choose the one which maximizes the BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' That is: �Yi = arg max �Yij,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=',M BLEU(�Yij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' � Xi), i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}, (9) where �Y = {�Yi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}} denotes the desired set of sentences generated at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Training the System Model Let X be the set of all possible sentences and pX(x), pλ(�x), and pµ(�y) denote the probability distributions of sentences X ∈ X, �X ∈ � X, and �Y ∈ �Y , respectively, for all x, �x, �y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='3 The sentences �X and �Y are generated by the sentence genera- tors in transmitter and receiver, respectively, parameterized by λ and µ, respectively, with the help of shared knowledge K 3Note that �x, �y ∈ X ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' To obtain the closed-form expressions for certain quantities, we relax the condition and assume X = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Note that, since �X and �Y are generated with the help of knowledge K, throughout the paper, the conditional distributions pλ(·|·) and pµ(·|·) are conditioned on the event K = k, ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let H(X) and DKL(p||q) represent, respectively, the en- tropy of a random variable X whose probability distribution is p and the Kullback Leibler (KL) divergence between the probability distributions p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' These quantities are defined as follows [26]: H(X) ≜ − � x∈X p(x) log p(x), (10) DKL(p||q) ≜ � x∈X p(x) log p(x) q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (11) The overall cross entropy (CE) loss measures the difference between the actual probability distribution at the input and the estimated probability distribution at the output and it can be minimized using the stochastic gradient descent (SGD) methods [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' So the overall cross entropy (CE) loss is defined as follows [28]: LCE(µ) ≜ H(X) + DKL(pX(x)||pµ(�y|k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (12) By using the expressions of (10) and (11), we simplify the expression for overall CE loss as follows: LCE(µ) = − � x∈X pX(x) log pµ(�y|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (13) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1, we observe that there are information losses at auto-encoder, auto-decoder, and sentence generation blocks both in transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We aim to minimize the summation of all these losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The characterization of these losses are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The loss of information between sentences X and �X at the transmitter is measured using the CE loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', LCE 1 (λ) = H(X) + DKL(pX(x)||pλ(�x|k)) (14) = − � x∈X pX(x) log pλ(�x|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (15) Similarly, the loss of information between sentences �X and �Y at the receiver is also measured using the CE loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', LCE 2 (µ, λ) = H(�X|K) + DKL(pλ(�x|k)||pµ(�y|k)) (16) = − � �x∈X pλ(�x|k) log pµ(�y|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (17) Lastly, the loss of information in the channel is measured in terms of mutual information (MI) between transmitted symbols and received symbols, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', LMI 3 (θe, φe, θd, φd) = I(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (18) Now, we define the overall semantic distortion function as follows:4 L(θe, φe, θd, φd, λ, µ) ≜ LCE 1 (·) + LCE 2 (·) − γLMI 3 (·), (19) where γ ≥ 0 is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In Theorem 1, we show that the overall semantic distortion L(·) attains an upper bound which can be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We aim to compute the optimal parameters of auto-encoder, auto-decoder, and sentence gener- ation blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' These blocks are characterized by the parameters θe, φe, θd, φd, λ, µ and are obtained by training the system model, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1(a), by minimizing the overall loss function defined in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The overall semantic distortion function attains the following upper-bound: L(θe, φe, θd, φd, λ, µ) ≤ LCE(·) − γLMI 3 (·) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (20) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The proof is given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The upper-bound B, provided in Theorem 1, is proved to be optimized using the SGD algorithms [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Also, in the published works [10], [17], the semantic distortion of the overall system is minimized using a similar expression as that of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hence, to minimize the overall semantic distortion defined in (19), we seek to minimize the upper-bound B provided in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The loss due to mutual information LMI 3 (·) can be estimated using state-of-the-art mutual information neural estimator (MINE) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 4We use (·) in place of parameters, wherever convenient, for ease of representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Accuracy versus Overhead Reduction Trade-off Given the limited size of knowledge base, though the accuracy of the reconstructed sentences in �Y may not be sufficiently high, the useful content in those sentences is summarized and conveyed to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This novel approach saves a significant amount of overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' There exists a trade-off between overhead reduction and the accuracy that depends on the size of the knowledge base K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' For example, if the set K is small, only a few keywords are extracted, encoded, and transmitted from the given input sentences in X, implying a higher amount of average overhead reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' On average, this results in a large amount of missing information, so the accuracy of the reconstructed sentences in �Y is expected to be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' On the other hand, if the set K is large, a significant number of keywords are extracted from the given sentences in X, encoded, and transmitted, implying a lower average overhead reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' On average, this results in a small amount of missing information, so the accuracy of the reconstructed sentences in �Y is expected to be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This phenomenon is numerically shown in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' So, we aim at minimizing the transmission of average number of words per sentence (equivalent to maximizing the average overhead reduction) by keeping a certain minimum accuracy information τ in the received sentence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', min 1 N N � i=1 |Ωi| (21) BLEU(�Yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xi) ≥ τ, i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' , N}, (22) where |Ωi| denotes the number of keywords in Ωi that corre- sponds to sentence Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Shared Knowledge Base We generate the shared knowledge base K by using the keywords from a limited dataset Ω which consists of only the relevant words of a particular event, like that of a football game in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We assume that both the transmitter and receiver have access to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' It is shown in [30] that the capacity of the channel can be increased beyond Shannon’s limit by using a semantic encoder with low semantic ambiguity and a semantic decoder with strong inference ability and a large shared knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' From Section III, recall that in every sentence, only the words w ∈ Ω are uniquely encoded and transmitted to the receiver in their corresponding time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' At other time slots, a common symbol is transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' By utilizing K, the receiver reconstructs the sentence based on the received symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' To improve the accuracy of the reconstructed sentences, we can increase the size of K by adding more keywords from the vocabulary generated using X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This result is shown using simulations in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' SIMULATION RESULTS First, we evaluate the performance of the text data trans- mission in terms of accuracy using BLEU score [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='5 In our work, we use the dataset provided in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We parse the 5We defined the BLEU score in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' TABLE I: Simulation parameters Number of matches used in training 1580 Number of matches used in evaluation 340 Number of epochs during training 10 SNR 6 dB Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='001 Batch Size 64 Channel AWGN football commentary data of 1920 matches from the website goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The considered football matches are from Union of European Football Associations (UEFA) Champions League, UEFA Europa League, and Premier League between 2016 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The simulation parameters used for plots in this section are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The simulations are performed in a computer with NVIDIA GeForce RTX 3090 GPU and Intel Core i9-10980XE CPU with 256GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let ρ be the fraction of the total vocabulary V , which contains all the dataset words, to be added to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ρ = 0 indicates that no additional vocabulary is added and the system is evaluated only with the initial keyword set Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Based on the way of adding the vocabulary words to Ω0, we propose two types of schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In the first type, ρ|V | vocabulary words are uniformly chosen at random from V and added to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In the second type, the words in V are first arranged in the decreasing order of the frequency of appearances in the dataset, and then the first ρ|V | vocabulary words are added to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We call these schemes as ‘RANDOM’ and ‘ORDERED’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' The accuracy performances of both the schemes and a deep learning based SemCom system method named DeepSC [10], in terms of BLEU score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ρ, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' From the plot we can infer that even with ρ = 0, the initial keyword set can produce a BLEU score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='55 (for 1-gram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This shows that the context-related keywords produce good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Also, we see that as we add more vocabulary words to Ω0, the BLEU score increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' For the same value of ρ and n, the ORDERED scheme performs better than the RANDOM scheme because of the addition of high frequency words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' And, in terms of different n-grams, BLEU score decreases as n increases, which is an expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In comparison to the DeepSC scheme, the proposed schemes perform poorly in terms of accuracy but outperform it in terms of overhead reduction, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Next, we evaluate the performance of the proposed schemes, in terms of the transmission of average number of words per sentence, with respect to DeepSC [10] and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let W denote the average number of words per sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' From the plot we observe that both the schemes outperform DeepSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Among the proposed schemes, for a given ρ the RANDOM scheme outperforms the ORDERED scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This is because, in the ORDERED scheme high frequency words are added which increases the number of words to be encoded in the input data as compared with the RANDOM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Now, we solve the optimization problem presented in (21) and (22) using both the proposed schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' For this purpose, we evaluate W vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' τ and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' From the plot we observe that both the schemes outperform DeepSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2: This plot shows the BLEU score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ρ for different values of n-grams, where n = {1, 2, 3, 4}, for the proposed schemes and the DeepSC scheme [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 3: These plots show the average number of words per sentence vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' ρ in the left plot and vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' τ in the right plot, respectively, for the proposed schemes and the DeepSC scheme [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Also, we see that the performance of both the schemes is same for a given accuracy threshold τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' This is because, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2, for a given value of ρ ∈ (0, 1), the ORDERED scheme outperforms the RANDOM scheme in terms of accuracy, whereas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 3a, the RANDOM scheme outperforms the ORDERED scheme in terms of overhead reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hence, we can choose any one of the proposed methods to solve the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' CONCLUSIONS In this paper, we first extracted relevant keywords from the dataset using the shared knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Then, using the received keywords and the shared knowledge, we designed an auto-encoder and auto-decoder that only transmit these keywords and, respectively, recover the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We proved that the overall semantic distortion function has an upper bound, which is shown to be optimized using the SGD algorithms in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We computed the accuracy of the reconstructed sentences at the receiver quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We demonstrated through simulations that the proposed methods outperform a state-of-the-art method in terms of average number of words per sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Furthermore, the proposed approach makes no new hardware modifications to the existing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' We focused solely on the text dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' however, similar approaches can be used in the future for other types of datasets such as image, audio, and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='8 BLEU Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='4 1-gram(RANDOM) 3-gram(ORDERED) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='2 2-gram (RANDOM) 4-gram(ORDERED) 3-gram(RANDOM) 1-gram (DeepSC) 4-gram(RANDOM) 2-gram(DeepSC) 1-gram(ORDERED) 3-gram(DeepSC) 2-gram(ORDERED 4-gram(DeepSC) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='8 120 15 M 10 RANDOM ORDERED DeepsC 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='8 1 p20 RANDOM 15 ORDERED M DeepsC 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='9 1 TAPPENDIX Now we provide the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Let us define the following term, parameterized by λ, µ, and k ∈ K: δ(λ, µ, k) ≜ log �pµ(�y|k) pλ(�x|k) � , ∀�x, �y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (23) From (19), we know that L(·) = LCE 1 (·) + LCE 2 (·) − γLMI 3 (·) (24a) = − � x∈X pX(x) log pλ(�x|k) − � �x∈X pλ(�x|k) log pµ(�y|k) − γI(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ω) (24b) = − � x∈X pX(x) log � pλ(�x|k)pµ(�y|k) pµ(�y|k) � − � �x∈X pλ(�x|k) log � pµ(�y|k)pλ(�x|k) pλ(�x|k) � − γI(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ω) (24c) = − � x∈X pX(x) log pµ(�y|k) + δ(λ, µ, k) − � �x∈X pλ(�x|k) log pλ(�x|k) − δ(λ, µ, k) − γI(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ω) (24d) = LCE(·) + H(�X|K) − γI(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ω) (24e) ≤ LCE(·) − γI(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' (24f) In (24b), we expand the loss function expressions using their respective definitions provided in (15), (17), and (18), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' In (24c), we multiply and divide pµ(�y|k) and pλ(�x|k) in the first and second terms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Using (23) and algebraic simplifications, we get (24d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' By using (12) and the definition of entropy, we write (24e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' And, finally the inequality in (24f) is due to H(�X|K) ≥ 0 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' REFERENCES [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Rajatheva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Atzeni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Bj¨ornson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Bourdoux, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Buzzi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Dor´e, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Erkucuk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Fuentes, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Guan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=', “White paper on broadband connectivity in 6G,” 6G Research Visions, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 10, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Liew, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Du, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Lim, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xiong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Niyato, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Miao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Kim, “Economics of Semantic Communication System: An Auction Approach,” arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='05040, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ismail, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Niyato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Erol-Kantarci, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Miao, “Semantic Information Market For The Metaverse: An Auction Based Approach,” arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='04878, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Liew, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Lim, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xiong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Niyato, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Chi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Cao, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Letaief, “Semantic communication meets edge intelligence,” arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='06471, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [5] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Chen, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Guo, “Semantic communications: Overview, open issues, and future research directions,” IEEE Wireless Communi- cations, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [6] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Weng and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Qin, “Semantic communication systems for speech transmission,” IEEE Journal on Selected Areas in Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2434–2444, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [7] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Weng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Qin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Li, “Semantic communications for speech signals,” in ICC 2021-IEEE International Conference on Communica- tions, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1–6, IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Tong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Saad, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yin, “Federated learning based audio semantic communication over wireless networks,” in 2021 IEEE Global Communications Conference (GLOBECOM), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1–6, IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [9] “Football/ soccer english vocabulary, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='cl/english/football-soccer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='htm.” [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xie, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Qin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Li, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Juang, “Deep learning enabled semantic communication systems,” IEEE Transactions on Signal Pro- cessing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 69, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2663–2675, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xie and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Qin, “A lite distributed semantic communication system for internet of things,” IEEE Journal on Selected Areas in Communica- tions, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 142–153, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Du, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Liew, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Lim, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xiong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Niyato, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Chi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Shen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Miao, “Semantic communications for 6G future internet: Fundamentals, applications, and challenges,” arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='00427, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [13] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Qin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Tao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Lu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Li, “Semantic communications: Principles and challenges,” arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='01389, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [14] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Lan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zeng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Popovski, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Huang, “What is semantic communication?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A view on conveying meaning in the era of machine intelligence,” Journal of Communications and Information Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 336–371, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Niu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Dai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Si, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Qin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhang, “Towards Semantic Communications: A Paradigm Shift,” arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='06692, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Seo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Park, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Bennis, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Debbah, “Semantics-native commu- nication with contextual reasoning,” arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='05681, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Sana and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Strinati, “Learning semantics: An opportunity for effective 6G communications,” in 2022 IEEE 19th Annual Consumer Communications & Networking Conference (CCNC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 631–636, IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Luo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Saad, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Niyato, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Poor, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Cui, “Performance optimization for semantic communications: An attention- based reinforcement learning approach,” IEEE Journal on Selected Areas in Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2598–2613, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ji, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Pan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Cambria, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Marttinen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Philip, “A survey on knowledge graphs: Representation, acquisition, and applications,” IEEE Transactions on Neural Networks and Learning Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 494–514, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ji, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Jiang, “A survey of knowledge-enhanced text generation,” ACM Computing Surveys (CSUR), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zong, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' He, “Keywords-guided ab- stractive sentence summarization,” Proceedings of the AAAI conference on artificial intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 05, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 8196–8203, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wang, “Knowledge Graph-Augmented Abstrac- tive Summarization with Semantic-Driven Cloze Reward,” in Proceed- ings of the 58th Annual Meeting of the Association for Computational Linguistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 5094–5107, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [23] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Lei, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hu, “Cognitive semantic communication systems driven by knowledge graph,” arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='11958, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Liang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Shi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Bennis, “Life-long learn- ing for reasoning-based semantic communication,” arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='01952, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Papineni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Roukos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ward, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhu, “BLEU: A method for automatic evaluation of machine translation,” in Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 311–318, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Cover, Elements of information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' John Wiley & Sons, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content='-l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Jiang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Yu, “Negative log likelihood ratio loss for deep neural network classification,” in Proceedings of the Future Technologies Conference (FTC) 2019: Volume 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 276–282, Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [28] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Goodfellow, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Bengio, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Courville, Deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' MIT press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Belghazi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Baratin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Rajeshwar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Ozair, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Bengio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Courville, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hjelm, “Mutual Information Neural Estimation,” in International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 531–540, PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Bao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Basu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Dean, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Partridge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Swami, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Leland, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Hendler, “Towards a theory of semantic communication,” in 2011 IEEE Network Science Workshop, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 110–117, IEEE, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Zhang and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' Eickhoff, “SOCCER: An Information-Sparse Discourse State Tracking Collection in the Sports Commentary Domain,” in Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} +page_content=' 4325–4333, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfvy-O/content/2301.12161v1.pdf'} diff --git a/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf b/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9ac1dfa06ec976b30771d97210bedfa5d5ca5cd6 --- /dev/null +++ b/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:daef3ffc0697cf2afcfa880f9f20742ec1eed350e4d0b75f19aaebe62be345bd +size 5589025 diff --git a/_dAzT4oBgHgl3EQfFvqG/vector_store/index.faiss b/_dAzT4oBgHgl3EQfFvqG/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4cd5a61368bfcf226e54edfc8edca6364d77f66b --- /dev/null +++ b/_dAzT4oBgHgl3EQfFvqG/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a3f88ce0418ad9fe2264e4a187bbcafbec9dec096de82bb952912112641acb20 +size 5177389 diff --git a/adE2T4oBgHgl3EQfFAZQ/content/tmp_files/2301.03641v1.pdf.txt b/adE2T4oBgHgl3EQfFAZQ/content/tmp_files/2301.03641v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7eba4ad6a298ee7be122b14617f1e3be9f00969 --- /dev/null +++ b/adE2T4oBgHgl3EQfFAZQ/content/tmp_files/2301.03641v1.pdf.txt @@ -0,0 +1,895 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 00, NO. 0, 00 2022 +1 +SatNetOps: Toward Multi-Layer Networking for +Satellite Network Operations +Peng Hu, Senior Member, IEEE +Abstract—Recent advancements in low-Earth-orbit (LEO) +satellites aim to bring resilience, ubiquitous, and high-quality +service to future Internet infrastructure. However, the soaring +number of space assets, increasing dynamics of LEO satellites and +expanding dimensions of network threats call for an enhanced +approach to efficient satellite operations. To address these press- +ing challenges, we propose an approach for satellite network +operations based on multi-layer satellite networking (MLSN), +called “SatNetOps”. Two SatNetOps schemes are proposed, +referred to as LEO-LEO MLSN (LLM) and GEO-LEO MLSN +(GLM). The performance of the proposed schemes is evaluated +in 24-hr satellite scenarios with typical payload setups in simu- +lations, where the key metrics such as latency and reliability are +discussed with the consideration of the Consultative Committee +for Space Data Systems (CCSDS) standard-compliant telemetry +and telecommand missions. Although the SatNetOps approach is +promising, we analyze the factors affecting the performance of +the LLM and GLM schemes. The discussions on the results and +conclusive remarks are made in the end. +Index Terms—Satellites, Telecommand, Telemetry, Space, Data +Link, Network Operations +I. INTRODUCTION +With the recent advancements and deployments of low- +Earth-orbit (LEO) satellites, the upcoming space assets will +approach a unprecedented volume in the coming years. These +space assets, while providing much convenience and resiliency +for global telecommunications networks, demand efficient, +reliable, and robust operations for the satellite networks in +“New Space” [1] ecosystems. +One of the essential services to support the new satellite +operations is telecommand (TC) and telemetry (TM) missions, +which rely on the efficient transmissions of packets for space- +craft operations. From a communications perspective, TC and +TM missions share a similar process where TC messages are +transferred from a ground station (GS) to a target satellite, +while TM messages are transferred in the reverse direction +and the packet transmission follow a similar pattern based on +the Consultative Committee for Space Data Systems (CCSDS) +standards. We therefore focus on the discussion of represen- +tative TC missions. In the traditional approach, TC messages +are sent or received when a direct contact opportunity occurs +between an operations GS and a target satellite. CCSDS [2] +has standardized the transmissions of TC packets for space- +to-space and ground-to-space communication links. The TC +packets are usually small in size but require high reliability. +However, such a traditional approach faces challenges that +significantly reduce the efficiency of satellite operations. For +example, the TC message transmissions depend on the access +states. Based on our 24-hr access analysis of typical LEO +Fig. 1. Access analysis of typical LEO satellite constellations +satellite constellations shown in Fig. 1, the average access +opportunity per satellite in three typical constellation is very +low for mostly < 2% chance of access to two typical GS +locations (one in northern Canada and one in western Canada +where most LEO satellites have coverage) with a minimum +elevation 25◦ over a 24-hr mission, which clearly shows the +limitations of the traditional operations approach in terms of +efficiency and real-time communication capabilities. Although +satellites are traditionally operated in isolation from their +orbiting counterparts, communicating only with ground-based +infrastructure with high delays leaves them susceptible to the +consequences of various anomalies. Further, the space network +has not been fully utilized for operations. These challenges +call for efficient and resilient operations of a spacecraft and +the entire satellite network. +Recently, using a legacy geostationary (GEO) satellite net- +work for satellite operations missions has been considered. +The constant positions of GEO satellites relative to ground +components at a high altitude provide broad coverage of LEO +satellites at a low altitude. For example, in November 2020, +Inmarsat reportedly started a new service called Inter-satellite +Data Relay System (IDRS), providing real-time links between +GEO and LEO satellites. With the recent launch of I-6F1 GEO +satellites with Ka/L-band payloads, new GEO satellites are +expected to be deeply integrated into other medium-Earth- +orbit (MEO) or LEO satellites into the 5G network as part +of an “ORCHESTRA” network. Although the space industry +shows intensive interest in an option of employing legacy +or new satellite fleet for operations, there are no technical +details or formal studies available about established schemes +or assessments. In fact, such an option would face practical +efficiency and timing challenges considering the distance +between GEO and LEO. To reduce the distance for packet +transmission, LEO satellite networks may be considered in +combination with GEO satellites. The idea falls into the multi- +layer satellite networking (MLSN) concept where satellites +in different orbits are viewed in multiple layers. The layers +may include various shells of a large constellation, different +constellations, and different Earth orbits. However, MLSN has +been mainly considered for data traffic routing, which differs +arXiv:2301.03641v1 [cs.NI] 9 Jan 2023 + +CA +1.5 +.6 +Iqaluit, +2.4 + Station Access + Satellite (%) +I Station: +2 +round +0.5 +60 +20 +A0 +Per s +150 +200 +2 +round +Time +1.6 +Station: +1.2 +0.5 +0.5 +20 +40 +100 +50 +60 +500 +1000 +3000 +3500 +Sat. ID (Telesat Polar) +Sat. ID (Iridium NEXT) +Sat. ID (Starlink)JOURNAL OF LATEX CLASS FILES, VOL. 00, NO. 0, 00 2022 +2 +from the TM/TC transmissions in practice for security and +reliability considerations. The solutions to utilizing MLSN for +operations missions are lacking. +This paper aims to address the new challenges of the +operational needs faced by the recent growth of space assets +in a timely manner. We propose a new approach considering +MLSN, collectively called “SatNetOps”, which can support +efficient operations of non-geostationary (NGSO) satellite op- +erations and enable the design of MLSN schemes. To the +best of our knowledge, this is the first work addressing the +new challenges for NGSO satellite network operations and +providing a formal evaluation of MLSN-based schemes. The +main contributions of the paper are summarized as follows: +• We identify the pressing challenges of the current satellite +operations and propose the “SatNetOps” approach with +two generic schemes devised for TC missions, called +GEO-LEO MLSN (GLM) and LEO-LEO MLSN (LLM). +• We evaluate the performance in terms of latency and +reliability in different scenarios considering the feasible +configurations. +• We adopt the popular communications payloads in the +recent satellites, i.e., the radio-frequency (RF) and free- +space optical (FSO) communications technologies in +space-to-space and space-to-ground communications. +The remainder of the paper is structured as follows. The re- +lated work is discussed in Section II. The proposed SatNetOps +schemes for TC message transfer are discussed in Section III. +The evaluation of the proposed schemes is discussed in Section +IV. The conclusive remarks are presented in Section V. +II. RELATED WORK +Compared to GEO satellites, throughput and latency are +usually considered the two significant advantages of the +current LEO satellite systems. The communications between +layers of satellite networks can be linked to the concept of +MLSN [3]. Recent work and deployment plans of LEO satel- +lite constellations are centered on throughput improvements. +The multi-layer networking may occur through multiple shells +of a large constellation with LEO satellites, such as Starlink’s +mega-constellation [4], although the cross-shell networking is +considered complex [5]. Pachler et al. [4] showed the use of +optical inter-satellite links (ISLs) on Telesat, Amazon Kuiper, +and Starlink constellations can almost double the system +throughput of each system compared to the non-optical ISLs. +A cooperative communication multi-access scheme in MLSN +was recently proposed in [6]. For the LEO satellite operations, +a real-world solution has recently realized by Inmarsat in late +2021 for using the GEO satellites for operations missions of +LEO satellites and the SES also implied the benefits of using +their upcoming o3b mPOWER fleet for satellite networking. +However, MLSN schemes and the discussion of the related +factors such as satellite and GS communications, MLSN +strategies, and mission parameters are not available. +CCSDS has developed the space data link protocols for TC +missions which may be used with other upper layer CCSDS +protocols. The latest issue of TC space data link protocol (TC- +SDLP) released in October 2021 [2] supports two types of +services: sequence-controlled (Type-A) and expedited (Type- +B) services for missions with different priorities. Although +both services have the same frame format, addressing, seg- +mentation and blocking mechanisms, Type-A services support +the Automatic Repeat Request (ARQ) mechanism for flow +control, while Type-B services do not have flow control. Type- +B services are used in “exceptional operational circumstances” +such as spacecraft recovery, or flow control is provided at the +upper layers [2]. Furthermore, these services in TC-SDLP only +support unidirectional and asynchronous services where no +predefined timing rules are specified. There is also no existing +study on the recommendations for the timing performance +for the operations of LEO satellite constellations. The timing +performance of satellite communications needs to consider +the recent developments of the satellite platforms. The perfor- +mance of the TC missions also depends on the communication +payload, where the Ka-band and S/L bands are broadly used. +Since the first test of optical links for space missions in +November 2014 [7], FSO communication is expected to be +well adopted by the space industry in the future. The recent +tests initiated by the European Space Agency (ESA), the +National Aeronautics and Space Administration (NASA) and +the commercial LEO satellite constellations further indicate +the increasingly planned use of FSO on space-to-space and +ground-to-space links. However, the study considering these +payloads for TC missions is lacking in the literature. +III. SATNETOPS AND SYSTEM MODEL +A. SatNetOps +The SatNetOps approach aims to enhance the efficiency of a +number of spacecraft operations through satellite networking. +SatNetOps intends to improve real-time and reliable commu- +nications for satellite operations. An example SatNetOps case +is shown in Fig. 2, where a SatNetOps Center attempts to +initiate a TC mission by sending a TC packet to a destination +NGSO satellite. The packet traverses in multiple hops through +one or more satellite networks, where the path established can +also be used for transferring TM packets or control messages. +B. Proposed SatNetOps Schemes +Here we propose two general schemes considering GEO +and LEO satellites, called GLM and LLM. GLM uses GEO +satellites to relay packets to the destination satellite, while +LLM uses LEO satellites to forward packets to the destination +satellite. GLM uses the GS-GEO, LEO-GEO, and GEO-GEO +links, while LLM uses GS-LEO and LEO-LEO links. The +brief descriptions of the schemes are shown in Fig. 3. For +both schemes, the TC mission preparation will determine the +TC message, destination/target LEO satellite in a constella- +tion, and the GEO satellites to be used. For LLM, the path +calculation can occur dynamically due to constant satellite +movements, where the inter- and intra-plane satellites are +calculated based on the well-adopted configuration that each +LEO satellite can access up to four neighbouring satellites in +the same direction. Each LEO/GEO satellite will calculate a +minimum elevation angle to ensure the next satellite is in line +of sight when choosing the next hop. + +JOURNAL OF LATEX CLASS FILES, VOL. 00, NO. 0, 00 2022 +3 +Fig. 2. Illustration of the proposed approach for SatNetOps. +Fig. 3. Flowcharts for the proposed LLM and GLM schemes +A SatNetOps scheme removes the need for direct contact +times between a GS and a target satellite and ensures timing +performance. Therefore, let us discuss the latency measure for +GLM/LLM-based TC missions. The latency 𝐷 of a TC mes- +sage transfer mission consists of four standard components: +propagation delay, 𝐷 𝑝𝑎, transmission delay, 𝐷𝑡, processing +delay, 𝐷 𝑝𝑐, and queuing delay, 𝐷𝑞. Since the TC packets +are small, the average queuing delay ¯𝐷𝑞 per satellite can be +assumed as a constant, 𝑚, and the average ¯𝐷 𝑝𝑐 per satellite +assumed to be a constant, 𝑘. The size of the TC packet in +bytes is 𝑀. The speed of light in FSO links is 𝑐, 2.998 × 108 +m/s. We also assume the data rate, 𝑟, for RF link using Ka +and L bands, and FSO are different, denoted as 𝑟𝑘𝑎, 𝑟𝑘𝑢, 𝑟𝑙, +and 𝑟𝑜. To simplify the notations, the data rate on the Ka-band +link is denoted as 𝑟𝑘. +The distances between a GS as a SatNetOps Center and the +source NGSO satellite and source GEO satellite are 𝑑0 and +𝑑1, respectively. The distance from a source GEO satellite to a +target GEO satellite is 𝑑2, and the distance from the destination +GEO satellite to a destination LEO satellite is 𝑑4. The distance +from the source LEO satellite to the destination LEO satellite +is 𝑑3. Let the total path length of these hops be 𝐿ℎ. In the GLM +scheme, 𝐿ℎ = 𝑑1+𝑑2+𝑑4, and the LLM scheme, 𝐿ℎ = 𝑑0+𝑑3. +Suppose the number of hops in a SatNetOps scheme is 𝑛ℎ. The +latency measure can be expressed as follows: +𝐷 = +𝑛ℎ +∑︁ +𝑖=1 +� 𝑑(𝑖) +𝑐 ++ 𝑀/𝑟(𝑖) + 𝑚 + 𝑘 +� +, +(1) +where 𝑑(𝑖) and 𝑟(𝑖) are the distance and data rate of the 𝑖th +hop. +The reliability measure is defined as the overall reliability +of all links: +Φ = (1 − +𝑛ℎ +� +𝑖=1 +(1 − 𝜙(𝑖))), +(2) +where 𝜙(𝑖) is the reliability of a link on a path. +In an NGSO satellite constellation with a homogeneous +platform, we can assume the reliability of a link has the +same reliability. The reliability can be modeled with multiple +factors, such as satellite system dependability, mean-time-to- +failure (MTTF), link stability, etc. Here we mainly consider +the general factors affecting the link reliability from the +communications perspective (e.g., propagation characteristics +of RF and FSO signals, etc.) A GEO satellite usually has +a longer design lifespan than an NGSO satellite. The GEO +satellites’ links (i.e., ground-to-space and space-to-space) are +readily accessible. To these reasons, the reliability of a GEO +satellite link is assumed to be higher than that of an NGSO link +[8], i.e., 𝜙(𝑖)𝐺𝐸𝑂 > 𝜙(𝑖)𝑁 𝐺𝑆𝑂. Due to the different designs +and deployment of an FSO system, the reliability between RF +and FSO systems will not be compared. +TABLE I +EVALUATION SCENARIOS S1-S4 +GS-LEO +GS-GEO +LEO-LEO +GEO-GEO +GEO-LEO +S1 +RF (Ka) +RF (Ka) +RF (Ka) +RF (Ka) +RF (Ka) +S2 +FSO +FSO +FSO +FSO +FSO +S3 +FSO +FSO +RF (Ka) +RF (Ka) +RF (Ka) +S4 +FSO +FSO +RF (Ka) +RF (Ka) +RF (L) +TABLE II +KEY SIMULATION PARAMETERS +Parameter +Value +Notes +𝑟𝑜,𝑔𝑙 +1.8 Gbps [7] +Rate of the FSO GEO-LEO link +𝑟𝑟,𝑔𝑙 +324 Mbps [8] +Rate of the RF GEO-LEO link +𝑟𝑘 +324 Mbps +Rate of the Ka-band link +𝑟𝑙 +150 kbps +Rate of of the L-band link +𝑀 +{512, 1024} B +TF size of a TC packet +𝑇 +24 hr +Mission duration +𝑇𝑠𝑡𝑎𝑟𝑡 +2022-01-01 22:23:24 +Mission start date and time +𝑇𝑒𝑛𝑑 +2022-01-02 22:23:24 +Mission end date and time +𝑇𝑠𝑎𝑚𝑝𝑙𝑒 +600 s +Sample time +𝑘 +100 𝜇s +Avg. processing delay +𝑚 +{0, 100} 𝜇s +Avg. queuing delay +𝜙1(𝑖) +0.998 +Reliability of a LEO ISL +𝜙2(𝑖) +0.999 +Reliability of a GEO ISL +𝜙3(𝑖) +0.999 +Reliability of a GEO-LEO link +IV. PERFORMANCE EVALUATION +The performance of the proposed SatNetOps TC missions +is evaluated in MATLAB simulations based on a satellite +scenario shown in Fig. 4. In the simulations, we generate the +ephemeris data from the existing Inmarsat-4 GEO satellites +and Telesat LEO satellites in polar orbit from the public + +SatNetOpsCenterJOURNAL OF LATEX CLASS FILES, VOL. 00, NO. 0, 00 2022 +4 +Fig. 4. Satellite scenario view +filing. The GEO satellites ephemeris are propagated based on +the public two-line element (TLE) data. The Telesat polar +constellation can cover the polar regions with the optical +payloads, where the inclination is 98.98◦ and altitude is 1015 +km with 78 satellites in 6 orbits. The GS chosen is in +Iqaluit, the capital city of Nunavut in northern Canada, with a +minimum elevation angle of 30◦. To generalize the simulation +scenarios, we consider the RF and FSO options on space- +to-space and ground-to-space links in four typical scenarios, +named S1-S4. These scenarios are displayed in Table I, where +S4 considers the L-band satellite link for GLM considering the +legacy satellite communication payloads; and RF (Ka) and RF +(L) indicate a link uses Ka and L band, respectively. +The key simulation parameters are shown in Table II, +where The data rates are based on the published results. +Our simulations aim to obtain generalizable results. All LEO +satellites in a constellation will get a chance to be a target +satellite in iterations, and our results are averaged over all +iterations over a 24-hr mission to obtain sufficient data. +The assumptions we made in the simulations are based +on the following justifications. We consider RF and optical +payloads in this case. For the RF payload, we consider L-band +and Ka-band, and adopted the parameters reported in [7], [8]. +For the GLM solution, we assume the communication link(s) +exist between GEO and LEO networks. The L-band has been +widely available on legacy GEO/LEO satellites, and it can +provide a good indication for the use of the S-band in some +communications satellites. The Ka-band has been a popular +option for recent satellites for broadband access. Based on the +CCSDS standards for space packet protocol (SPP) and TC- +SDLP [2], [9], an SPP packet can have a maximum length of +65542 B, and transfer frame (TF) in TC-SDLP has a maximum +length of 1024 B. Thus, we let 𝑀 = {512, 1024} B in the +experimentation. +When transmitting the 64 KB SPP packet, there is a seg- +mentation process where the packet will be split into multiple +TFs with no re-transmission. This process is compatible with +Type-A and Type-B services in TC-SDLP, as Type-A service +allows re-transmissions but Type-B does not support it. For the +processing delay per satellite note, we consider the parameter +from the experimental study from [10], where the mean delay +for UDP/ICMP payload size ranging from 32 B to 1450 B is +around 100 𝜇s. Since these payload sizes match the values of +𝑀 for our simulations, we let 𝑘 = 100𝜇s. As for the queuing +delay 𝑚, although it depends on various factors, such as traffic, +buffer configuration, and algorithmic implementations, it is +reasonable to assume zero to a slight latency in the same scale +of 𝑘, in this case, 𝑚 = {0, 100}𝜇s. +(a) +(b) +Fig. 5. Latency performance in S1-S4 where 𝑘 = 100𝜇s and 𝑚 = 0𝜇s. (a) +𝑀=512 B (b) 𝑀=1024 B +Fig. 6. Overall latency performance in S1-S4 (𝑀 = 1024) for three cases: +(1) 𝑚 = 0, 𝑘 = 0; (2) 𝑚 = 0, 𝑘 = 100; and (3) 𝑚 = 100, 𝑘 = 100 +A. Simulation Results +As shown in Fig. 5, when 𝑀=512 B, S2 has the lowest +latency in LLM and GLM. The mean latency values for all +destination LEO satellites of LLM and GLM are 6.9 ms and +45.6 ms, respectively. Due to the small size of the TC packet, +the latency variations for LLM and GLM in S1-S3 are small. +For GLM, the latency in S4 increases significantly due to the +low data rate on the last GEO-LEO link, where the mean +latency value is 72.9 ms. We can also see the latency in LLM +has a correlation to the hop count (as shown in Fig. 9). When +𝑀 is increased to 1024 B, LLM maintains its overall low +latency compared to GLM, and S2 has the lowest latency in +all scenarios, where its mean latency is similar to the case of +𝑀 = 512 due to the high data rate on the FSO links. The +latency of GLM in S4 is increased to 100.2 ms due to the + +INMARSAT 4-F1 +INMARSAT 4-F2 +INMARSAT +4-F30.08 +0.07 +0.06 +0.05 +ncy +0.04 +_ater +LLM +(S1) +GLM +(S1) +0.03 +LLM +(S2) +GLM +(S2) + LLM +(S3) +0.02 +GLM +(S3) + LLM +(S4) +GLM +(S4) +0.01 +0 +0. +10 +20 +30 +40 +50 +60 +70 +80 +Dest. LEO Satellite ID0.1 +0.09 +LLM +(S1) +GLM +(S1) +0.08 +LLM +(S2) +GLM +(S2) +LLM +(S3) +0.07 +GLM +(S3) +LLM +(S4) +GLM +(S4) +0.06 +tency +0.05 +0.04 +0.03 +0.02 +0.01 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Dest. LEO Satellite ID0.1 +0.08 +S +0.06 +Latency ( +0.04 +0.02 +0 +S1 +S2 +S3 +S4 +■LLM(m=0, k=0) +■LLM (m=0, k=100) +■LLM (m=100, k=100) +GLM(m=0.k=0) +■GLM(m=0.k=100) +■GLM(m=100.k=100)JOURNAL OF LATEX CLASS FILES, VOL. 00, NO. 0, 00 2022 +5 +(a) +(b) +Fig. 7. Latency performance in S1-S4 for an SPP packet transfer in (a) LLM +and (b) GLM schemes where 𝑘 = 100, 𝑚 = 0, 𝑀 = 1024 +Fig. 8. Overall latency performance for an SPP packet transmission where +𝑀 = 512 and 𝑀 = 1024 +increased size of the TC packet. +In Fig. 6, the average latency performance for all satellites +is shown, where we can see the case when 𝑚 = 0, 𝑘 = 100 +has lower latency than the case when 𝑚 = 100, 𝑘 = 100. To +demonstrate a lower bound we can achieve in the proposed +schemes, we plot the case when 𝑚 = 0, 𝑘 = 0, indicating there +is no process and queuing delays on satellite nodes, and this +case shows the lowest latency than the previous two cases. +Now let us evaluate the scenario when an SPP packet with +the size of 65542 B is transmitted. There is a segmentation +process where the packet is split into TC packets subject to the +value of 𝑀. In Fig. 7, we can see LLM has the lowest latency +than GLM. In Fig. 7(b), two subplots show the latency values +for S1-S3 and S4, respectively, where the use of the L-band +on the GEO-LEO link in S4 still results in the lowest latency +performance. In Fig. 8, the overall latency performance for +all destination LEO satellites is shown for LLM and GLM, +where we can see that the average latency decreases when 𝑀 +increases due to less segmented packets. LLM in S2 has the +lowest latency of 881.3 ms, and GLM in S4 has the worst +latency of 9.3343 s. When 𝑀 = 512 B, GLM takes 6.624 to +10.51 times longer than LLM to transfer the SPP packet, while +when 𝑀 = 1024 B, GLM takes 6.612 to 14.31 times longer +than LLM to transmit the SPP packet. +Fig. 9. Mean hop count, path length, and reliability in all scenarios +Fig. 10. Reliability of LLM when 𝜙1(𝑖) decreases +Fig. 9 shows the mean hop count, mean path length, and +reliability. We can see in Fig. 9, the mean hop count and path +length of GLM are steadier than those of LLM with regard to +the destination LEO satellites. This is due to the constant GEO +satellite availability to GS as their orbital period matches the +Earth’s rotation. The overall mean path length values of LLM +and GLM are 1.8682e+07 m and 1.3539e+08 m, respectively. +This means the path length of LLM is 7.2475 times shorter +than GLM. The reliability performance is related to the type +of links in the path based on the assumption that the reliability +of an LEO-LEO ISL is slightly less than that of a GEO- +LEO/GEO ISL. In addition, the maximum and minimum mean +hop counts per destination LEO satellite in LLM are 8.5798 +and 4.6723, respectively. For all events, the maximum and +minimum hop counts are 17 and 0, respectively, where 0 hop +indicates the GS can directly contact the destination satellite. +The hop count is expected to increase for a large constellation +of LEO satellites due to the shorter distance between satellites. +To see the extended scheme where LLM and GLM are used +in parallel, referred to as “LLM-GLM” in the reliability plot, +we can see the reliability can be significantly increased to over +99.99%. +In order to show how 𝜙1(𝑖) changes the overall reliability +of LLM, Fig. 10 is shown. The results in Fig. 10 indicate that +although LLM can reduce the latency compared to GLM, the +cost is the overall reliability, which may result in the trans- +mission impairments that may lead to the need for additional +mechanisms for mitigation. + +10 +9 +8 +7 +S +6 +Latency +5 +4 +3 +2 +1 +0 +S1 +S2 +S3 +S4 +■LLM (m=0,k=100,M=512) +LLM (m=0, k=100, M=1024) +GLM (m=0, k=100,M=512) +■GLM (m=0, k=100, M=1024)Mean Hop Count +0 +10 +20 +30 +40 +50 +60 +70 +80 +("T) +X107 +15 +Mean Path Length +10 +LLM --GLM +LLM-GLM +5 +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.995 +liability +0.99 +0.985 +R +0.98 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Dest. LEO Satellite IDLLM (Φ = 0.999) +LLM (=0.998) +LLM (Φ=0.996) +...- LLM (Φ=0.994) +0.995 +0.99 +Reliability of LLM +0.985 +0.98 +0.975 +0.97 +0.965 +0.96 +0.955 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Dest. LEO Satellite IDLLM(S1) +LLM(S2) +-LLM(S4) +0.6 +0.55 +Latency +0.5 +0.45 +0.4 +0.35 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Dest. LEO Satellite ID3 +GL.M +(S1) +GLM +(S2) +GLM (S3) +@ 2.95 +Latency +2.9 +2.85 +0 +10 +20 +30 +40 +50 +60 +70 +80 +6.5 +GLM (S4) +@ 6.45 +Latency +6.4 +6.35 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Dest. LEO Satellite IDJOURNAL OF LATEX CLASS FILES, VOL. 00, NO. 0, 00 2022 +6 +B. Discussion of the Results +To make the proposed SatNetOps schemes work, we need +to ensure access status between a GS and the satellites. The +probability of contact between a GS and a direct satellite +determines the chances of a successful scheme execution. +When evaluating the LLM scheme, there are 11310 data +entries where we noticed that there are 2028 occurrences +that GS has no immediate contact with LEO satellites. These +occurrences are ruled out in data analysis to have a fair +comparison between the GLM and LLM schemes. Although +we used the GS in the northern region where other LEO +constellations may have poor or no coverage, our proposed +schemes can work in different satellite constellations (such as +another LEO/MEO constellation) and GS locations. +For compatibility with the heterogeneous payload and con- +figurations on satellites, a relatively conservative data rate for +Ka-band links is adopted in our evaluation. We noticed there +the data rate can be higher, e.g., 600 Mbps, as reported in [11], +where our results can still be used as an performance indication +of the proposed schemes in this case. Subject to a specific +configuration for an application, e.g., beams configuration and +a coding & modulating scheme, a fine-grained performance +can be further derived. Such fine-grained results may be +generated on a specific FSO setup due to the currently non- +standard configurations on different satellite platforms. +The selection of 𝜙 in a real network will also have an +impact on the reliability. When modelling the reliability with +the considerations of system-level component dependability, a +lower bound performance result may be obtained. The results +shown in Fig. 10 also indicate the additional measures that may +be required to compensate for the cost of reduced reliability for +LLM. This also indicates that the hop count on a path for a TC +message transfer should be maintained at a reasonable level. +In a larger constellation than the one used in experimentation, +when a path contains many hops, the reliability may be +reduced and the 𝑘 and 𝑚 on the satellite nodes may further +introduce latency in the LLM scheme. Therefore, there is a +trade-off in an LLM scheme that requires careful analysis +before tailoring the LLM parameters for a specific mission +and LEO satellite constellation. +For the generality, the single flow scenario is considered +in our evaluation. Our results can be extended to multi-flow +scenarios subject to the implementations considering RF and +FSO for space-to-space and ground-to-space links. In addition, +extra delays may be introduced if access and specific routing +schemes are employed. +V. CONCLUSION +The proposed SatNetOps approach provides a new way +of addressing the increasing operations challenges imposed +by the upcoming NGSO satellite constellations. This paper +validates the effectiveness of the proposed approach with two +feasible schemes. These schemes can be applied or extended to +other scenarios for TM/TC and network management missions +where timing and reliability performance needs to be assured. +There is still much room for future contributions. For example, +using RF/FSO channel models, data rates subject to coding and +modulation scheme, and additional scenarios using different +NGSO satellite constellations will be explored in future work. +ACKNOWLEDGMENT +This work was supported by the High-Throughput and +Secure Networks Challenge program of National Research +Council Canada. We also acknowledge the support of the +Natural Sciences and Engineering Research Council of Canada +(NSERC), [funding reference number RGPIN-2022-03364]. +REFERENCES +[1] D. Paikowsky, “What Is New Space? The Changing Ecosystem of Global +Space Activity,” New Space, vol. 5, no. 2, pp. 84–88, 2017. +[2] “Recommendation for Space Data System Standards – TC SPACE +DATA LINK PROTOCOL,” The Consultative Committee for Space Data +Systems, Washington, DC, USA, Standard, 2021. +[3] H. Nishiyama, Y. Tada, N. Kato et al., “Toward optimized traffic +distribution for efficient network capacity utilization in two-layered +satellite networks,” IEEE Transactions on Vehicular Technology, vol. 62, +no. 3, pp. 1303–1313, March 2013. +[4] N. Pachler, I. del Portillo, E. F. Crawley et al., “An updated comparison +of four low earth orbit satellite constellation systems to provide global +broadband,” in 2021 IEEE International Conference on Communications +Workshops (ICC Workshops), June 2021, pp. 1–7. +[5] S. +Cakaj, +“The +parameters +comparison +of +the +“starlink” +leo +satellites +constellation +for +different +orbital +shells,” +Frontiers +in +Communications and Networks, vol. 2, 2021. [Online]. Available: +https://www.frontiersin.org/article/10.3389/frcmn.2021.643095 +[6] R. Ge, D. Bian, K. An et al., “Performance analysis of cooperative +nonorthogonal multiple access scheme in two-layer geo/leo satellite +network,” IEEE Systems Journal, pp. 1–11, 2021. +[7] H. Zech, F. Heine, D. Tr¨ondle et al., “LCT for EDRS: LEO to +GEO optical communications at 1,8 Gbps between Alphasat and +Sentinel 1a,” in Proc.SPIE, vol. 9647, oct 2015. [Online]. Available: +https://doi.org/10.1117/12.2196273 +[8] E. B. Clements, “Probabilistic methods for systems engineering with +application to nanosatellite laser communications,” Ph.D. dissertation, +Dept. of Aeronautics and Astronautics, MIT, Cambridge, MA, USA, +2018. +[9] “Recommendation for Space Data System Standards – SPACE PACKET +PROTOCOL,” The Consultative Committee for Space Data Systems, +Washington, DC, USA, Standard, 2020. +[10] P. Carlsson, D. Constantinescu, A. D. Popescu et al., “Delay perfor- +mance in ip routers,” 2004. +[11] J. Sedin, L. Feltrin, and X. Lin, “Throughput and capacity evaluation of +5g new radio non-terrestrial networks with leo satellites,” in GLOBE- +COM 2020 - 2020 IEEE Global Communications Conference, 2020, pp. +1–6. +Peng Hu received his Ph.D. degree in Electrical Engineering from Queen’s +University, Canada. He is currently a Research Officer at the National +Research Council of Canada, and Adjunct Professor at the Cheriton School +of Computer Science and the Department of Statistics and Actuarial Science, +University of Waterloo, Canada. He has served as an associate editor of the +Canadian Journal of Electrical and Computer Engineering, and as a member +on the IEEE Sensors Standards committee and on the organizing and tech- +nical boards/committees of industry consortia and international conferences +including AllSeen Alliance, DASH7, IEEE PIMRC’17, and IEEE AINA’15. +His current research interests include satellite-terrestrial integrated networks, +autonomous networking, and Internet of Things systems. + diff --git a/adE2T4oBgHgl3EQfFAZQ/content/tmp_files/load_file.txt b/adE2T4oBgHgl3EQfFAZQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..81c2bf4c8d280a4828d930f223427c16b26eb5ca --- /dev/null +++ b/adE2T4oBgHgl3EQfFAZQ/content/tmp_files/load_file.txt @@ -0,0 +1,422 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf,len=421 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 00, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 0, 00 2022 1 SatNetOps: Toward Multi-Layer Networking for Satellite Network Operations Peng Hu, Senior Member, IEEE Abstract—Recent advancements in low-Earth-orbit (LEO) satellites aim to bring resilience, ubiquitous, and high-quality service to future Internet infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' However, the soaring number of space assets, increasing dynamics of LEO satellites and expanding dimensions of network threats call for an enhanced approach to efficient satellite operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To address these press- ing challenges, we propose an approach for satellite network operations based on multi-layer satellite networking (MLSN), called “SatNetOps”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Two SatNetOps schemes are proposed, referred to as LEO-LEO MLSN (LLM) and GEO-LEO MLSN (GLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The performance of the proposed schemes is evaluated in 24-hr satellite scenarios with typical payload setups in simu- lations, where the key metrics such as latency and reliability are discussed with the consideration of the Consultative Committee for Space Data Systems (CCSDS) standard-compliant telemetry and telecommand missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Although the SatNetOps approach is promising, we analyze the factors affecting the performance of the LLM and GLM schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The discussions on the results and conclusive remarks are made in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Index Terms—Satellites, Telecommand, Telemetry, Space, Data Link, Network Operations I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' INTRODUCTION With the recent advancements and deployments of low- Earth-orbit (LEO) satellites, the upcoming space assets will approach a unprecedented volume in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' These space assets, while providing much convenience and resiliency for global telecommunications networks, demand efficient, reliable, and robust operations for the satellite networks in “New Space” [1] ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' One of the essential services to support the new satellite operations is telecommand (TC) and telemetry (TM) missions, which rely on the efficient transmissions of packets for space- craft operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' From a communications perspective, TC and TM missions share a similar process where TC messages are transferred from a ground station (GS) to a target satellite, while TM messages are transferred in the reverse direction and the packet transmission follow a similar pattern based on the Consultative Committee for Space Data Systems (CCSDS) standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We therefore focus on the discussion of represen- tative TC missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In the traditional approach, TC messages are sent or received when a direct contact opportunity occurs between an operations GS and a target satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' CCSDS [2] has standardized the transmissions of TC packets for space- to-space and ground-to-space communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The TC packets are usually small in size but require high reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' However, such a traditional approach faces challenges that significantly reduce the efficiency of satellite operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For example, the TC message transmissions depend on the access states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Based on our 24-hr access analysis of typical LEO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Access analysis of typical LEO satellite constellations satellite constellations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 1, the average access opportunity per satellite in three typical constellation is very low for mostly < 2% chance of access to two typical GS locations (one in northern Canada and one in western Canada where most LEO satellites have coverage) with a minimum elevation 25◦ over a 24-hr mission, which clearly shows the limitations of the traditional operations approach in terms of efficiency and real-time communication capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Although satellites are traditionally operated in isolation from their orbiting counterparts, communicating only with ground-based infrastructure with high delays leaves them susceptible to the consequences of various anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Further, the space network has not been fully utilized for operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' These challenges call for efficient and resilient operations of a spacecraft and the entire satellite network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Recently, using a legacy geostationary (GEO) satellite net- work for satellite operations missions has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The constant positions of GEO satellites relative to ground components at a high altitude provide broad coverage of LEO satellites at a low altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For example, in November 2020, Inmarsat reportedly started a new service called Inter-satellite Data Relay System (IDRS), providing real-time links between GEO and LEO satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' With the recent launch of I-6F1 GEO satellites with Ka/L-band payloads, new GEO satellites are expected to be deeply integrated into other medium-Earth- orbit (MEO) or LEO satellites into the 5G network as part of an “ORCHESTRA” network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Although the space industry shows intensive interest in an option of employing legacy or new satellite fleet for operations, there are no technical details or formal studies available about established schemes or assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In fact, such an option would face practical efficiency and timing challenges considering the distance between GEO and LEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To reduce the distance for packet transmission, LEO satellite networks may be considered in combination with GEO satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The idea falls into the multi- layer satellite networking (MLSN) concept where satellites in different orbits are viewed in multiple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The layers may include various shells of a large constellation, different constellations, and different Earth orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' However, MLSN has been mainly considered for data traffic routing, which differs arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='03641v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='NI] 9 Jan 2023 CA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='6 Iqaluit, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='4 Station Access Satellite (%) I Station: 2 round 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='5 60 20 A0 Per s 150 200 2 round Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='6 Station: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='5 20 40 100 50 60 500 1000 3000 3500 Sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' ID (Telesat Polar) Sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' ID (Iridium NEXT) Sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' ID (Starlink)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 00, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 0, 00 2022 2 from the TM/TC transmissions in practice for security and reliability considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The solutions to utilizing MLSN for operations missions are lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' This paper aims to address the new challenges of the operational needs faced by the recent growth of space assets in a timely manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We propose a new approach considering MLSN, collectively called “SatNetOps”, which can support efficient operations of non-geostationary (NGSO) satellite op- erations and enable the design of MLSN schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To the best of our knowledge, this is the first work addressing the new challenges for NGSO satellite network operations and providing a formal evaluation of MLSN-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The main contributions of the paper are summarized as follows: We identify the pressing challenges of the current satellite operations and propose the “SatNetOps” approach with two generic schemes devised for TC missions, called GEO-LEO MLSN (GLM) and LEO-LEO MLSN (LLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We evaluate the performance in terms of latency and reliability in different scenarios considering the feasible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We adopt the popular communications payloads in the recent satellites, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', the radio-frequency (RF) and free- space optical (FSO) communications technologies in space-to-space and space-to-ground communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The remainder of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The re- lated work is discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The proposed SatNetOps schemes for TC message transfer are discussed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The evaluation of the proposed schemes is discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The conclusive remarks are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' RELATED WORK Compared to GEO satellites, throughput and latency are usually considered the two significant advantages of the current LEO satellite systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The communications between layers of satellite networks can be linked to the concept of MLSN [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Recent work and deployment plans of LEO satel- lite constellations are centered on throughput improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The multi-layer networking may occur through multiple shells of a large constellation with LEO satellites, such as Starlink’s mega-constellation [4], although the cross-shell networking is considered complex [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Pachler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [4] showed the use of optical inter-satellite links (ISLs) on Telesat, Amazon Kuiper, and Starlink constellations can almost double the system throughput of each system compared to the non-optical ISLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' A cooperative communication multi-access scheme in MLSN was recently proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For the LEO satellite operations, a real-world solution has recently realized by Inmarsat in late 2021 for using the GEO satellites for operations missions of LEO satellites and the SES also implied the benefits of using their upcoming o3b mPOWER fleet for satellite networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' However, MLSN schemes and the discussion of the related factors such as satellite and GS communications, MLSN strategies, and mission parameters are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' CCSDS has developed the space data link protocols for TC missions which may be used with other upper layer CCSDS protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The latest issue of TC space data link protocol (TC- SDLP) released in October 2021 [2] supports two types of services: sequence-controlled (Type-A) and expedited (Type- B) services for missions with different priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Although both services have the same frame format, addressing, seg- mentation and blocking mechanisms, Type-A services support the Automatic Repeat Request (ARQ) mechanism for flow control, while Type-B services do not have flow control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Type- B services are used in “exceptional operational circumstances” such as spacecraft recovery, or flow control is provided at the upper layers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Furthermore, these services in TC-SDLP only support unidirectional and asynchronous services where no predefined timing rules are specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' There is also no existing study on the recommendations for the timing performance for the operations of LEO satellite constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The timing performance of satellite communications needs to consider the recent developments of the satellite platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The perfor- mance of the TC missions also depends on the communication payload, where the Ka-band and S/L bands are broadly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Since the first test of optical links for space missions in November 2014 [7], FSO communication is expected to be well adopted by the space industry in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The recent tests initiated by the European Space Agency (ESA), the National Aeronautics and Space Administration (NASA) and the commercial LEO satellite constellations further indicate the increasingly planned use of FSO on space-to-space and ground-to-space links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' However, the study considering these payloads for TC missions is lacking in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' SATNETOPS AND SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' SatNetOps The SatNetOps approach aims to enhance the efficiency of a number of spacecraft operations through satellite networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' SatNetOps intends to improve real-time and reliable commu- nications for satellite operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' An example SatNetOps case is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 2, where a SatNetOps Center attempts to initiate a TC mission by sending a TC packet to a destination NGSO satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The packet traverses in multiple hops through one or more satellite networks, where the path established can also be used for transferring TM packets or control messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Proposed SatNetOps Schemes Here we propose two general schemes considering GEO and LEO satellites, called GLM and LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' GLM uses GEO satellites to relay packets to the destination satellite, while LLM uses LEO satellites to forward packets to the destination satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' GLM uses the GS-GEO, LEO-GEO, and GEO-GEO links, while LLM uses GS-LEO and LEO-LEO links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The brief descriptions of the schemes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For both schemes, the TC mission preparation will determine the TC message, destination/target LEO satellite in a constella- tion, and the GEO satellites to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For LLM, the path calculation can occur dynamically due to constant satellite movements, where the inter- and intra-plane satellites are calculated based on the well-adopted configuration that each LEO satellite can access up to four neighbouring satellites in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Each LEO/GEO satellite will calculate a minimum elevation angle to ensure the next satellite is in line of sight when choosing the next hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 00, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 0, 00 2022 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Illustration of the proposed approach for SatNetOps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Flowcharts for the proposed LLM and GLM schemes A SatNetOps scheme removes the need for direct contact times between a GS and a target satellite and ensures timing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Therefore, let us discuss the latency measure for GLM/LLM-based TC missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The latency 𝐷 of a TC mes- sage transfer mission consists of four standard components: propagation delay, 𝐷 𝑝𝑎, transmission delay, 𝐷𝑡, processing delay, 𝐷 𝑝𝑐, and queuing delay, 𝐷𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Since the TC packets are small, the average queuing delay ¯𝐷𝑞 per satellite can be assumed as a constant, 𝑚, and the average ¯𝐷 𝑝𝑐 per satellite assumed to be a constant, 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The size of the TC packet in bytes is 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The speed of light in FSO links is 𝑐, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='998 × 108 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We also assume the data rate, 𝑟, for RF link using Ka and L bands, and FSO are different, denoted as 𝑟𝑘𝑎, 𝑟𝑘𝑢, 𝑟𝑙, and 𝑟𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To simplify the notations, the data rate on the Ka-band link is denoted as 𝑟𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The distances between a GS as a SatNetOps Center and the source NGSO satellite and source GEO satellite are 𝑑0 and 𝑑1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The distance from a source GEO satellite to a target GEO satellite is 𝑑2, and the distance from the destination GEO satellite to a destination LEO satellite is 𝑑4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The distance from the source LEO satellite to the destination LEO satellite is 𝑑3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Let the total path length of these hops be 𝐿ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In the GLM scheme, 𝐿ℎ = 𝑑1+𝑑2+𝑑4, and the LLM scheme, 𝐿ℎ = 𝑑0+𝑑3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Suppose the number of hops in a SatNetOps scheme is 𝑛ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The latency measure can be expressed as follows: 𝐷 = 𝑛ℎ ∑︁ 𝑖=1 � 𝑑(𝑖) 𝑐 + 𝑀/𝑟(𝑖) + 𝑚 + 𝑘 � , (1) where 𝑑(𝑖) and 𝑟(𝑖) are the distance and data rate of the 𝑖th hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The reliability measure is defined as the overall reliability of all links: Φ = (1 − 𝑛ℎ � 𝑖=1 (1 − 𝜙(𝑖))), (2) where 𝜙(𝑖) is the reliability of a link on a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In an NGSO satellite constellation with a homogeneous platform, we can assume the reliability of a link has the same reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The reliability can be modeled with multiple factors, such as satellite system dependability, mean-time-to- failure (MTTF), link stability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Here we mainly consider the general factors affecting the link reliability from the communications perspective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', propagation characteristics of RF and FSO signals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=') A GEO satellite usually has a longer design lifespan than an NGSO satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The GEO satellites’ links (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', ground-to-space and space-to-space) are readily accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To these reasons, the reliability of a GEO satellite link is assumed to be higher than that of an NGSO link [8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', 𝜙(𝑖)𝐺𝐸𝑂 > 𝜙(𝑖)𝑁 𝐺𝑆𝑂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Due to the different designs and deployment of an FSO system, the reliability between RF and FSO systems will not be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' TABLE I EVALUATION SCENARIOS S1-S4 GS-LEO GS-GEO LEO-LEO GEO-GEO GEO-LEO S1 RF (Ka) RF (Ka) RF (Ka) RF (Ka) RF (Ka) S2 FSO FSO FSO FSO FSO S3 FSO FSO RF (Ka) RF (Ka) RF (Ka) S4 FSO FSO RF (Ka) RF (Ka) RF (L) TABLE II KEY SIMULATION PARAMETERS Parameter Value Notes 𝑟𝑜,𝑔𝑙 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='8 Gbps [7] Rate of the FSO GEO-LEO link 𝑟𝑟,𝑔𝑙 324 Mbps [8] Rate of the RF GEO-LEO link 𝑟𝑘 324 Mbps Rate of the Ka-band link 𝑟𝑙 150 kbps Rate of of the L-band link 𝑀 {512, 1024} B TF size of a TC packet 𝑇 24 hr Mission duration 𝑇𝑠𝑡𝑎𝑟𝑡 2022-01-01 22:23:24 Mission start date and time 𝑇𝑒𝑛𝑑 2022-01-02 22:23:24 Mission end date and time 𝑇𝑠𝑎𝑚𝑝𝑙𝑒 600 s Sample time 𝑘 100 𝜇s Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' processing delay 𝑚 {0, 100} 𝜇s Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' queuing delay 𝜙1(𝑖) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='998 Reliability of a LEO ISL 𝜙2(𝑖) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='999 Reliability of a GEO ISL 𝜙3(𝑖) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='999 Reliability of a GEO-LEO link IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' PERFORMANCE EVALUATION The performance of the proposed SatNetOps TC missions is evaluated in MATLAB simulations based on a satellite scenario shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In the simulations, we generate the ephemeris data from the existing Inmarsat-4 GEO satellites and Telesat LEO satellites in polar orbit from the public SatNetOpsCenterJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 00, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 0, 00 2022 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Satellite scenario view filing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The GEO satellites ephemeris are propagated based on the public two-line element (TLE) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The Telesat polar constellation can cover the polar regions with the optical payloads, where the inclination is 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='98◦ and altitude is 1015 km with 78 satellites in 6 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The GS chosen is in Iqaluit, the capital city of Nunavut in northern Canada, with a minimum elevation angle of 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To generalize the simulation scenarios, we consider the RF and FSO options on space- to-space and ground-to-space links in four typical scenarios, named S1-S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' These scenarios are displayed in Table I, where S4 considers the L-band satellite link for GLM considering the legacy satellite communication payloads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' and RF (Ka) and RF (L) indicate a link uses Ka and L band, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The key simulation parameters are shown in Table II, where The data rates are based on the published results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Our simulations aim to obtain generalizable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' All LEO satellites in a constellation will get a chance to be a target satellite in iterations, and our results are averaged over all iterations over a 24-hr mission to obtain sufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The assumptions we made in the simulations are based on the following justifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We consider RF and optical payloads in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For the RF payload, we consider L-band and Ka-band, and adopted the parameters reported in [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For the GLM solution, we assume the communication link(s) exist between GEO and LEO networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The L-band has been widely available on legacy GEO/LEO satellites, and it can provide a good indication for the use of the S-band in some communications satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The Ka-band has been a popular option for recent satellites for broadband access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Based on the CCSDS standards for space packet protocol (SPP) and TC- SDLP [2], [9], an SPP packet can have a maximum length of 65542 B, and transfer frame (TF) in TC-SDLP has a maximum length of 1024 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Thus, we let 𝑀 = {512, 1024} B in the experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' When transmitting the 64 KB SPP packet, there is a seg- mentation process where the packet will be split into multiple TFs with no re-transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' This process is compatible with Type-A and Type-B services in TC-SDLP, as Type-A service allows re-transmissions but Type-B does not support it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For the processing delay per satellite note, we consider the parameter from the experimental study from [10], where the mean delay for UDP/ICMP payload size ranging from 32 B to 1450 B is around 100 𝜇s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Since these payload sizes match the values of 𝑀 for our simulations, we let 𝑘 = 100𝜇s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' As for the queuing delay 𝑚, although it depends on various factors, such as traffic, buffer configuration, and algorithmic implementations, it is reasonable to assume zero to a slight latency in the same scale of 𝑘, in this case, 𝑚 = {0, 100}𝜇s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Latency performance in S1-S4 where 𝑘 = 100𝜇s and 𝑚 = 0𝜇s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' (a) 𝑀=512 B (b) 𝑀=1024 B Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Overall latency performance in S1-S4 (𝑀 = 1024) for three cases: (1) 𝑚 = 0, 𝑘 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' (2) 𝑚 = 0, 𝑘 = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' and (3) 𝑚 = 100, 𝑘 = 100 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Simulation Results As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 5, when 𝑀=512 B, S2 has the lowest latency in LLM and GLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The mean latency values for all destination LEO satellites of LLM and GLM are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='9 ms and 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='6 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Due to the small size of the TC packet, the latency variations for LLM and GLM in S1-S3 are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For GLM, the latency in S4 increases significantly due to the low data rate on the last GEO-LEO link, where the mean latency value is 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='9 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We can also see the latency in LLM has a correlation to the hop count (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' When 𝑀 is increased to 1024 B, LLM maintains its overall low latency compared to GLM, and S2 has the lowest latency in all scenarios, where its mean latency is similar to the case of 𝑀 = 512 due to the high data rate on the FSO links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The latency of GLM in S4 is increased to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='2 ms due to the INMARSAT 4-F1 INMARSAT 4-F2 INMARSAT 4-F30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='05 ncy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='04 _ater LLM (S1) GLM (S1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='03 LLM (S2) GLM (S2) LLM (S3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='02 GLM (S3) LLM (S4) GLM (S4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 Dest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' LEO Satellite ID0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='09 LLM (S1) GLM (S1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='08 LLM (S2) GLM (S2) LLM (S3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='07 GLM (S3) LLM (S4) GLM (S4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='06 tency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='01 0 10 20 30 40 50 60 70 80 Dest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' LEO Satellite ID0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='08 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='06 Latency ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='02 0 S1 S2 S3 S4 ■LLM(m=0, k=0) ■LLM (m=0, k=100) ■LLM (m=100, k=100) GLM(m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='k=0) ■GLM(m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='k=100) ■GLM(m=100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='k=100)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 00, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 0, 00 2022 5 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Latency performance in S1-S4 for an SPP packet transfer in (a) LLM and (b) GLM schemes where 𝑘 = 100, 𝑚 = 0, 𝑀 = 1024 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Overall latency performance for an SPP packet transmission where 𝑀 = 512 and 𝑀 = 1024 increased size of the TC packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 6, the average latency performance for all satellites is shown, where we can see the case when 𝑚 = 0, 𝑘 = 100 has lower latency than the case when 𝑚 = 100, 𝑘 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To demonstrate a lower bound we can achieve in the proposed schemes, we plot the case when 𝑚 = 0, 𝑘 = 0, indicating there is no process and queuing delays on satellite nodes, and this case shows the lowest latency than the previous two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Now let us evaluate the scenario when an SPP packet with the size of 65542 B is transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' There is a segmentation process where the packet is split into TC packets subject to the value of 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 7, we can see LLM has the lowest latency than GLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 7(b), two subplots show the latency values for S1-S3 and S4, respectively, where the use of the L-band on the GEO-LEO link in S4 still results in the lowest latency performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 8, the overall latency performance for all destination LEO satellites is shown for LLM and GLM, where we can see that the average latency decreases when 𝑀 increases due to less segmented packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' LLM in S2 has the lowest latency of 881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='3 ms, and GLM in S4 has the worst latency of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='3343 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' When 𝑀 = 512 B, GLM takes 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='624 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='51 times longer than LLM to transfer the SPP packet, while when 𝑀 = 1024 B, GLM takes 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='612 to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='31 times longer than LLM to transmit the SPP packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Mean hop count, path length, and reliability in all scenarios Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Reliability of LLM when 𝜙1(𝑖) decreases Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 9 shows the mean hop count, mean path length, and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 9, the mean hop count and path length of GLM are steadier than those of LLM with regard to the destination LEO satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' This is due to the constant GEO satellite availability to GS as their orbital period matches the Earth’s rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The overall mean path length values of LLM and GLM are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='8682e+07 m and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='3539e+08 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' This means the path length of LLM is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='2475 times shorter than GLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The reliability performance is related to the type of links in the path based on the assumption that the reliability of an LEO-LEO ISL is slightly less than that of a GEO- LEO/GEO ISL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In addition, the maximum and minimum mean hop counts per destination LEO satellite in LLM are 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='5798 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='6723, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For all events, the maximum and minimum hop counts are 17 and 0, respectively, where 0 hop indicates the GS can directly contact the destination satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The hop count is expected to increase for a large constellation of LEO satellites due to the shorter distance between satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' To see the extended scheme where LLM and GLM are used in parallel, referred to as “LLM-GLM” in the reliability plot, we can see the reliability can be significantly increased to over 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In order to show how 𝜙1(𝑖) changes the overall reliability of LLM, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 10 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 10 indicate that although LLM can reduce the latency compared to GLM, the cost is the overall reliability, which may result in the trans- mission impairments that may lead to the need for additional mechanisms for mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 10 9 8 7 S 6 Latency 5 4 3 2 1 0 S1 S2 S3 S4 ■LLM (m=0,k=100,M=512) LLM (m=0, k=100, M=1024) GLM (m=0, k=100,M=512) ■GLM (m=0, k=100, M=1024)Mean Hop Count 0 10 20 30 40 50 60 70 80 ("T) X107 15 Mean Path Length 10 LLM --GLM LLM-GLM 5 0 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='995 liability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='985 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='98 0 10 20 30 40 50 60 70 80 Dest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' LEO Satellite IDLLM (Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='999) LLM (=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='998) LLM (Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='996) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='- LLM (Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='994) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='99 Reliability of LLM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='955 0 10 20 30 40 50 60 70 80 Dest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' LEO Satellite IDLLM(S1) LLM(S2) LLM(S4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='55 Latency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='35 0 10 20 30 40 50 60 70 80 Dest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' LEO Satellite ID3 GL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='M (S1) GLM (S2) GLM (S3) @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='95 Latency 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='85 0 10 20 30 40 50 60 70 80 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='5 GLM (S4) @ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='45 Latency 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='35 0 10 20 30 40 50 60 70 80 Dest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' LEO Satellite IDJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 00, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 0, 00 2022 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Discussion of the Results To make the proposed SatNetOps schemes work, we need to ensure access status between a GS and the satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The probability of contact between a GS and a direct satellite determines the chances of a successful scheme execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' When evaluating the LLM scheme, there are 11310 data entries where we noticed that there are 2028 occurrences that GS has no immediate contact with LEO satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' These occurrences are ruled out in data analysis to have a fair comparison between the GLM and LLM schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Although we used the GS in the northern region where other LEO constellations may have poor or no coverage, our proposed schemes can work in different satellite constellations (such as another LEO/MEO constellation) and GS locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For compatibility with the heterogeneous payload and con- figurations on satellites, a relatively conservative data rate for Ka-band links is adopted in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We noticed there the data rate can be higher, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', 600 Mbps, as reported in [11], where our results can still be used as an performance indication of the proposed schemes in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Subject to a specific configuration for an application, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', beams configuration and a coding & modulating scheme, a fine-grained performance can be further derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Such fine-grained results may be generated on a specific FSO setup due to the currently non- standard configurations on different satellite platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The selection of 𝜙 in a real network will also have an impact on the reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' When modelling the reliability with the considerations of system-level component dependability, a lower bound performance result may be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 10 also indicate the additional measures that may be required to compensate for the cost of reduced reliability for LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' This also indicates that the hop count on a path for a TC message transfer should be maintained at a reasonable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In a larger constellation than the one used in experimentation, when a path contains many hops, the reliability may be reduced and the 𝑘 and 𝑚 on the satellite nodes may further introduce latency in the LLM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Therefore, there is a trade-off in an LLM scheme that requires careful analysis before tailoring the LLM parameters for a specific mission and LEO satellite constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For the generality, the single flow scenario is considered in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Our results can be extended to multi-flow scenarios subject to the implementations considering RF and FSO for space-to-space and ground-to-space links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' In addition, extra delays may be introduced if access and specific routing schemes are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' CONCLUSION The proposed SatNetOps approach provides a new way of addressing the increasing operations challenges imposed by the upcoming NGSO satellite constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' This paper validates the effectiveness of the proposed approach with two feasible schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' These schemes can be applied or extended to other scenarios for TM/TC and network management missions where timing and reliability performance needs to be assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' There is still much room for future contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' For example, using RF/FSO channel models, data rates subject to coding and modulation scheme, and additional scenarios using different NGSO satellite constellations will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported by the High-Throughput and Secure Networks Challenge program of National Research Council Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' We also acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2022-03364].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Paikowsky, “What Is New Space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' The Changing Ecosystem of Global Space Activity,” New Space, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 84–88, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [2] “Recommendation for Space Data System Standards – TC SPACE DATA LINK PROTOCOL,” The Consultative Committee for Space Data Systems, Washington, DC, USA, Standard, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Nishiyama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Tada, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', “Toward optimized traffic distribution for efficient network capacity utilization in two-layered satellite networks,” IEEE Transactions on Vehicular Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 1303–1313, March 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Pachler, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' del Portillo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Crawley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', “An updated comparison of four low earth orbit satellite constellation systems to provide global broadband,” in 2021 IEEE International Conference on Communications Workshops (ICC Workshops), June 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Cakaj, “The parameters comparison of the “starlink” leo satellites constellation for different orbital shells,” Frontiers in Communications and Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 2, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='frontiersin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='org/article/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='3389/frcmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='643095 [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Ge, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Bian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', “Performance analysis of cooperative nonorthogonal multiple access scheme in two-layer geo/leo satellite network,” IEEE Systems Journal, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 1–11, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Zech, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Heine, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Tr¨ondle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', “LCT for EDRS: LEO to GEO optical communications at 1,8 Gbps between Alphasat and Sentinel 1a,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='SPIE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 9647, oct 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='2196273 [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Clements, “Probabilistic methods for systems engineering with application to nanosatellite laser communications,” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' dissertation, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' of Aeronautics and Astronautics, MIT, Cambridge, MA, USA, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [9] “Recommendation for Space Data System Standards – SPACE PACKET PROTOCOL,” The Consultative Committee for Space Data Systems, Washington, DC, USA, Standard, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Carlsson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Constantinescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Popescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=', “Delay perfor- mance in ip routers,” 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Sedin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Feltrin, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Lin, “Throughput and capacity evaluation of 5g new radio non-terrestrial networks with leo satellites,” in GLOBE- COM 2020 - 2020 IEEE Global Communications Conference, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' Peng Hu received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' degree in Electrical Engineering from Queen’s University, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' He is currently a Research Officer at the National Research Council of Canada, and Adjunct Professor at the Cheriton School of Computer Science and the Department of Statistics and Actuarial Science, University of Waterloo, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' He has served as an associate editor of the Canadian Journal of Electrical and Computer Engineering, and as a member on the IEEE Sensors Standards committee and on the organizing and tech- nical boards/committees of industry consortia and international conferences including AllSeen Alliance, DASH7, IEEE PIMRC’17, and IEEE AINA’15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} +page_content=' His current research interests include satellite-terrestrial integrated networks, autonomous networking, and Internet of Things systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE2T4oBgHgl3EQfFAZQ/content/2301.03641v1.pdf'} diff --git a/btAzT4oBgHgl3EQfLfs5/vector_store/index.faiss b/btAzT4oBgHgl3EQfLfs5/vector_store/index.faiss new file mode 100644 index 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of a machine learning (ML) model’s train- +ing data. It has been shown that in scenarios where each class +corresponds to a different individual, such as face classifiers, +this represents a severe privacy risk. In this work, we explore a +new application for MI: the extraction of speakers’ voices from +a speaker recognition system. We present an approach to (1) re- +construct audio samples from a trained ML model and (2) ex- +tract intermediate voice feature representations which provide +valuable insights into the speakers’ biometrics. +Therefore, we propose an extension of MI attacks which +we call sliding model inversion. Our sliding MI extends stan- +dard MI by iteratively inverting overlapping chunks of the au- +dio samples and thereby leveraging the sequential properties of +audio data for enhanced inversion performance. We show that +one can use the inverted audio data to generate spoofed audio +samples to impersonate a speaker, and execute voice-protected +commands for highly secured systems on their behalf. To the +best of our knowledge, our work is the first one extending MI +attacks to audio data, and our results highlight the security risks +resulting from the extraction of the biometric data in that setup. +Index Terms: speaker recognition, model inversion, privacy +1. Introduction +Privacy analysis of audio data has shown that speech parame- +ters, such as accent, rhythm, or acoustic properties of speech in- +herently carry biometric information about the speakers, such as +their age, gender, physical health, and geographical origin [1]. +Therefore, it is important for machine learning (ML) models in +speaker recognition not to leak information about their training +data. However, recent research [2, 3, 4] suggests that ML mod- +els are, in general, vulnerable to privacy attacks. One particu- +lar attack is model inversion (MI) [2] which allows an attacker +to retrieve abstract representations for individual classes of the +target model’s training data. With speaker recognition systems +treating each individual as their own class, MI attacks have the +potential to cause severe privacy breaches [2]. So far, the fea- +sibility of MI attacks on speaker recognition systems and audio +data has never been tested, thus, the question if information on +the speakers can be maliciously retrieved remained open. +We are the first to show how to adapt and apply MI attacks +for audio data. We do so by targeting SincNet, a state-of-the-art +neural network (NN)-based speaker recognition model [5, 6]. +We show that MI attacks are able to infer both entire audio sam- +ples and d-vectors as intermediate representations of the speak- +ers’ voice characteristics from the trained target model. Further, +we propose the sliding model inversion, a novel form of the +standard MI attack that leverages sequential processing prop- +erties of the audio data to improve inversion success. While +with standard MI, the target model successfully identifies up +∗Authors contributed equally. +to 54% of the inverted audio samples as their correct speaker +class, with our novel attack, we achieve to up to 90% accu- +racy. Also, our sliding MI manages to decrease the distance +between original and inverted sample in the d-vector represen- +tation, hence, yielding higher fidelity inversions. For directly +inverting d-vectors, our experiments show that even standard +MI achieves 100% identification success. These results high- +light the vulnerability of speaker recognition models to privacy +attacks. As a proof-of-concept to showcase that our MI can be +exploited as a departure point for further attacks against speaker +recognition, we, furthermore, explore using inverted audio sam- +ples as inputs for deepfake generation. Such deepfakes could be +used to fool voice identification with arbitrary speech samples +or to execute any speech command on behalf of the speakers un- +der attack. While our generated deepfakes do not perfectly fool +a human listener, as an informal evaluation conducted by the au- +thors shows, they illustrate that privacy attacks can not only be +used to disclose sensitive information about the individuals the +model was trained on; additionally, they can severely threaten +the security of systems relying on voice biometrics. Our contri- +butions can be summarized as follows: +• We successfully apply MI attacks on speaker recognition +models to invert entire audio samples and d-vectors and +experimentally evaluate what kind of random initializa- +tion works best as an input for MI attacks on audio data. +• We introduce a novel sliding MI which exploits proper- +ties of sequential and chunk-wise audio processing. +• We show the feasibility of generating deepfakes based +on the inferred audio samples. +2. Background and Related Work +The following section provides background information on +speaker recognition systems and attacks against their privacy. +Speaker recognition. +In this paper, we use a SincNet- +based [5] text-independent speaker recognition system. This +system uses NNs to extract voice features into so-called d- +vectors and adds a classification layer on top of these. The in- +put to the system consists of raw audio waves and the outputs +is a per-class probability score over all possible classes (i.e., +speakers). Overall, the system is composed of three submodels +(see Figure 1) 1) SincNet, a convolutional NN that resembles a +band-pass filter; 2) a multi-layer perceptron (MLP) calculating +the d-vectors [7]; and 3) a fully connected layer to calculate the +probabilities per speaker. It achieves a reported classification er- +ror1 of 5.772 · 10−3 on the TIMIT [8] test data set (measured at +sentence level). In its current version, SincNet does not provide +any dedicated privacy-preserving mechanisms. +1See +https://pythonlang.dev/repo/ +mravanelli-sincnet/. +arXiv:2301.03206v1 [cs.SD] 9 Jan 2023 + +Exp. 1 +SincNet +MLP +1-Layer +… +d +Exp. 2 +Figure 1: Speaker Recognition Model. The speaker recognition +model and its three submodels: SincNet obtains 1 raw audio +input and generates 2 features. These features are input to an +MLP which generates the 3 d-vectors. A single layer performs +4 classification on them. In experiment 1, we invert full audio +samples, while in experiment 2, we invert the d-vectors. +Algorithm 1 Standard Model Inversion Attack [2] +function MI(input vector x0, target class t, iterations α, pa- +tience β, minimum cost threshold γ, learning rate for gradient +descent λ) +for i ← 1, . . . , α do +xi ← xi−1 − λ · ∇c(xi−1) +if c(xi) ≥ max(c(xi−1), . . . , c(xi−β)) then +break +if c(xi) ≤ γ then +break +return [argminxi c(xi), minxi c(xi)] +Privacy +in +speaker +recognition. +The +ISO/IEC +norm +24745:2011 proposes three general requirements to ensure +individual privacy: irreversibility, renewability, and unlinka- +bility of the protected data [9]. To reach this goal in speaker +recoginition systems, several solutions have been proposed +and discussed [9]. However, aiming for any anonymization to +protect the privacy of speakers in a speaker recognition model +goes directly against the purpose of the system, which is to +identify speakers based on their individual characteristics. +MI attacks. +In MI attacks [2], the attacker exploits an ML +model’s prediction confidence for inverting individual training +classes (see Algorithm 1). More formally speaking, a MI attack +can be expressed as follows: let f be the target model under at- +tack. It is trained to map from an n-dimensional input data point +x to an m-dimensional vector p indicating the probability per +class, such that f : x �→ p, with Rn → [0, 1]m. To invert the +model, we define an objective function in order to use gradient +descent. This function is called cost function c(x) and basically +defines how close we are to the information we would like to +reconstruct. We set c(x) = 1 − pt, where t denotes the target +class we would like to gain information about. Starting from +a randomly initialized input sample x0, we calculate its cost +c(x0). With this at hand, we apply the gradient descent algo- +rithm for α iterations with a learning rate λ to alter the original +input. The aim is to minimize the costs for a specific class [2], +such that the resulting data sample is a representation of that +class. In speaker recognition, every speaker denotes their own +class. Hence, MI can reveal representations of the data from +every single individual in the training data set. Particularly, this +data can encode biometric as well as paralinguistic features. +Algorithm 2 Sliding Model Inversion Attack +function SMI(target class t, length l, stride s, windowsize w, +α, β, γ, λ as in Algorithm 1) +inverted[0,. . . ,l] ← [N(µ, σ2)]l +for k ← 0, . . . , (l − s) according to stride s do +x ← inverted[k : (k + w)] +for i ← 1, . . . , α do +xi ← xi−1 − λ · ∇c(xi−1) +if c(xi) ≥ max(c(xi−1), . . . , c(xi−β)) then +break +if c(xi) ≤ γ then +break +inverted[k : (k + w)] ← argminx c(x) +return inverted[ w +2 : (l − w +2 )] +3. Sliding MI Attack +In this section, we present our novel sliding MI attack. It ex- +tends standard MI (see Algorithm 1) to sequentially and chunk- +wise processed data, e.g., audio data. Instead of using MI to +invert every chunk of data separately, our sliding ML iteratively +inverts overlapping chunks. This way, some of the input to the +MI is already inverted, and hence, in wave-form similar to ac- +tual speech data. Thereby, MI can more successfully invert it +into representatives of the original speech data. +Our sliding MI consists of the following steps: (1) We invert +the first window of a randomly initialized input vector. Note +that different random initializations yield inversions of different +quality. We experiment with several different types of initial- +ization settings as our first main experiment, described in more +detail in Section 4.1. (2) Then, we replace the first window of +the input vector by the resulting inverted data. (3) Next, we iter- +atively calculate the inverted data for the subsequent input win- +dow. This input’s first part consists of the previously inverted +data, its second part stems from the random input vector. Note +that the amount of overlap for the sliding window determines +the proportion of the previously inverted data and the randomly +initialized input vector that are used for calculating the inver- +sion during the MI attack. Its value depends on the stride of our +inversion. For our experiments, we use a stride of 500 samples +(roughly 30ms). Since in this new method, updates rely on the +output of the previous inversion, we cannot use parallelization +as a speed-up. Instead, the stride determines the computational +overhead. By increasing the stride value, computational time +can be decreased. +For a visualisation of our novel approach, see Figure 2. +Note that in addition to the hyperparameters of the standard MI, +we need to specify the length of our input data l, the stride s, and +the window size w. While s specifies the overlap between sub- +sequent inversions, w determines the lengths of the data chunks +that are inverted. For each chunk, inversion is performed as an +iterative process as in the standard MI. Since the beginning and +the end of the inverted vector are iterated less, we cut the re- +turned vector to half the window size. See Algorithm 2 for a +formal introduction of our novel sliding ML. +4. Experiments +We conduct three experiments: The first experiment is similar to +MI in other domains, i.e., it inverts random input vectors back +to the original input data domain. In the second experiment +we do not invert the whole NN, but only the layers up to the +d-vectors, which provide unique voice features of an individ- + +234Input Vector +Inv. 1 +Inv. 2 +Figure 2: Sliding MI. 1 We initialize a random input vector. +2 Starting from the beginning, we invert the first window based +on this vector and replace the vector’s first part by our inverted +data. +3 For the all subsequent windows, we use parts of the +previously inverted vector and fill the remainder with the input +vector to apply MI. 4 We then iteratively replace the input vec- +tor with our inverted data. +ual (see Figure 1). Inverting these vectors instead of full audio +samples reduces the computational costs of the attack. Our third +experiment shows that in speaker recognition, our MI attack en- +ables us to impersonate individual speakers, and to synthesize +speech samples for them. The spoofing is performed based on +the inverted audio samples from the experiment one. +We attack the NN-based speaker recognition system using +SincNet [5], trained on the TIMIT dataset [8], and we use the +pretrained model provided by Ravanelli et al. [5]. To perform +our MI attack, we assume the attacker to have white-box access +to the target model. This is the case, for example, when the +speaker recognition model is deployed to a user-device, e.g. for +biometric identification. Further, the attacker needs a unique +identifier of their target individual under attack to know which +class of the training data to invert. This can be, for example, the +name or some pseudonymized combination of characters, in the +case of the TIMIT dataset, e.g. “FGMB0”. +We quantify the success of our experiments as follows: +• Percentage of correctly classified inverted samples. We +quantify the classification accuracy of the original target +speaker model on both the inverted audio data and the in- +verted d-vectors. An inverted sample is “correctly clas- +sified” if it is classified as the correct original speaker. +• Euclidean distance between original and reconstructed +d-vector. We measure the Euclidean distance between +both d-vectors to specify the similarity between the re- +spective samples. +The first metric allows us to analyse if the MI may be consid- +ered successful with respect to the target model, i.e., it answers +the question of how successfully this model can be fooled. The +second metric, in contrast, focuses on the inverted samples’ sim- +ilarity to the original samples. Hence, it quantifies the similarity +from the perspective of a human listener. +Since MI generates average representations of training +classes, we use the target model’s classification accuracy on av- +eraged per-speaker samples as a baseline (97.84% and 75.97% +of correct classification on train and test data, respectively). In +the following, we present our overall experimental setup to then +describe every experiment in more detail. +4.1. Experiment 1: Invert Audio Samples +In the first experiment, we use MI to calculate full inverted au- +dio samples which could be used to trick the speaker recogni- +tion model under attack without human listeners present. The +experiment is designed to answer the following three questions: +(1) Is it possible to successfully generate inverted audio samples +for speaker recognition? (2) Which kind of randomly initialized +input vector to the MI attack produces the most successful in- +verted audio samples (with respect to the classification as the +original speaker)? (3) How does our new sliding MI approach +improve the results in comparison to standard MI? +Experiment. +Over all experiments, the audio data chunks are +3200 samples (or 200ms) long, and our sliding MI uses a stride +of 500 samples (roughly 30ms). We evaluate the following (ran- +dom) initializations: +• Plain inputs: all zeros, all ones, and all minus ones; +• Noises: white, pink, brown, violet, and blue noise. We +generated them with methods pre-implemented in the +python-acoustics library and applied the tanh- +function to transform them to the interval range [−1, 1] +with zero mean in a non-linear manner; +• Samplings from distributions, such as uniform (ranges +[0, 1] or [−1, 1]), Gaussian (with µ = 0 or σ = 0.2), +Laplace (with µ = 0 or b = 0.07), Gumbel (with µ = 0 +or β = 0.1), and von Mises (with µ = 0 or κ = 0.1); +• Samples from another dataset: Librispeech [10], used as +a plain input or averaged over 50, 100, 150 input vectors +or with white noise (0.85 · input vector + 0.15 · noise). +For the optimization process within the MI algorithm +(see Section 2), we optimize two parameters, namely the max- +imum number of iterations α and the learning rate l. For all +experiments, setting α = 1000 showed to be sufficient. The +optimal l depends on the experiment and is reported in the re- +sults. In our evaluation, for every input with its settings and +optimal l, we report the following metrics: +• MI Accuracy: the percentage of correctly classified in- +verted samples; +• # Correct Speakers: the number of correctly classified +speakers; +• Avg. Eucl. Distance with Std. Deviation: the average Eu- +clidean distance of the inverted sample to original sam- +ples of the same speaker in the d-vector space, calculated +on the successfully inverted audio samples. +For the Euclidean distances within the d-vector space, the aver- +age within-speaker distance in the original training samples can +serve as a baseline (1.923 · 10−1). +Results. +(1) Is it possible to successfully generate inverted au- +dio samples for speaker recognition? By looking at the dis- +tribution of the inverted samples, we observe that it does not +fully match the distribution of the original data. Since the in- +verted samples also sound differently from the original ones— +they do not necessary sound like speech—hence, they do not +allow to fool a human listener. However, the results suggest +that standard MI is good enough to fool the classification of +the automatic speaker recognition system with an accuracy of +up to 54.76%. The classification accuracy can be significantly +improved to 90.48% through our new sliding MI. For speaker +recognition models in charge of identity control for a highly se- +cured system, this accuracy on inverted data would be beyond +acceptable. Our novel sliding MI, also reduces the Euclidean +distance between inverted and original samples for some input +vectors, in comparison to a standard MI. However, despite this +decrease in distance, the reconstructed speech samples still do +not sound very close to the original speaker. + +234(2) Which kind of randomly initialized input vector to the +MI attack produces the most successful inverted audio samples +(with respect to the classification as the original speaker)? We +can also conclude that not all input vectors to the MI are equally +suited to create inverted audio samples which successfully fool +the speaker model: plain input vectors achieve the lowest qual- +ity in inversion with respect to classification accuracy. We as- +sume that this is due to the difficulty of transforming constant +vectors into speech-like wave forms through optimization. It +seems that random initialization or data that is already in wave +form are more suited inputs to MI for audio data: The best +classification accuracy can be achieved with white noise and +tanh activation. Brown noise exhibits the poorest performance, +yet it exhibits a relatively small mean Euclidean distance. The +Laplace distributions achieves the overall highest results. See +Table 1 for an overview of results. +(3) How does our new sliding MI approach improve the re- +sults in comparison to standard MI? We observe from the re- +sults in Table 1 that sliding MI exhibits a higher performance +than standard MI. While with standard MI, the accuracy of the +target model on the inverted data is 54%, depending on the ran- +dom initialization, our sliding MI yields above 90% accuracy. +4.2. Experiment 2: Partial MI to Invert d-Vectors +While in previous applications on other data types, only a com- +plete MI back to the original input domain is valuable, this is +different for speaker recognition: the d-vectors, which are fea- +ture representations of the voice samples, already carry impor- +tant paralinguistic information that can, for example, be used +to generate spoofed audio samples [11]. Inverting simple d- +vectors instead of full audio samples reduces the computational +costs of the attack (since it does not need to be performed se- +quentially), and can be performed with standard MI attacks. +Therefore, in our second experiment we set out to invert a model +on the intermediate layers with the aim to answer the follow- +ing two questions: (1) Is it possible to successfully invert d- +vector? (2) Which input vector produces the most successful +inverted d-vector (with respect to the classification as the origi- +nal speaker)? +Experiment. +We apply partial inversion by removing the +SincNet and MLP part of the network and only focusing on the +submodel 3 for the d-vector inversion (see Figure 1). +Results. +(1) Is it possible to successfully invert d-vector? Our +findings show that we can reconstruct d-vectors that are success- +fully classified as the original speaker (classification accuracy +of the target model on them reaches 100%). This outperforms +the baseline where we measure accuracy of the target model on +averaged per-speaker samples (97.84%). However, even for the +best-performing input (zeros), the Euclidean distance of 0.84 +± 0.028 is clearly above the baseline average within speaker +distance (1.923 · 10−1). To evaluate whether the inverted d- +vectors still leak the individual speakers’ privacy, we perform a +principal component analysis (PCA) and train a binary classifier +to predict the individuals’ gender. We fit the PCA to the TIMIT +test dataset and transform the inverted d-vectors. Our results are +visualized in Figure 3. They suggest that the inverted d-vector +leak gender privacy. +(2) Which input vector produces the most successful in- +verted d-vector (with respect to the classification as the original +speaker)? Since d-vectors and audio data have different prop- +erties, they require different input vectors for successful inver- +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +First Component +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Second Component +TIMIT test data F +TIMIT test data M +Inverted laplace M +Inverted laplace F +Figure 3: PCA on d-Vectors. PCA fitted on the TIMIT test +dataset (blue: female; green: male) and used to transform the +inverted d-vectors (purple: female; red: male). Results indicate +that inverted d-vectors reveals the individuals’ gender. +sion. Sound noises are good inputs for audio samples. However, +our observations suggest that initializing the input vector with +sound noises does not yield high-quality inverted d-vector rep- +resentations. Instead, we found plain zeros perform best when +inverting d-vectors. +4.3. Experiment 3: Create Deepfakes +Even though the inverted samples are classified correctly by the +target model, they do not necessarily carry useful information +for human listeners. With the following experiments, we focus +on the question: (1) Based on the inverted audio samples, is +it possible to generate audio data that resembles the original +speaker for a human listener? The experiment can be considered +as a proof-of-concept to demonstrate further security risks in +speaker recognition systems made possible by our attack. +Experiment. +To generate the deepfakes, we use the work +from [11]. Their architecture consist of three parts: (1) speaker +encoder, (2) speech synthesizer, and (3) vocoder. The speaker +encoder is used to create a d-vector out of an audio file, which +characterizes the audio sample in vector space. With this infor- +mation, the speech synthesizer creates the mel-spectogram by +using the d-vectors. Finally, vocoder grabs the mel-spectogram +to perform frequency to time domain conversion of. The un- +derlying speech synthesizer is Tacotron 2 [12] and vocoder is +Wavenet [13]. We use the inverted audio samples from our slid- +ing MI, and the inverted d-vectors as input for the method. In +principle, inverted audio samples can be fed directly into the +speaker encoder for the deepfake generation. However, to use +our inverted d-vectors (2048 dimensions), we have to transform +them to match the deepfake model’s speaker encoding, as it ex- +pects d-vectors with 256 dimensions. To do so, we train an MLP +to map one vector space to another. The MLP has two hiddens +layers with 1024 and 512 neurons and an output layer with 256 +neurons. We use tanh activation in the first two layers. We cre- +ate the training set for this transformation by feeding sound files +to our and the deepfake’s speaker encoder. Our encodings are +used as input while their encodings are treated as the outputs to +be learned. + +Sample Type +Learning R. +MI Accuracy +# Correct Speakers +Avg. E. Dist. ± Std. Dev. +plain +inputs +ones +1 · 10−05 +0.43% +2 +0.800 ± 0.0416 +0.2 +5.41% +25 +0.696 ± 0.0663 +zeros +1 · 10−08 +0.65% +3 +0.792 ± 0.0377 +0.5 +8.87% +41 +0.720 ± 0.0682 +dists +Gumbel +0.01 +54.76% +253 +0.797 ± 0.0562 +0.01 +89.83% +415 +0.748 ± 0.0561 +Laplace +0.005 +54.33% +251 +0.793 ± 0.0555 +0.005 +90.48% +418 +0.752 ± 0.0550 +noise +white-tanh +0.2 +54.11% +250 +0.798 ± 0.0549 +0.2 +88.31% +408 +0.757 ± 0.0561 +brown +0.01 +5.84% +27 +0.676 ± 0.0938 +0.05 +31.82% +147 +0.681 ± 0.0679 +Librispeech +samples +Librispeech sample +0.001 +6.277% +29 +0.751 ± 0.0768 +0.2 +22.73% +105 +0.696 ± 0.0558 +Sample noise + Librispeech sample +0.005 +52.6% +243 +0.779 ± 0.0673 +0.01 +90.26% +417 +0.757 ± 0.0595 +Librispeech mean +0.001 +33.55% +155 +0.750 ± 0.0665 +0.01 +56.06% +259 +0.707 ± 0.0713 +Sample noise + Libspeech mean +0.005 +45.89% +212 +0.776 ± 0.0604 +0.01 +80.30% +371 +0.757 ± 0.0600 +Table 1: Results for standard MI (gray) and sliding MI (black) calculated for inverted audio samples of the 462 speakers in the TIMIT +dataset (326 men, 136 women). Sliding MI improves standard MI (higher accuracy and lower average Euclidean distance.) +Results. +(1) Based on the inverted audio samples, is it possi- +ble to generate audio data that resembles the original speaker +for a human listener? +As reported in experiment 2, our d- +vectors, though correctly classified as the original speakers, +were far away from the speaker’s original d-vectors in the Eu- +clidean space. The question if a generated sample sounds simi- +lar to an original one is a semantic question and depends on the +sensitivity of the context. From the authors’ perspective on in- +spection case-by-case, the d-vector-based deepfakes did not al- +low individual speakers characteristics to be recognized. How- +ever, based on the inverted audio samples from our novel sliding +MI, we were able to generate a few good quality spoofed audio +samples that resembled the original speaker.2 With such sam- +ples at hand, an attacker could, hence, spoof someone’s iden- +tity solely based the inverted data from the pre-trained NN. We +expect this to become even much more prevalent with more so- +phisticated deepfake generation systems in the future. +5. Countermeasures and Discussion +Speaker recognition systems heavily rely on learning individ- +ual per-speaker characteristics in order to fulfill the task they +are designed for. Therefore, these systems always and neces- +sarily contain information about the speaker data that they were +trained on. Noising out individual speaker characteristics will +result in drastically decreased performance of the systems. In +particular, pseudonomization [14] and privacy methods that are +used in general speech systems (e.g., [15, 16, 17, 18, 19, 20]) +render speaker recognition unusable for their original purpose. +Protecting against MI attacks. +As an alternative, one can +consider privacy protection methods that aim at impeding MI +attacks. Most existing defenses from other domains focus on +2Examples for the original audio data, averaged and inverted sam- +ples, and the spoofed audio data generated based on the inverted au- +dio samples, are available at https://www.dropbox.com/sh/ +ge6xx90laqmru9b/\AABUsS3p4EwaN0n7g4Rgq8rwa. +suppressing the model confidence score or reducing their util- +ity. This can be done by injecting uniform noise to them [21], +reducing their precision [2] or their dispersion [22]. The lat- +ter one leads to a decrease in the correlation between the in- +put data and the scores, which renders MI attacks more inaccu- +rate. With a similar aim, the use of regularization in the training +loss function has been reported as a defense [23]. Additionally, +hardware-oriented solutions to prevent an attacker from access- +ing the model parameters to decrease MI success, or at least +preventing the extraction of intermediate features (see our ex- +periment 2) can be applied [24]. +Differential privacy. +Initial work empirically showed that +Differential Privacy (DP) [25]can reduce the success of MI at- +tack’s [2] when using a very large amount of noise, which, in re- +turn, drastically degrades the model’s performance. Later work +suggests that DP training for ML models cannot at all prevent +MI attacks [26] because its aim is to dissimulate the presence of +a data point in a specific data set and not to protect privacy over +classes of data. +Limitations. +So far, MI attacks require the availability of +an NN’s confidence scores, and the attack’s success depends +largely on the random initializations. Especially, many speaker +recognition tools depend on the cosine similarity [27, 28] for +which the algorithm would need to be updated. Also, the qual- +ity of the spoofed audio samples is limited by the deepfake cre- +ation methods. As a consequence, the practical impact of our +attack might currently still be limited. However, with new and +ever more powerful privacy attacks and deepfake methods be- +ing proposed, the threat space of exploiting privacy attacks to +violate security of speaker recognition systems will gain impor- +tance. It is, hence, important to create awareness and to consider +and protect privacy and security jointly, rather than separately. + +6. Conclusion +In this work, for the first time, we successfully perform MI at- +tacks on audio data. Therefore, we introduce a novel sliding MI +method which leverages the sequential properties of the audio +data for improved inversion. We experimentally evaluate the at- +tack’s success on a state-of-the-art speaker recognition system. +Our results indicate that our inverted audio samples can be used +as a departure point for further attacks against the security of +the target system. Thereby, we highlight the importance of im- +plementing adequate privacy protection in such systems. +7. Acknowledgements +This research was supported by the Bavarian Ministry of Eco- +nomic Affairs, Regional Development and Energy. The authors +acknowledge J. Williams for comments on the manuscript. +8. References +[1] J. L. Kr¨oger, O. H.-M. Lutz, and P. Raschke, “Privacy implica- +tions of voice and speech analysis–information disclosure by in- +ference,” in IFIP International Summer School on Privacy and +Identity Management. +Springer, 2019, pp. 242–258. +[2] M. Fredrikson, S. Jha, and T. Ristenpart, “Model inversion attacks +that exploit confidence information and basic countermeasures,” +in Proceedings of the 22nd ACM SIGSAC Conference on Com- +puter and Communications Security, 2015, pp. 1322–1333. +[3] R. Shokri, M. Stronati, C. Song, and V. Shmatikov, “Membership +inference attacks against machine learning models,” in 2017 IEEE +Symposium on Security and Privacy (SP). IEEE, 2017, pp. 3–18. +[4] K. Ganju, Q. Wang, W. Yang, C. A. Gunter, and N. Borisov, +“Property inference attacks on fully connected neural networks +using permutation invariant representations,” in Proceedings of +the 2018 ACM SIGSAC Conference on Computer and Commu- +nications Security, 2018, pp. 619–633. +[5] M. Ravanelli and Y. Bengio, “Speaker recognition from raw wave- +form with sincnet,” in 2018 IEEE Spoken Language Technology +Workshop (SLT). +IEEE, 2018, pp. 1021–1028. +[6] M. Ravanelli, T. Parcollet, A. Rouhe, P. Plantinga, E. Rastorgueva, +L. Lugosch, N. Dawalatabad, C. Ju-Chieh, A. Heba, F. Grondin, +W. Aris, C.-F. Liao, S. Cornell, S.-L. Yeh, H. Na, Y. Gao, S.- +W. Fu, C. Subakan, R. De Mori, and Y. Bengio, “Speechbrain,” +https://github.com/speechbrain/speechbrain, 2021. +[7] E. Variani, X. Lei, E. McDermott, I. L. Moreno, and J. Gonzalez- +Dominguez, “Deep neural networks for small footprint text- +dependent speaker verification,” in 2014 IEEE international con- +ference on acoustics, speech and signal processing (ICASSP). +IEEE, 2014, pp. 4052–4056. +[8] J. S. Garofolo, “Timit acoustic phonetic continuous speech cor- +pus,” Linguistic Data Consortium, 1993, 1993. +[9] A. Nautsch, A. Jim´enez, A. Treiber, J. Kolberg, C. Jasserand, +E. Kindt, H. Delgado, M. Todisco, M. A. Hmani, A. Mtibaa et al., +“Preserving privacy in speaker and speech characterisation,” Com- +puter Speech & Language, vol. 58, pp. 441–480, 2019. +[10] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, “Lib- +rispeech: an asr corpus based on public domain audio books,” +in 2015 IEEE international conference on acoustics, speech and +signal processing (ICASSP). +IEEE, 2015, pp. 5206–5210. +[11] Y. Jia, Y. Zhang, R. J. Weiss, Q. Wang, J. Shen, F. Ren, Z. Chen, +P. Nguyen, R. Pang, I. L. Moreno et al., “Transfer learning +from speaker verification to multispeaker text-to-speech synthe- +sis,” arXiv preprint arXiv:1806.04558, 2018. +[12] J. Shen, R. Pang, R. J. Weiss, M. Schuster, N. Jaitly, Z. Yang, +Z. Chen, Y. Zhang, Y. Wang, R. Skerrv-Ryan et al., “Natural +tts synthesis by conditioning wavenet on mel spectrogram pre- +dictions,” in 2018 IEEE international conference on acoustics, +speech and signal processing (ICASSP). +IEEE, 2018, pp. 4779– +4783. +[13] A. v. d. Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, +A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu, +“Wavenet: A generative model for raw audio,” arXiv preprint +arXiv:1609.03499, 2016. +[14] P.-G. No´e, A. Nautsch, N. Evans, J. Patino, J.-F. Bonastre, +N. Tomashenko, and D. Matrouf, “Towards a unified assessment +framework of speech pseudonymisation,” Computer Speech & +Language, vol. 72, p. 101299, 2022. +[15] R. Aloufi, H. Haddadi, and D. Boyle, “A tandem framework +balancing privacy and security for voice user interfaces,” arXiv +preprint arXiv:2107.10045, 2021. +[16] Q. Jin, A. R. Toth, T. Schultz, and A. W. Black, “Voice conver- +gin: Speaker de-identification by voice transformation,” in 2009 +IEEE International Conference on Acoustics, Speech and Signal +Processing. +IEEE, 2009, pp. 3909–3912. +[17] B. M. L. Srivastava, N. Vauquier, M. Sahidullah, A. Bellet, +M. Tommasi, and E. Vincent, “Evaluating voice conversion-based +privacy protection against informed attackers,” in ICASSP 2020- +2020 IEEE International Conference on Acoustics, Speech and +Signal Processing (ICASSP). +IEEE, 2020, pp. 2802–2806. +[18] B. M. L. Srivastava, N. Tomashenko, X. Wang, E. Vincent, J. Ya- +magishi, M. Maouche, A. Bellet, and M. Tommasi, “Design +choices for x-vector based speaker anonymization,” arXiv preprint +arXiv:2005.08601, 2020. +[19] J. Qian, H. Du, J. Hou, L. Chen, T. Jung, and X.-Y. Li, “Hide- +behind: +Enjoy voice input with voiceprint unclonability and +anonymity,” in Proceedings of the 16th ACM Conference on Em- +bedded Networked Sensor Systems, 2018, pp. 82–94. +[20] B. M. L. Srivastava, A. Bellet, M. Tommasi, and E. Vincent, +“Privacy-preserving adversarial representation learning in asr: +Reality or illusion?” arXiv preprint arXiv:1911.04913, 2019. +[21] A. Salem, A. Bhattacharya, M. Backes, M. Fritz, and Y. Zhang, +“Updates-leak: +Data set inference and reconstruction attacks +in online learning,” in 29th {USENIX} Security Symposium +({USENIX} Security 20), 2020, pp. 1291–1308. +[22] Z. Yang, B. Shao, B. Xuan, E.-C. Chang, and F. Zhang, “Defend- +ing model inversion and membership inference attacks via predic- +tion purification,” arXiv preprint arXiv:2005.03915, 2020. +[23] T. Wang, Y. Zhang, and R. Jia, “Improving robustness to model +inversion attacks via mutual information regularization,” arXiv +preprint arXiv:2009.05241, 2020. +[24] Q. Xu, M. T. Arafin, and G. Qu, “Midas: Model inversion de- +fenses using an approximate memory system,” in 2020 Asian +Hardware Oriented Security and Trust Symposium (AsianHOST). +IEEE, 2020, pp. 1–4. +[25] C. Dwork, “Differential privacy,” in International Colloquium on +Automata, Languages, and Programming. +Springer, 2006, pp. +1–12. +[26] Y. Zhang, R. Jia, H. Pei, W. Wang, B. Li, and D. Song, “The +secret revealer: Generative model-inversion attacks against deep +neural networks,” in Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, 2020, pp. 253–261. +[27] Y. Zhang, Z. Lv, H. Wu, S. Zhang, P. Hu, Z. Wu, H.-y. Lee, +and H. Meng, “Mfa-conformer: +Multi-scale feature aggrega- +tion conformer for automatic speaker verification,” arXiv preprint +arXiv:2203.15249, 2022. +[28] B. Desplanques, J. Thienpondt, and K. Demuynck, “Ecapa- +tdnn: +Emphasized channel attention, +propagation and ag- +gregation in tdnn based speaker verification,” arXiv preprint +arXiv:2005.07143, 2020. + diff --git a/etE1T4oBgHgl3EQfegQF/content/tmp_files/load_file.txt b/etE1T4oBgHgl3EQfegQF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ccd3c041e2694ecfbb940917b5146768ece8a79 --- /dev/null +++ b/etE1T4oBgHgl3EQfegQF/content/tmp_files/load_file.txt @@ -0,0 +1,670 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf,len=669 +page_content='Introducing Model Inversion Attacks on Automatic Speaker Recognition Karla Pizzi∗,1,2, Franziska Boenisch∗,1, Ugur Sahin∗,1,2, Konstantin B¨ottinger1 1Fraunhofer AISEC, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 2Technical University Munich, Germany [firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='lastname]@aisec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='de Abstract Model inversion (MI) attacks allow to reconstruct average per- class representations of a machine learning (ML) model’s train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' It has been shown that in scenarios where each class corresponds to a different individual, such as face classifiers, this represents a severe privacy risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In this work, we explore a new application for MI: the extraction of speakers’ voices from a speaker recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We present an approach to (1) re- construct audio samples from a trained ML model and (2) ex- tract intermediate voice feature representations which provide valuable insights into the speakers’ biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Therefore, we propose an extension of MI attacks which we call sliding model inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our sliding MI extends stan- dard MI by iteratively inverting overlapping chunks of the au- dio samples and thereby leveraging the sequential properties of audio data for enhanced inversion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We show that one can use the inverted audio data to generate spoofed audio samples to impersonate a speaker, and execute voice-protected commands for highly secured systems on their behalf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' To the best of our knowledge, our work is the first one extending MI attacks to audio data, and our results highlight the security risks resulting from the extraction of the biometric data in that setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Index Terms: speaker recognition, model inversion, privacy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Introduction Privacy analysis of audio data has shown that speech parame- ters, such as accent, rhythm, or acoustic properties of speech in- herently carry biometric information about the speakers, such as their age, gender, physical health, and geographical origin [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Therefore, it is important for machine learning (ML) models in speaker recognition not to leak information about their training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, recent research [2, 3, 4] suggests that ML mod- els are, in general, vulnerable to privacy attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' One particu- lar attack is model inversion (MI) [2] which allows an attacker to retrieve abstract representations for individual classes of the target model’s training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' With speaker recognition systems treating each individual as their own class, MI attacks have the potential to cause severe privacy breaches [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' So far, the fea- sibility of MI attacks on speaker recognition systems and audio data has never been tested, thus, the question if information on the speakers can be maliciously retrieved remained open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We are the first to show how to adapt and apply MI attacks for audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We do so by targeting SincNet, a state-of-the-art neural network (NN)-based speaker recognition model [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We show that MI attacks are able to infer both entire audio sam- ples and d-vectors as intermediate representations of the speak- ers’ voice characteristics from the trained target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Further, we propose the sliding model inversion, a novel form of the standard MI attack that leverages sequential processing prop- erties of the audio data to improve inversion success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' While with standard MI, the target model successfully identifies up ∗Authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' to 54% of the inverted audio samples as their correct speaker class, with our novel attack, we achieve to up to 90% accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Also, our sliding MI manages to decrease the distance between original and inverted sample in the d-vector represen- tation, hence, yielding higher fidelity inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For directly inverting d-vectors, our experiments show that even standard MI achieves 100% identification success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' These results high- light the vulnerability of speaker recognition models to privacy attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' As a proof-of-concept to showcase that our MI can be exploited as a departure point for further attacks against speaker recognition, we, furthermore, explore using inverted audio sam- ples as inputs for deepfake generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Such deepfakes could be used to fool voice identification with arbitrary speech samples or to execute any speech command on behalf of the speakers un- der attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' While our generated deepfakes do not perfectly fool a human listener, as an informal evaluation conducted by the au- thors shows, they illustrate that privacy attacks can not only be used to disclose sensitive information about the individuals the model was trained on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' additionally, they can severely threaten the security of systems relying on voice biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our contri- butions can be summarized as follows: We successfully apply MI attacks on speaker recognition models to invert entire audio samples and d-vectors and experimentally evaluate what kind of random initializa- tion works best as an input for MI attacks on audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We introduce a novel sliding MI which exploits proper- ties of sequential and chunk-wise audio processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We show the feasibility of generating deepfakes based on the inferred audio samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Background and Related Work The following section provides background information on speaker recognition systems and attacks against their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Speaker recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In this paper, we use a SincNet- based [5] text-independent speaker recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This system uses NNs to extract voice features into so-called d- vectors and adds a classification layer on top of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The in- put to the system consists of raw audio waves and the outputs is a per-class probability score over all possible classes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', speakers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Overall, the system is composed of three submodels (see Figure 1) 1) SincNet, a convolutional NN that resembles a band-pass filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 2) a multi-layer perceptron (MLP) calculating the d-vectors [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' and 3) a fully connected layer to calculate the probabilities per speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' It achieves a reported classification er- ror1 of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='772 · 10−3 on the TIMIT [8] test data set (measured at sentence level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In its current version, SincNet does not provide any dedicated privacy-preserving mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1See https://pythonlang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='dev/repo/ mravanelli-sincnet/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='03206v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='SD] 9 Jan 2023 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1 SincNet MLP 1-Layer … d Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 2 Figure 1: Speaker Recognition Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The speaker recognition model and its three submodels: SincNet obtains 1 raw audio input and generates 2 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' These features are input to an MLP which generates the 3 d-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' A single layer performs 4 classification on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In experiment 1, we invert full audio samples, while in experiment 2, we invert the d-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Algorithm 1 Standard Model Inversion Attack [2] function MI(input vector x0, target class t, iterations α, pa- tience β, minimum cost threshold γ, learning rate for gradient descent λ) for i ← 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' , α do xi ← xi−1 − λ · ∇c(xi−1) if c(xi) ≥ max(c(xi−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' , c(xi−β)) then break if c(xi) ≤ γ then break return [argminxi c(xi), minxi c(xi)] Privacy in speaker recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The ISO/IEC norm 24745:2011 proposes three general requirements to ensure individual privacy: irreversibility, renewability, and unlinka- bility of the protected data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' To reach this goal in speaker recoginition systems, several solutions have been proposed and discussed [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, aiming for any anonymization to protect the privacy of speakers in a speaker recognition model goes directly against the purpose of the system, which is to identify speakers based on their individual characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' MI attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In MI attacks [2], the attacker exploits an ML model’s prediction confidence for inverting individual training classes (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' More formally speaking, a MI attack can be expressed as follows: let f be the target model under at- tack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' It is trained to map from an n-dimensional input data point x to an m-dimensional vector p indicating the probability per class, such that f : x �→ p, with Rn → [0, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' To invert the model, we define an objective function in order to use gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This function is called cost function c(x) and basically defines how close we are to the information we would like to reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We set c(x) = 1 − pt, where t denotes the target class we would like to gain information about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Starting from a randomly initialized input sample x0, we calculate its cost c(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' With this at hand, we apply the gradient descent algo- rithm for α iterations with a learning rate λ to alter the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The aim is to minimize the costs for a specific class [2], such that the resulting data sample is a representation of that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In speaker recognition, every speaker denotes their own class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Hence, MI can reveal representations of the data from every single individual in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Particularly, this data can encode biometric as well as paralinguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Algorithm 2 Sliding Model Inversion Attack function SMI(target class t, length l, stride s, windowsize w, α, β, γ, λ as in Algorithm 1) inverted[0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' ,l] ← [N(µ, σ2)]l for k ← 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' , (l − s) according to stride s do x ← inverted[k : (k + w)] for i ← 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' , α do xi ← xi−1 − λ · ∇c(xi−1) if c(xi) ≥ max(c(xi−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' , c(xi−β)) then break if c(xi) ≤ γ then break inverted[k : (k + w)] ← argminx c(x) return inverted[ w 2 : (l − w 2 )] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Sliding MI Attack In this section, we present our novel sliding MI attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' It ex- tends standard MI (see Algorithm 1) to sequentially and chunk- wise processed data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Instead of using MI to invert every chunk of data separately, our sliding ML iteratively inverts overlapping chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This way, some of the input to the MI is already inverted, and hence, in wave-form similar to ac- tual speech data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Thereby, MI can more successfully invert it into representatives of the original speech data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our sliding MI consists of the following steps: (1) We invert the first window of a randomly initialized input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Note that different random initializations yield inversions of different quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We experiment with several different types of initial- ization settings as our first main experiment, described in more detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (2) Then, we replace the first window of the input vector by the resulting inverted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (3) Next, we iter- atively calculate the inverted data for the subsequent input win- dow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This input’s first part consists of the previously inverted data, its second part stems from the random input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Note that the amount of overlap for the sliding window determines the proportion of the previously inverted data and the randomly initialized input vector that are used for calculating the inver- sion during the MI attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Its value depends on the stride of our inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For our experiments, we use a stride of 500 samples (roughly 30ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Since in this new method, updates rely on the output of the previous inversion, we cannot use parallelization as a speed-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Instead, the stride determines the computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' By increasing the stride value, computational time can be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For a visualisation of our novel approach, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Note that in addition to the hyperparameters of the standard MI, we need to specify the length of our input data l, the stride s, and the window size w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' While s specifies the overlap between sub- sequent inversions, w determines the lengths of the data chunks that are inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For each chunk, inversion is performed as an iterative process as in the standard MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Since the beginning and the end of the inverted vector are iterated less, we cut the re- turned vector to half the window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' See Algorithm 2 for a formal introduction of our novel sliding ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Experiments We conduct three experiments: The first experiment is similar to MI in other domains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', it inverts random input vectors back to the original input data domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In the second experiment we do not invert the whole NN, but only the layers up to the d-vectors, which provide unique voice features of an individ- 234Input Vector Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1 Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 2 Figure 2: Sliding MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1 We initialize a random input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 2 Starting from the beginning, we invert the first window based on this vector and replace the vector’s first part by our inverted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 3 For the all subsequent windows, we use parts of the previously inverted vector and fill the remainder with the input vector to apply MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 4 We then iteratively replace the input vec- tor with our inverted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' ual (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Inverting these vectors instead of full audio samples reduces the computational costs of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our third experiment shows that in speaker recognition, our MI attack en- ables us to impersonate individual speakers, and to synthesize speech samples for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The spoofing is performed based on the inverted audio samples from the experiment one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We attack the NN-based speaker recognition system using SincNet [5], trained on the TIMIT dataset [8], and we use the pretrained model provided by Ravanelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' To perform our MI attack, we assume the attacker to have white-box access to the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This is the case, for example, when the speaker recognition model is deployed to a user-device, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' for biometric identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Further, the attacker needs a unique identifier of their target individual under attack to know which class of the training data to invert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This can be, for example, the name or some pseudonymized combination of characters, in the case of the TIMIT dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' “FGMB0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We quantify the success of our experiments as follows: Percentage of correctly classified inverted samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We quantify the classification accuracy of the original target speaker model on both the inverted audio data and the in- verted d-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' An inverted sample is “correctly clas- sified” if it is classified as the correct original speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Euclidean distance between original and reconstructed d-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We measure the Euclidean distance between both d-vectors to specify the similarity between the re- spective samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The first metric allows us to analyse if the MI may be consid- ered successful with respect to the target model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', it answers the question of how successfully this model can be fooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The second metric, in contrast, focuses on the inverted samples’ sim- ilarity to the original samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Hence, it quantifies the similarity from the perspective of a human listener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Since MI generates average representations of training classes, we use the target model’s classification accuracy on av- eraged per-speaker samples as a baseline (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='84% and 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='97% of correct classification on train and test data, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In the following, we present our overall experimental setup to then describe every experiment in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Experiment 1: Invert Audio Samples In the first experiment, we use MI to calculate full inverted au- dio samples which could be used to trick the speaker recogni- tion model under attack without human listeners present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The experiment is designed to answer the following three questions: (1) Is it possible to successfully generate inverted audio samples for speaker recognition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (2) Which kind of randomly initialized input vector to the MI attack produces the most successful in- verted audio samples (with respect to the classification as the original speaker)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (3) How does our new sliding MI approach improve the results in comparison to standard MI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Over all experiments, the audio data chunks are 3200 samples (or 200ms) long, and our sliding MI uses a stride of 500 samples (roughly 30ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We evaluate the following (ran- dom) initializations: Plain inputs: all zeros, all ones, and all minus ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Noises: white, pink, brown, violet, and blue noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We generated them with methods pre-implemented in the python-acoustics library and applied the tanh- function to transform them to the interval range [−1, 1] with zero mean in a non-linear manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Samplings from distributions, such as uniform (ranges [0, 1] or [−1, 1]), Gaussian (with µ = 0 or σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2), Laplace (with µ = 0 or b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='07), Gumbel (with µ = 0 or β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1), and von Mises (with µ = 0 or κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Samples from another dataset: Librispeech [10], used as a plain input or averaged over 50, 100, 150 input vectors or with white noise (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='85 · input vector + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='15 · noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For the optimization process within the MI algorithm (see Section 2), we optimize two parameters, namely the max- imum number of iterations α and the learning rate l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For all experiments, setting α = 1000 showed to be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The optimal l depends on the experiment and is reported in the re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In our evaluation, for every input with its settings and optimal l, we report the following metrics: MI Accuracy: the percentage of correctly classified in- verted samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' # Correct Speakers: the number of correctly classified speakers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Distance with Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Deviation: the average Eu- clidean distance of the inverted sample to original sam- ples of the same speaker in the d-vector space, calculated on the successfully inverted audio samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For the Euclidean distances within the d-vector space, the aver- age within-speaker distance in the original training samples can serve as a baseline (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='923 · 10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (1) Is it possible to successfully generate inverted au- dio samples for speaker recognition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' By looking at the dis- tribution of the inverted samples, we observe that it does not fully match the distribution of the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Since the in- verted samples also sound differently from the original ones— they do not necessary sound like speech—hence, they do not allow to fool a human listener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, the results suggest that standard MI is good enough to fool the classification of the automatic speaker recognition system with an accuracy of up to 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='76%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The classification accuracy can be significantly improved to 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='48% through our new sliding MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' For speaker recognition models in charge of identity control for a highly se- cured system, this accuracy on inverted data would be beyond acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our novel sliding MI, also reduces the Euclidean distance between inverted and original samples for some input vectors, in comparison to a standard MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, despite this decrease in distance, the reconstructed speech samples still do not sound very close to the original speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 234(2) Which kind of randomly initialized input vector to the MI attack produces the most successful inverted audio samples (with respect to the classification as the original speaker)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We can also conclude that not all input vectors to the MI are equally suited to create inverted audio samples which successfully fool the speaker model: plain input vectors achieve the lowest qual- ity in inversion with respect to classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We as- sume that this is due to the difficulty of transforming constant vectors into speech-like wave forms through optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' It seems that random initialization or data that is already in wave form are more suited inputs to MI for audio data: The best classification accuracy can be achieved with white noise and tanh activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Brown noise exhibits the poorest performance, yet it exhibits a relatively small mean Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The Laplace distributions achieves the overall highest results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' See Table 1 for an overview of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (3) How does our new sliding MI approach improve the re- sults in comparison to standard MI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We observe from the re- sults in Table 1 that sliding MI exhibits a higher performance than standard MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' While with standard MI, the accuracy of the target model on the inverted data is 54%, depending on the ran- dom initialization, our sliding MI yields above 90% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Experiment 2: Partial MI to Invert d-Vectors While in previous applications on other data types, only a com- plete MI back to the original input domain is valuable, this is different for speaker recognition: the d-vectors, which are fea- ture representations of the voice samples, already carry impor- tant paralinguistic information that can, for example, be used to generate spoofed audio samples [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Inverting simple d- vectors instead of full audio samples reduces the computational costs of the attack (since it does not need to be performed se- quentially), and can be performed with standard MI attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Therefore, in our second experiment we set out to invert a model on the intermediate layers with the aim to answer the follow- ing two questions: (1) Is it possible to successfully invert d- vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (2) Which input vector produces the most successful inverted d-vector (with respect to the classification as the origi- nal speaker)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We apply partial inversion by removing the SincNet and MLP part of the network and only focusing on the submodel 3 for the d-vector inversion (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (1) Is it possible to successfully invert d-vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our findings show that we can reconstruct d-vectors that are success- fully classified as the original speaker (classification accuracy of the target model on them reaches 100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This outperforms the baseline where we measure accuracy of the target model on averaged per-speaker samples (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='84%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, even for the best-performing input (zeros), the Euclidean distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='028 is clearly above the baseline average within speaker distance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='923 · 10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' To evaluate whether the inverted d- vectors still leak the individual speakers’ privacy, we perform a principal component analysis (PCA) and train a binary classifier to predict the individuals’ gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We fit the PCA to the TIMIT test dataset and transform the inverted d-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our results are visualized in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' They suggest that the inverted d-vector leak gender privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (2) Which input vector produces the most successful in- verted d-vector (with respect to the classification as the original speaker)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Since d-vectors and audio data have different prop- erties, they require different input vectors for successful inver- −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='4 First Component −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='4 Second Component TIMIT test data F TIMIT test data M Inverted laplace M Inverted laplace F Figure 3: PCA on d-Vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' PCA fitted on the TIMIT test dataset (blue: female;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' green: male) and used to transform the inverted d-vectors (purple: female;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' red: male).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Results indicate that inverted d-vectors reveals the individuals’ gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Sound noises are good inputs for audio samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, our observations suggest that initializing the input vector with sound noises does not yield high-quality inverted d-vector rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Instead, we found plain zeros perform best when inverting d-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Experiment 3: Create Deepfakes Even though the inverted samples are classified correctly by the target model, they do not necessarily carry useful information for human listeners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' With the following experiments, we focus on the question: (1) Based on the inverted audio samples, is it possible to generate audio data that resembles the original speaker for a human listener?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The experiment can be considered as a proof-of-concept to demonstrate further security risks in speaker recognition systems made possible by our attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' To generate the deepfakes, we use the work from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Their architecture consist of three parts: (1) speaker encoder, (2) speech synthesizer, and (3) vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The speaker encoder is used to create a d-vector out of an audio file, which characterizes the audio sample in vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' With this infor- mation, the speech synthesizer creates the mel-spectogram by using the d-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Finally, vocoder grabs the mel-spectogram to perform frequency to time domain conversion of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The un- derlying speech synthesizer is Tacotron 2 [12] and vocoder is Wavenet [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We use the inverted audio samples from our slid- ing MI, and the inverted d-vectors as input for the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In principle, inverted audio samples can be fed directly into the speaker encoder for the deepfake generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, to use our inverted d-vectors (2048 dimensions), we have to transform them to match the deepfake model’s speaker encoding, as it ex- pects d-vectors with 256 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' To do so, we train an MLP to map one vector space to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The MLP has two hiddens layers with 1024 and 512 neurons and an output layer with 256 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We use tanh activation in the first two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We cre- ate the training set for this transformation by feeding sound files to our and the deepfake’s speaker encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our encodings are used as input while their encodings are treated as the outputs to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Sample Type Learning R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' MI Accuracy # Correct Speakers Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' ± Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' plain inputs ones 1 · 10−05 0.' metadata={'source': 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+page_content='0938 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='05 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='82% 147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='681 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0679 Librispeech samples Librispeech sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='001 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='277% 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='751 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='73% 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='696 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0558 Sample noise + Librispeech sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='005 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Librispeech mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='001 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='55% 155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='750 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='01 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='06% 259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='707 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0713 Sample noise + Libspeech mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='005 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='89% 212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='776 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='01 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='30% 371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='757 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='0600 Table 1: Results for standard MI (gray) and sliding MI (black) calculated for inverted audio samples of the 462 speakers in the TIMIT dataset (326 men, 136 women).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Sliding MI improves standard MI (higher accuracy and lower average Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=') Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' (1) Based on the inverted audio samples, is it possi- ble to generate audio data that resembles the original speaker for a human listener?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' As reported in experiment 2, our d- vectors, though correctly classified as the original speakers, were far away from the speaker’s original d-vectors in the Eu- clidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The question if a generated sample sounds simi- lar to an original one is a semantic question and depends on the sensitivity of the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' From the authors’ perspective on in- spection case-by-case, the d-vector-based deepfakes did not al- low individual speakers characteristics to be recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' How- ever, based on the inverted audio samples from our novel sliding MI, we were able to generate a few good quality spoofed audio samples that resembled the original speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='2 With such sam- ples at hand, an attacker could, hence, spoof someone’s iden- tity solely based the inverted data from the pre-trained NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We expect this to become even much more prevalent with more so- phisticated deepfake generation systems in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Countermeasures and Discussion Speaker recognition systems heavily rely on learning individ- ual per-speaker characteristics in order to fulfill the task they are designed for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Therefore, these systems always and neces- sarily contain information about the speaker data that they were trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Noising out individual speaker characteristics will result in drastically decreased performance of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' In particular, pseudonomization [14] and privacy methods that are used in general speech systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', [15, 16, 17, 18, 19, 20]) render speaker recognition unusable for their original purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Protecting against MI attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' As an alternative, one can consider privacy protection methods that aim at impeding MI attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Most existing defenses from other domains focus on 2Examples for the original audio data, averaged and inverted sam- ples, and the spoofed audio data generated based on the inverted au- dio samples, are available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='com/sh/ ge6xx90laqmru9b/\\AABUsS3p4EwaN0n7g4Rgq8rwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' suppressing the model confidence score or reducing their util- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' This can be done by injecting uniform noise to them [21], reducing their precision [2] or their dispersion [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The lat- ter one leads to a decrease in the correlation between the in- put data and the scores, which renders MI attacks more inaccu- rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' With a similar aim, the use of regularization in the training loss function has been reported as a defense [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Additionally, hardware-oriented solutions to prevent an attacker from access- ing the model parameters to decrease MI success, or at least preventing the extraction of intermediate features (see our ex- periment 2) can be applied [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Initial work empirically showed that Differential Privacy (DP) [25]can reduce the success of MI at- tack’s [2] when using a very large amount of noise, which, in re- turn, drastically degrades the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Later work suggests that DP training for ML models cannot at all prevent MI attacks [26] because its aim is to dissimulate the presence of a data point in a specific data set and not to protect privacy over classes of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' So far, MI attacks require the availability of an NN’s confidence scores, and the attack’s success depends largely on the random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Especially, many speaker recognition tools depend on the cosine similarity [27, 28] for which the algorithm would need to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Also, the qual- ity of the spoofed audio samples is limited by the deepfake cre- ation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' As a consequence, the practical impact of our attack might currently still be limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' However, with new and ever more powerful privacy attacks and deepfake methods be- ing proposed, the threat space of exploiting privacy attacks to violate security of speaker recognition systems will gain impor- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' It is, hence, important to create awareness and to consider and protect privacy and security jointly, rather than separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Conclusion In this work, for the first time, we successfully perform MI at- tacks on audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Therefore, we introduce a novel sliding MI method which leverages the sequential properties of the audio data for improved inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' We experimentally evaluate the at- tack’s success on a state-of-the-art speaker recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Our results indicate that our inverted audio samples can be used as a departure point for further attacks against the security of the target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Thereby, we highlight the importance of im- plementing adequate privacy protection in such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Acknowledgements This research was supported by the Bavarian Ministry of Eco- nomic Affairs, Regional Development and Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' The authors acknowledge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Williams for comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Kr¨oger, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Lutz, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Raschke, “Privacy implica- tions of voice and speech analysis–information disclosure by in- ference,” in IFIP International Summer School on Privacy and Identity Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Springer, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 242–258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Fredrikson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jha, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Ristenpart, “Model inversion attacks that exploit confidence information and basic countermeasures,” in Proceedings of the 22nd ACM SIGSAC Conference on Com- puter and Communications Security, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1322–1333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Shokri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Stronati, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Song, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Shmatikov, “Membership inference attacks against machine learning models,” in 2017 IEEE Symposium on Security and Privacy (SP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 3–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Ganju, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Gunter, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Borisov, “Property inference attacks on fully connected neural networks using permutation invariant representations,” in Proceedings of the 2018 ACM SIGSAC Conference on Computer and Commu- nications Security, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 619–633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Ravanelli and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Bengio, “Speaker recognition from raw wave- form with sincnet,” in 2018 IEEE Spoken Language Technology Workshop (SLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1021–1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Ravanelli, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Parcollet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Rouhe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Plantinga, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Rastorgueva, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Lugosch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Dawalatabad, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Ju-Chieh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Heba, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Grondin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Aris, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Liao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Cornell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Yeh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Na, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Gao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='- W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Fu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Subakan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' De Mori, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Bengio, “Speechbrain,” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='com/speechbrain/speechbrain, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Variani, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Lei, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' McDermott, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Moreno, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Gonzalez- Dominguez, “Deep neural networks for small footprint text- dependent speaker verification,” in 2014 IEEE international con- ference on acoustics, speech and signal processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 4052–4056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Garofolo, “Timit acoustic phonetic continuous speech cor- pus,” Linguistic Data Consortium, 1993, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Nautsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jim´enez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Treiber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Kolberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jasserand, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Kindt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Delgado, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Todisco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Hmani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Mtibaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', “Preserving privacy in speaker and speech characterisation,” Com- puter Speech & Language, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 58, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 441–480, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Panayotov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Povey, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Khudanpur, “Lib- rispeech: an asr corpus based on public domain audio books,” in 2015 IEEE international conference on acoustics, speech and signal processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 5206–5210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Weiss, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Shen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Ren, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Nguyen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Pang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', “Transfer learning from speaker verification to multispeaker text-to-speech synthe- sis,” arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='04558, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Shen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Pang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Weiss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Schuster, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jaitly, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Skerrv-Ryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=', “Natural tts synthesis by conditioning wavenet on mel spectrogram pre- dictions,” in 2018 IEEE international conference on acoustics, speech and signal processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 4779– 4783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Oord, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Dieleman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Simonyan, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Vinyals, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Graves, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Kalchbrenner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Senior, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Kavukcuoglu, “Wavenet: A generative model for raw audio,” arXiv preprint arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='03499, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' No´e, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Nautsch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Evans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Patino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Bonastre, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Tomashenko, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Matrouf, “Towards a unified assessment framework of speech pseudonymisation,” Computer Speech & Language, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 101299, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Aloufi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Haddadi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Boyle, “A tandem framework balancing privacy and security for voice user interfaces,” arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='10045, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [16] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Toth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Schultz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Black, “Voice conver- gin: Speaker de-identification by voice transformation,” in 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 3909–3912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Srivastava, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Vauquier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Sahidullah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Bellet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Tommasi, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Vincent, “Evaluating voice conversion-based privacy protection against informed attackers,” in ICASSP 2020- 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 2802–2806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Srivastava, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Tomashenko, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Vincent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Ya- magishi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Maouche, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Bellet, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Tommasi, “Design choices for x-vector based speaker anonymization,” arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='08601, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Qian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Du, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Hou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jung, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Li, “Hide- behind: Enjoy voice input with voiceprint unclonability and anonymity,” in Proceedings of the 16th ACM Conference on Em- bedded Networked Sensor Systems, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 82–94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [20] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Srivastava, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Bellet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Tommasi, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Vincent, “Privacy-preserving adversarial representation learning in asr: Reality or illusion?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='04913, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Salem, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Bhattacharya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Backes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Fritz, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, “Updates-leak: Data set inference and reconstruction attacks in online learning,” in 29th {USENIX} Security Symposium ({USENIX} Security 20), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1291–1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [22] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Shao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Xuan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Chang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, “Defend- ing model inversion and membership inference attacks via predic- tion purification,” arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='03915, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jia, “Improving robustness to model inversion attacks via mutual information regularization,” arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='05241, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [24] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Arafin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Qu, “Midas: Model inversion de- fenses using an approximate memory system,” in 2020 Asian Hardware Oriented Security and Trust Symposium (AsianHOST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Dwork, “Differential privacy,” in International Colloquium on Automata, Languages, and Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Springer, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Jia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Pei, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Li, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Song, “The secret revealer: Generative model-inversion attacks against deep neural networks,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' 253–261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Lv, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Meng, “Mfa-conformer: Multi-scale feature aggrega- tion conformer for automatic speaker verification,” arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='15249, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' [28] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Desplanques, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Thienpondt, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content=' Demuynck, “Ecapa- tdnn: Emphasized channel attention, propagation and ag- gregation in tdnn based speaker verification,” arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} +page_content='07143, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfegQF/content/2301.03206v1.pdf'} diff --git a/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf b/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..659442d69fdcc646d7acda40d45761f22034bc67 --- /dev/null +++ b/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:70551be9269739d664eab93fa5e29f309c68d61b7adff73d9a1ded658e13ec1a +size 1438234 diff --git a/fNE3T4oBgHgl3EQffQos/vector_store/index.pkl b/fNE3T4oBgHgl3EQffQos/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ccff11b197fd2ae37de549b5dcc83156866244ad --- /dev/null +++ b/fNE3T4oBgHgl3EQffQos/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:64a815d30d37f8b2bfa336f90d97d998d767d7d522e9dc7b05222651a1884666 +size 360117 diff --git a/gdE3T4oBgHgl3EQfIAlQ/content/tmp_files/2301.04329v1.pdf.txt b/gdE3T4oBgHgl3EQfIAlQ/content/tmp_files/2301.04329v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..98404dafe519f814e3556a807f1cd6cb2b800c35 --- /dev/null +++ b/gdE3T4oBgHgl3EQfIAlQ/content/tmp_files/2301.04329v1.pdf.txt @@ -0,0 +1,1206 @@ +Coexistence of zigzag antiferromagnetic order and +superconductivity in compressed NiPSe3 +Hualei Sun1, Liang Qiu1, Yifeng Han3, Enkui Yi1, Junlong Li4, Mengwu Huo1, Chaoxin +Huang1, Hui Liu1, Manrong Li3, Weiliang Wang2, Dao-Xin Yao1, Benjamin A. +Frandsen5, Bing Shen1,*, Yusheng Hou1,#,and Meng Wang1,† + +1Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, +School of Physics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China +2 School of Physics, Guangdong Province Key Laboratory of Display Material and Technology, Sun Yat-Sen University, Guangzhou, +Guangdong 510275, China +3 Key Laboratory of Bioinorganic and Synthetic Chemistry of Ministry of Education, School of Chemistry, Sun Yat-Sen University, +Guangzhou, Guangdong 510275, China +4Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China +5Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA +# houysh@mail.sysu.edu.cn +* shenbing@mail.sysu.edu.cn +† wangmeng5@mail.sysu.edu.cn + +Abstract +NiPSe3 is regarded as a bandwidth-controlled Mott insulator, distinct from the widely studied Mott +insulating magnetic graphene MPSe3 (M = Mn and Fe) family. By employing high-pressure synchrotron +X-ray diffraction, we observe two structural transitions as a function of pressure. With the help of first- +principles calculations, we discover the antiferromagnetic (AFM) moment directions of NiPSe3 switch +from out-of-plane to in-plane and the honeycomb layers slide relative to each other at the first structural +transition. The in-plane AFM order persists until the second structural transition, whereupon the two- +dimensional (2D) structure assumes a more three-dimensional (3D) character. A bandwidth-controlled +Mott insulator-metal transition (IMT) occurs between the two structural transitions at Pc≈8.0 GPa, +concomitant with the emergence of superconductivity with Tc≈4.8 K. The superconductivity in NiPSe3 +emerging in the 2D monoclinic phase coexists with the in-plane AFM order and continues into the 3D +trigonal phase. Our electronic structure calculations reveal that the Mott IMT and superconductivity in +NiPSe3 are both closely related to the enhanced Se2- 4p and Ni2+ 3d electronic hybridizations under +pressure. From these results, we construct the temperature-pressure electronic phase diagram of NiPSe3, +revealing rich physics and many similarities with copper oxide and iron-based superconductors. + +Introduction +The MPSe3 family, where M is a transition metal, has attracted extensive attention for their unique +magnetic properties and potential for spintronic device applications. The long-range AFM order in +MPSe3 can be maintained even down to the monolayer scale, such that these materials have been +described as “magnetic graphene”1,2,3,4,5. This is possible because the magnetocrystalline anisotropic +energy of magnetism counteracts the tendency of thermal fluctuations to destroy 2D magnetic order6. +The magnetocrystalline anisotropic energy gap results from distortions of the MSe6 octahedra and the +hexagonal honeycomb structure7,8. The anisotropic magnetism also leads to different types of magnetic + +interactions and diverse magnetic ordered ground states. For example, MnPSe3 exhibits a Néel type AFM +with spins pointing parallel to the van der Waals (vdW) plane9, while FePSe3 possesses zig-zag type AFM +with spins perpendicular to the vdW plane10. +For MPSe3 (M = Mn, Fe, and Ni) under atmospheric pressure, the MSe6 octahedra have a nearly regular +coordination structure. The transition metals Mn, Fe, and Ni are all in the high spin magnetic states. +Pressure can induce distortions of the MSe6 octahedra and is expected to tune the crystal field, possibly +resulting in spin state transitions of the 3d metals. MnPSe3 and FePSe3 both have partially filled eg and +t2g states. The pressure-induced spin state transitions, called spin crossover transitions, are indeed found +coincident with a dramatic decrease of the ionic radius of Mn2+ and Fe2+ ions. Concomitantly, the d-d +overlap of the two t2g orbitals between the nearest two Mn2+ or Fe2+ ions causes a Mn or Fe dimer to +form9,10,11. Thus, the pressure-induced spin state transition and the formation of metallic bonds in MnPSe3 +and FePSe3 are accompanied by a structural transition12,13. In addition, superconductivity (SC) was found +in the non-magnetic state of FePSe3, where the spin of Fe2+ is S=0. Previous calculations predicted the +lattice structures of NiPSe3 to undergo different behaviors during the IMT under pressure, due to the +different occupation states of the eg and t2g orbitals of Ni ions12,13. Therefore, it is of highly interesting to +explore the properties of NiPSe3 under pressure. However, high pressure studies on NiPSe3 are absent +due to the lack of available single crystals. +Here, we report comprehensive high-pressure studies on NiPSe3 single crystals up to 34.0 GPa utilizing +synchrotron X-ray diffraction (XRD), electrical resistance measurements, and first-principles +calculations. NiPSe3 undergoes two structural transitions at pressures of ~4.0 and ~15.0 GPa, respectively. +The first structural transition corresponds to a sliding of the honeycomb layers accompanied by a +reorientation of the moments in the AFM zigzag order from out-of-plane to in-plane. The second one is +a transition from the monoclinic symmetry high-pressure I (HP-I) phase to the nonmagnetic trigonal +symmetry HP-II phase, coincident with a 2D to 3D structural transition. The in-plane AFM order is +suppressed gradually in the 2D monoclinic HP-I phase. An IMT occurs at ~8.0 GPa between the two +structural transitions, consistent with a bandwidth-controlled Mott transition. Superconductivity appears +immediately following the IMT, coexisting with the zigzag AFM order in the HP-I phase and persisting +into the nonmagnetic HP-II phase. + +Results +Pressure-induced structural transitions in NiPSe3. The Ni2+ ions in NiPSe3 form a 2D honeycomb +layered lattice. Each Ni2+ ion is octahedrally coordinated by six nearest-neighbor Se atoms. The +honeycomb sublattices stack along the c axis. Figure 1a shows the high-pressure powder XRD patterns +taken at room temperature up to 25.7 GPa. Two distinct phase transitions can be identified in this pressure +range. The first transition occurs at ~4.0 GPa with a new peak appearing at ~13.2°. We call this the HP- +I phase. The peak at ~13.7° in the low pressure (LP) phase is suppressed with further increasing pressure. +The second transition occurs around 15.0 GPa, as evidenced by a new peak emerging at ~14.4°. The peak +at ~13.6° from the HP-I phase disappears quickly under additional pressure. Rietveld refinements +performed with TOPAS-Academic +14 are shown for representative pressures in the HP-II, HP-I, and LP +phases in Figs. 1b, 1c, and 1d, respectively. Figure 2 displays the refined structures. The detailed +structural parameters are listed in Supplementary Table S1. +The structural transition at ~4.0 GPa is an isomorphic structural transition, where the honeycomb +layers shift relative to each other in a sliding motion of ~a/3 along the a-axis, resulting in the β angle of +the monoclinic unit cell contracting to nearly 90°. Such a large sliding of the honeycomb sublattices is + +possible due to the weak interlayer vdW interactions. The unit cell remains in the monoclinic space group +C2/m. The second structural transition to a trigonal symmetry (space group P-31m) at ~15 GPa is non- +isomorphic. There is an obvious reduction of the interlayer distance. At 25.7 GPa, the distance between +the two nearest P ions is 2.224(3) Å, which is much less than the distance between two vdW-coupled P +ions (3.8 Å) and quite close to the phosphate dimer distance (2.21 Å)15. This collapse along the c axis +therefore corresponds to a transition from the quasi-2D layered structure at lower pressure to a genuinely +3D structure at higher pressure. A similar structural transition and 2D-3D crossover in NiPSe3 were +recently reported16. + +Fig. 1 a High-pressure XRD patterns from 2.2 to 25.7 GPa. The X-ray wavelength is λ=0.6199 Å. Two structural transitions +occur, one between 3.2 and 4.5 GPa, and the other between 14.3 and 15.5 GPa. b-d Rietveld refinements at 25.7, 6.3 and 2.2 GPa, +corresponding to the HP-II, HP-I, and LP phases. + + +Fig. 2 Schematics of the structural phases of NiPSe3 under pressure. (a, d) Refined structure in the LP phase at 2.2 GPa, +displayed along a viewing axis perpendicular to and parallel to the vdW planes, respectively. The different orientations are drawn +to the same scale with respect to the lattice parameters. (b, e) Equivalent figures for the HP-I structure at 6.3 GPa. (c, f) Equivalent +figures for the HP-II structure at 25.7 GPa. + +Pressure-induced IMT and superconductivity. We now investigate the possibility that changes in +electrical transport properties of NiPSe3 accompany the observed structural transitions17–20. Figure 3a +shows the temperature dependence of the resistance for a single crystal of NiPSe3 measured at high + +a +P (GPa) +Observed +P = 25.7 GPa +Calculated +Difference +25.7 +HP-II phase +Intensity (arb.unit) +Observed +P = 6.3 GPa +15.5 +Calculated +Difference +14.3 +HP-I phase +Observed +P= 2.2 GPa +Calculated +Difference +4.5 +LP phase +3.2 +2.2 +10 +12 +14 +16 +18 +20 +22 +24 +26 10 +12 +14 +16 +1820222426 +20 (degree) +20 (degree)a +CORO +b +- +Q +b +b +b +Ni +P +Se +C +apressures up to 32.0 GPa. We observe a clear IMT under pressure. The resistance as a function of pressure +for selected temperatures is presented in Fig. 3b. At all measured temperatures, we observe an abrupt +decrease of the resistance at the pressure corresponding to the isomorphic structural transition from the +LP phase to the HP-I phase, with NiPSe3 becoming completely metallic at ~ 8.0 GPa, which is between +the two structural transitions. The electrical transport measurements have been repeated on several +samples (see Supplementary Figs. S2a and S2b). +In conjunction with this IMT, we observed a significant drop in resistance below 4.9 K for an applied +pressure of 8.6 GPa. We will show that this corresponds to a SC transition. The transition temperature +extracted from the resistance curves initially increases as a function of pressure across the HP-I and HP- +II phases, reaches a maximum of 5.9 K around 27.6 GPa, and then remains constant or decreases slightly +with higher pressures. This is illustrated in Fig. 3d. To determine whether the drop in resistance represents +a SC transition, we measured the resistance in an applied magnetic field in the HP-I phase at 14.0 GPa +(Figs. 3e and 3f) and the HP-II phase at 15.6 GPa (Figs. 3g and 3h). The drop in resistance is clearly +suppressed to lower temperature with increasing magnetic field, revealing a SC transition. We note that +the resistance remains nonzero at low temperature, but this is commonly observed for pressure-induced +superconductivity, possibly due to lattice distortions or inhomogeneous pressure21,22,23,24. The Ginzburg- +Landau formula 𝜇!𝐻"#(𝑇) = 𝜇!𝐻"#(0) (1 − + +$ +$!, +# +- is adopted to fit the upper critical field of the +superconductivity. The fitted values of 𝜇!𝐻"# are 0.91 T at 14.0 GPa and 3.95 T at 15.6 GPa, lower than +the Pauli limit. We see that both the transition temperature and the upper critical field are enhanced in +the HP-II phase (see Supplementary Figs. S2c-S2f). + + +Fig. 3 Electrical transport measurements under pressure and magnetic field. a Temperature dependence of the resistance for +single-crystal NiPSe3 at pressures between 2.5 to 32.0 GPa. The vertical axis is on a logarithmic scale. The color of each curve +indicates the corresponding pressure. b Pressure dependence of the resistance at various temperatures up to 300 K. The vertical +axis is on a logarithmic scale. The grey, blue, and yellow backgrounds indicate the LP, HP-I, and HP-II phases. The dashed line +indicates the IMT at ~ 8.0 GPa. c A zoomed-in plot of the resistance as a function of temperature from 2 to 8 K at pressures between +7.6 and 34.0 GPa. d SC transition temperature Tc extracted from the measurements of two single crystals at various pressures. e, g +Magnetic field dependence of Tc at 14.0 and 15.6 GPa. f, h Ginzburg-Landau fits to the experimentally determined Tc as a function +of magnetic field at 14.0 and 15.6 GPa. The values of the upper critical field 𝜇!𝐻"#(0) and Tc in the absence of an applied magnetic +field are labelled. + +DFT calculations of electronic structure and magnetic order. To obtain the comprehensive electronic +structure of NiPSe3 under different pressures, we first investigate the band structure using density + +a +105 +b +e +LP +HP-I +HP-II +1.0 +10* +P (GPa) +run2 +0.0550 +F +104 +(u) +50K +run2 +Resistance (2) +0.8 +103 +103 +- +Resistance +Resistance +100K +150K +P = 14.0 GPa +E +102 +0.6 +102 +2.5 +3.4 +5.5 +6.9 +IMT +8.0 +10.8 +15.6 +200K +0.0525 +OT +12.4 +0.4 +P = 14.0 GPa +101 +22.0 +32.0 +10 +- +250K +0.5 T +300K +1 T +T, = 5.0 K +10° +100 +1.5T +0.2 +TCT +μ,H(0) = 0.91T +10-1 +10-1 +0.0500 +35 +0.0 +0 +50 +100 +150 +200 +250 +300 +0 +5 +10 +20 +25 +30 +2 +4 +6 +8 +0 +2 +4 +6 +Temperature (K) +Pressure (GPa) +Temperature (K) +T。 (K) +0.12 +d +7.0 +LP +HP-I1 +HP-II +g +0.165 +h +4.0 +P (GPa) +run1 +() +6.0 +0.160 +P = 15.6 GPa +E +3.0 +Resistance +0.11 +Resistance ( +10 +0.5T +7.6 +8.6 +0.155 +1T +1.5T +10.6 +12.6 +P = 15.6 GPa +16.6 +18.6 +5.0 +2T +2.5T +20.8 +22.6 +· +T, run1 +3T +3.5T +T, = 5.3 K +0.150 +27.6 +29.2 +1.0 +31.2 +34.0 +T run2 +4T +4.5T +0.10 +μ,H(0) = 3.95 T +5 +4.0 +0.145, +0.05 +2 +4 +6 +8 +0 +5 +10 +15 +20 +25 +30 +35 +2 +4 +6 +8 +0 +2 +4 +6 +Temperature (K) +Pressure(GP) +Temperature (K) +T (K)functional theory (DFT) calculations. As shown in Fig. 4a, NiPSe3 remains insulating with an indirect +gap of 0.316 eV when the pressure is 6.3 GPa. As the pressure increases to 12.3 GPa, there are electronic +pockets between Γ and Y but holes between B and E in our DFT calculated band structure (Fig. 4b). +Such features indicate that NiPSe3 has become a metal, consistent with the experimentally observed +pressure-induced IMT at 8.0 GPa. When the pressure is further increased to 25.7 GPa, many bands cross +the Fermi level (Fig. 4c), revealing that NiPSe3 is a good metal. From the projected density of state +(PDOS) as shown in Figs. 4d-4f, we can see that the 3d states of Ni2+ ions are mainly located below the +Fermi level, irrespective of the pressure. It is worth noting that the PDOS from Ni2+ and Se2- ions +significantly increase near the Fermi level as the pressure increases. This is understandable because the +pressure shortens the bond lengths between Ni2+ and Se2- ions and thus enhances the p-d hybridization. +Therefore, the IMT and superconductivity in NiPSe3 are closely related to the enhanced p-d hybridization +between Ni2+ and Se2- ions under pressure. Previous calculations also show that the partially filled eg +orbital of Ni2+ is at the Fermi level and contributes to the metallic electrons25. + +Fig. 4 DFT-calculated electronic structure of NiPSe3 under different pressures. a-c Band structure with spin–orbit coupling +(SOC) included at 6.3, 12.3, and 25.7 GPa. d-f PDOS of Ni 3d, Se 4p, and P 3p orbitals at 6.3, 12.3, and 25.7 GPa. + +To understand the influence of pressure on the magnetic order and the Néel temperature TN in NiPSe3, +we investigate the magnetic exchange couplings based on the following spin model: +. +Considering the layered honeycomb structure of NiPSe3, the five nearest neighbor (NN) Heisenberg +exchange couplings (parameterized by Jij) are included. In Eq. (1), A is the single-ion magnetic anisotropy +parameter. The values of TN under different pressures could be identified from both resistance and +calculations (see Supplementary Figs. S3 and S4, and Table S2). The calculated TN at atmospheric +pressure is 189.9 K, close to 206 K determined by neutron scattering26. As the pressure increases, TN +obtained from the resistance curves decreases from 203.1 K at 2.0 GPa to 139.2 K at 7.6 GPa, as shown +in Fig. 5. DFT calculations and Monte Carlo simulations indicate that the magnetic structures are +different in the LP and HP-I phases, despite the isomorphic nature of the structural phase transition. For +the LP phase, the magnetic moment directions are perpendicular to the honeycomb layers and exhibit an +( +) +( ) +2 += +1 +z +ij +i +j +i +ij +i +H +J +A +S +× ++ +å +å +S S + +a +b +c +6.3 GPa + Ni +GPa +S +(eV) +(eV) +0 +Energy +Energy +0 +Z +r +Y +A +B +D +E +C +Z +Y +A +B +D +E +C +M K +r +A +L +H +A +d +6 +e +4 +6.3 GPa +6 +12.3GPa +6 +25.7 GPa +PDOS (a.u.) +(a.u.) +(a.u.) +4 +4 +Ni-d +Se-p +PDOS +PDOS +P-p +2 +2 +0 +0 +- +.4 +-3 +-2 +-1 +0 +? +3 +2 +.4 +.3 +-2 +-1 +0 +2 +3 +Energy (eV) +Energy (eV) +Energy (eV)AFM arrangement within the plane and a FM arrangement between planes (designated as zigzag-out, see +the inset in Fig. 5). The gap is 0.99 eV. For the HP-I phase, the magnetic moment directions are within +the honeycomb layers, displaying an AFM arrangement both within and between layers (designated as +zigzag-in, see the inset in Fig. 5). Lastly, our DFT calculations show that the HP-II phase is non-magnetic. +Pressure causes a significant decrease of the interlayer distance, resulting in further enhancement of the +metallization and the disappearance of the magnetism. This is confirmed by high pressure Raman +scattering in NiPS3, which showed a complete suppression of TN at the second structural transition27. + + +Fig. 5 Phase diagram of NiPSe3 under pressure. The insets illustrate the magnetic structures, i.e. zigzag-out for the LP phase and +zigzag-in for the HP-I phase. TN for the AFM order and Tc for the superconductivity are extracted from resistance measurements +and DFT calculations. + +Discussion +Bandwidth controlled insulator-metal transition. The IMT occurring at ~8.0 GPa in NiPSe3 is +separated from the structural transitions. This is distinct from the d4-d7 transition metal complexes, where +both the t2g and eg orbitals of the transition metals are partially filled. For FePSe3 and MnPSe3, the IMT, +spin crossover, and structural transition appear simultaneously due to the interorbital eg-t2g hopping. +However, Ni2+ (3d8) has fully filled t2g orbitals and partially filled eg orbitals, producing no ionic radius +collapse under pressure. Pressure induces the sliding of the vdW layers before the IMT. It was suggested +that NiPS3 exhibits negative charge-transfer character with a d9 L configuration where L indicates a ligand +(chalcogen) hole 28–30. The IMT can be ascribed to progressive enhancement of the p-d hybridization +between Ni2+ and Se2- ions and increase of the Ni-d9 configuration (S=1/2) under pressure12,13, thus +qualifying it as a bandwidth controlled IMT. The 2D monoclinic structure with vdW layers observed in +the LP and HP-I phases transforms into a 3D trigonal structure in the HP-II phase, in low-spin S=0 Ni +ions with itinerant electrons are favored. + +Coexistence of superconductivity and magnetic order. Our theoretical analysis reveals the LP phase +to be a Mott insulator with out-of-plane zigzag AFM order. With increasing pressure, the structure +transitions to the HP-I phase characterized by the honeycomb layers sliding relative to each other, and +the magnetic moments reorient to point in the plane. Superconductivity emerges from the zigzag AFM +parent phase in the HP-I structure. The magnetic order in the SC phase is dominated by the S=1/2 AFM + +300 +LP +HP-I +HP-1I +C2/m +C2/m +250 +P/31m +IMT +(K) +200 +5 × Tc run1 +Temperature +5 × Tc run2 +150 +T run1 +T run2 +100 +T calculation +50 +AFM +SC +0 +0 +5 +10 +15 +20 +25 +30 +Pressure (GPa)order, due to charge transfer from Se under pressure. The magnetism and superconductivity coexist and +compete reminiscent of copper oxide and iron-based superconductors. In the HP-II phase, the S=0 state +is favored, yielding a nonmagnetic phase. + +In summary, our work reveals that pressure induces structural and magnetic transitions in NiPSe3. A +bandwidth-controlled insulator-metal transition is observed, accompanied by the emergence of +superconductivity. Interestingly, the LP phase and HP-I phase have different zigzag antiferromagnetic +orders with magnetic moments perpendicular and parallel to the honeycomb layers. Pressure induces a +significant reduction of the interlayer distance, resulting in further enhancement of the metallization and +the disappearance of the magnetism. The HP-II phase shows 3D character with a nonmagnetic ground +state. The variation of the structural dimension and the cooperation between spin and lattice degrees of +freedom make NiPSe3 an interesting compound to explore intertwined superconductivity and magnetism. + +Methods +Material synthesis. NiPSe3 powders were synthesized by heating the stoichiometric Ni, P and Se +powders at 600 ℃ in sealed quartz ampoule for two weeks. High quality single crystals of NiPSe3 were +grown with pure NiPSe3 powders mixed with NaCl/AlCl3 using the chemical vapor transport method. +The starting materials were sealed in a quartz ampoule and placed in a two-zone furnace for one month. +The NiPSe3 single crystals were acquired by dissolving the flux by water. The crystals were naturally +cleaved along the (001) surface with shiny gray appearance. The crystals were irregular hexagons with +sides of ~1 mm in length and ~10-50 μm in thickness. + +High-pressure XRD measurements. High-pressure synchrotron radiation XRD patterns of NiPSe3 were +collected at 300 K with an X-ray wavelength of 0.6199 Å. A symmetric diamond anvil cell (DAC) with +a pair of 300-μm-diameter culets was used. A sample chamber with a diameter of 110 μm was laser- +drilled in a pre-indented steel gasket. The NiPSe3 powders were compressed into a pellet with a 60-μm +diameter and 20-μm thickness. The pellet was loaded into the middle of the sample chamber and silicone +oil was used as a pressure transmitting medium. A ruby sphere was also loaded into the sample chamber +and pressure was determined by measuring the shift of its fluorescence wavelength. The data were +initially integrated using Dioptas31 (with a CeO2 calibration) and the subsequent Rietveld refinements +were processed using TOPAS-Academic.14 + +High-pressure electrical property measurements. Magnetic and electrical measurements were taken +on a physical property measurement system (PPMS, Quantum Design). High-pressure electrical transport +measurements of NiPSe3 single crystals were carried out using a miniature DAC made from a Be–Cu +alloy on a PPMS. Diamond anvils with a 300-μm culet were used, and the corresponding sample chamber +(with a diameter of 110-μm) was made in an insulating gasket achieved by cubic boron nitride and epoxy +mixture. NaCl powders were employed as the pressure-transmitting medium, providing a quasi- +hydrostatic environment. The pressure was also calibrated by measuring the shift of the fluorescence +wavelength of the ruby sphere, which was loaded in the sample chamber. The standard four-probe +technique was adopted for these measurements. + +First-principles calculations. DFT calculations are performed using the Vienna ab initio Simulation +Package (VASP) at the level of the generalized gradient approximation.32,33 We adopted the projector + +augmented wave pseudopotentials and a plane-wave cutoff energy of 500 eV.34 The experimentally +measured lattice constants were used in our calculations and the positions of all atoms were fully relaxed +until the force on each atom was less than 0.01 eV/Å. We used U = 2.8 eV for atmospheric pressure but +U = 0.4 eV for higher pressures to describe the correlation among 3d electrons of Ni2+ ions. The values +of TN were obtained through parallel tempering Monte Carlo (MC) simulations.35,36 + +References +1. +Park, J. Opportunities and challenges of 2D magnetic van der Waals materials : magnetic +graphene ? J. Phys. Condens. Matter 28, 301001 (2016). +2. +Gong, C. & Zhang, X. Two-dimensional magnetic crystals and emergent heterostructure +devices. Science. 363, eaav4450 (2019). +3. +Liu, B. et al. Critical behavior of the quasi-two-dimensional semiconducting ferromagnet +CrSiTe3. Sci. Rep. 6, 33873 (2016). +4. +Kim, K. et al. Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer +NiPS3. Nat. Commun. 10, 345 (2019). +5. +Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the +monolayer limit. Nature 546, 270–273 (2017). +6. +Seyler, K. L. et al. Ligand-field helical luminescence in a 2D ferromagnetic insulator. Nat. +Phys. 14, 277–281 (2018). +7. +Sun, Y. J., Tan, Q. H., Liu, X. L., Gao, Y. F. & Zhang, J. Probing the Magnetic Ordering of +Antiferromagnetic MnPS3 by Raman Spectroscopy. J. Phys. Chem. Lett. 10, 3087–3093 +(2019). +8. +Calder, S. et al Magnetic exchange interactions in the van der Waals layered antiferromagnet +MnPSe3. Phys. Rev. B 103, 024414 (2021). +9. +Kawakami, T. et al. Spin transition in a four-coordinate iron oxide. Nat. Chem. 1, 371–376 +(2009). +10. +Struzhkin, V. V., Badro, J., Shu, J., Hemley, R. J. & Mao, H. K. Pressure-Induced High-Spin to +Low-Spin Transition in FeS Evidenced by X-Ray Emission Spectroscopy. Phys. Rev. Lett. 82, +3284–3287 (1999). +11. +Wang, Y. et al. Giant Pressure-Driven Lattice Collapse Coupled with Intermetallic Bonding +and Spin-State Transition in Manganese Chalcogenides. Angew. Chemie - Int. Ed. 55, 10350– +10353 (2016). +12. +Wang, Y. et al. Pressure-Driven Cooperative Spin-Crossover, Large-Volume Collapse, and +Semiconductor-to-Metal Transition in Manganese(II) Honeycomb Lattices. J. Am. Chem. Soc. +138, 15751 (2016). +13. +Wang, Y. et al. Emergent superconductivity in an iron-based honeycomb lattice initiated by +pressure-driven spin-crossover. Nat. Commun. 9, 1914 (2018). +14. +Coelho, A. A. TOPAS and TOPAS-Academic : an optimization program integrating computer +algebra and crystallographic objects written in C++. J. Appl. Crystallogr. 51, 210 (2018). +15. +Mann, J. B. Atomic. Structure Calculations II.Hartree-Fock Wavefunctions and Radial +Expectation Values: Hydrogen to Lawrencium. (1968). +16. +Ma, X. et al Dimensional Crossover Tuned by Pressure in Layered Magnetic NiPS3. Sci. China +Physics, Mech. Astron. 64, 297011 (2021). +17. +Sun, H. et al. Magnetism variation of the compressed antiferromagnetic topological insulator + +EuSn2As2. Sci. China Physics, Mech. Astron. 64, 118211 (2021). +18. +Sun, H. et al. Exchange field enhanced upper critical field of the superconductivity in +compressed antiferromagnetic EuTe2. (2022). +19. +Cai, W. et al. Pressure-induced superconductivity and structural transition in ferromagnetic +CrSiTe3. Phys. Rev. B 102, 144525 (2020). +20. +Liu, Z. & Sun, H. Evidence for charge and spin density wave in single crystals of La3Ni2O7 +and La3Ni2O6. Sci. China Physics, Mech. Astron. 66, 217411 (2023). +21. +Zhao, L. et al. Monoclinic EuSn2As2: A Novel High-Pressure Network Structure. Phys. Rev. +Lett. 126, 155701 (2021). +22. +Debessai, M., Matsuoka, T., Hamlin, J. J. & Schilling, J. S. Pressure-Induced Superconducting +State of Europium Metal at Low Temperatures. Phys. Rev. Lett. 102, 197002 (2009). +23. +Pfleiderer, C. et al. Coexistence of superconductivity and ferromagnetism in the d-band metal +ZrZn2. Nature 412, 58–61 (2001). +24. +Zhang, J. et al. Observation of two superconducting domes under pressure in tetragonal FeS. +npj Quantum Mater. 2, 49 (2017). +25. +Kim, H. S., Haule, K. & Vanderbilt, D. Mott Metal-Insulator Transitions in Pressurized +Layered Trichalcogenides. Phys. Rev. Lett. 123, 236401 (2019). +26. +Le Flem, G., Brec, R., Ouvard, G., Louisy, A. & Segransan, P. Magnetic interactions in the +layer compounds MPX3 (M = Mn, Fe, Ni; X = S, Se). J. Phys. Chem. Solids 43, 455–461 +(1982). +27. +Kim, S. Y. et al. Charge-Spin Correlation in van der Waals Antiferromagnet NiPS3. Phys. Rev. +Lett. 120, 136402 (2018). +28. +Takubo, K. et al. Unusual superexchange pathways in an NiS2 triangular lattice with negative +charge-transfer energy. Phys. Rev. Lett. 99, 037203 (2007). +29. +Patel, R. K. et al. Hole doping in a negative charge transfer insulator. Commun. Phys. 5, +s42005 (2022). +30. +Bisogni, V. et al. Ground-state oxygen holes and the metal-insulator transition in the negative +charge-transfer rare-earth nickelates. Nat. Commun. 7, 13017 (2016). +31. +Prescher, C. & Prakapenka, V. B. DIOPTAS : a program for reduction of two-dimensional X- +ray diffraction data and data exploration. High Press. Res. 7959, 223–230 (2015). +32. +Kresse, G. & Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations +using a plane-wave basis set. Phys. Rev. B 54, 169–186 (1996). +33. +Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. +Phys. Rev. Lett. 77, 3865–3868 (1996). +34. +Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. +Rev. B - Condens. Matter Mater. Phys. 59, 1758–1775 (1999). +35. +Lou, F. et al. PASP: Property analysis and simulation package for materials. J. Chem. Phys. +154, 114103 (2021). +36. +Hukushima, K. & Nemoto, K. Exchange Monte Carlo Method and Application to Spin Glass +Simulations. Journal of the Physical Society of Japan 65, 1604–1608 (1996). + +Acknowledgements +Work at Sun Yat-Sen University was supported by the National Natural Science Foundation of China +(Grant Nos. 12174454, 12104518, U213010013, 11974432, 92165204), Guangdong Basic and Applied + +Basic Research Funds (Grant Nos. 2021B1515120015, 2022A1515012643, 2022A1515010035, +2022B1212010008), Guangzhou Basic and Applied Basic Research Funds (Grant Nos. 202201011123, +202201011118, 202201011798), National Key Research and Development Program of China (Grant Nos. +2019YFA0705702, 2022YFA1402802, 2018YFA0306001), Shenzhen International Quantum Academy +(Grant No. SIQA202102), the Fundamental Research Funds for the Central Universities, Sun Yat-sen +University (Grant No. 22QNTD3004), DFT calculations are performed on Tianhe-II. We appreciate the +support of BSRF, IHEP, CAS for high pressure XRD measurements. + + + + + +Supplementary Materials for “Coexistence of zigzag +antiferromagnetic order and superconductivity in +compressed NiPSe3” +Hualei Sun1, Liang Qiu1, Yifeng Han3, Enkui Yi1, Junlong Li4, Mengwu Huo1, Chaoxin +Huang1, Hui Liu1, Manrong Li3, Weiliang Wang2, Dao-Xin Yao1, Benjamin A. +Frandsen5, Bing Shen1,*, Yusheng Hou1,#,and Meng Wang1,† + +1Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, +School of Physics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China +2 School of Physics, Guangdong Province Key Laboratory of Display Material and Technology, Sun Yat-Sen University, Guangzhou, +Guangdong 510275, China +3 Key Laboratory of Bioinorganic and Synthetic Chemistry of Ministry of Education, School of Chemistry, Sun Yat-Sen University, +Guangzhou, Guangdong 510275, China +4Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China +5Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA +# houysh@mail.sysu.edu.cn +* shenbing@mail.sysu.edu.cn +† wangmeng5@mail.sysu.edu.cn + +Properties of the single crystals of NiPSe3 at atmospheric pressure +An X-ray diffraction (XRD) pattern of a single crystal of NiPSe3 was measured at ambient pressure, +as shown in Fig. S1. The XRD pattern shows (00L) diffraction peaks. The obtained lattice parameter c is +6.8663(3) Å, which is consistent with the previous reports, confirming the high quality of the sample. + +Figure S1 XRD pattern of a single crystal of NiPSe3 with the corresponding Miller indices (00L) at atmospheric pressure. The +wavelength is λ=1.54 Å. The inset shows an image of a typical single crystal. + +Structural transitions of NiPSe3 under pressure +The in situ high pressure XRD patterns of both the LP and HP-I phases can be well indexed by the +monoclinic C2/m space group. During the isomorphic structural transition from LP to HP-I, there is an +obvious sliding of the honeycomb layers relative to each other. The transition occurring at about 15.0 +GPa is a non-isomorphic structural transition. At this transition, there is a large decrease of the interlayer + +(002) +2000 +Intensity (arb.unit) +(001) +1000 +(004) +(005) +(006) +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +20 (degree)spacing. The XRD patterns of the HP-II phase can be well indexed by the trigonal P-31m space group. +The related structural parameters are listed in Table S1. + +Table S1 Refined lattice parameters, atomic coordinates, and Wyckoff positions (WP) of NiPSe3 at 2.2, 6.3, and 25.7 GPa. +The LP phase at 2.2 GPa, space group: C2/m +a = 6.052(1), b = 10.440(1), and c = 6.705(1) Å, α =90˚, β = 108.45(1)°, γ =90˚, Rwp = 2.04%, Rp = 3.42% +atom +x +y +z +Occ. +WP +Ni +0 +0.331(1) +0 +1 +4g +P +0.102(7) +0 +0.161(8) +1 +4i +Se 1 +0.759(2) +0 +0.261(3) +1 +4i +Se 2 +0.251(5) +0.173(6) +0.247(2) +1 +8j +The HP-I phase at 6.3 GPa, space group: C2/m + a = 5.898(1), b = 10.348(1), and c = 6.303(1) Å, α = 90˚, β = 88.38(1)°, γ = 90˚, Rwp = 5.12%, Rp = 4.71% +atom +x +y +z +Occ +WP +Ni +0 +0.3038(5) +0 +1 +4g +P +0.04(1) +0 +0.18(1) +1 +4i +Se 1 +0.166(3) +0.181(2) +0.271(2) +1 +4i +Se 2 +0.632(3) +0 +0.309(3) +1 +8j +The HP-II phase at 25.7 GPa, space group: P-31m +a = b = 5.873(2) and c = 4.274(4) Å, α=90˚, β = 90°, γ= 120°, Rwp = 5.12%, Rp = 4.71% +atom +x +y +z +Occ +WP +Ni +1/3 +2/3 +0 +1 +2c +P +0 +0 +0 +1 +2e +Se +0.327(3) +0 +-0.330(2) +1 +6k + +High-pressure resistance and Ginzburg-Landau fitting of the 𝝁𝟎𝑯𝒄𝟐. +The electrical resistance as a function of pressure and temperature for single crystals of NiPSe3 is shown +in Figs. S2a and S2b, respectively. The resistance decreases dramatically at the first structural transition +and further decreases at ~8.0 GPa. The simple Ginzburg-Landau formula 𝜇!𝐻"#(𝑇) = 𝜇!𝐻"#(0) (1 − ++ +$ +$!, +# +- is adopted to fit the upper critical field of the superconductivity. The fitted values of 𝜇!𝐻"# are +1.26 T at 8.0 GPa and 2.80 T at 34.0 GPa, lower than the Pauli limit. + + +Figure S2 High-pressure resistance measurements. a Temperature dependence of the resistance under various pressures up to +34.0 GPa. The resistance is shown on a logarithmic scale. b Pressure dependence of the resistance at various temperatures up to +300 K shown on a logarithmic scale. The grey, blue and yellow backgrounds indicate the LP, HP-I, and HP-II phases. The vertical +dashed line indicates the IMT at ~ 8.0 GPa. c and e Magnetic field dependence of the SC transition at 8.0 and 34.0 GPa. d and f +Ginzburg-Landau formula fits to the experimentally determined Tc as a function of magnetic field at 8.0 and 34.0 GPa. + +The AFM transition in NiPSe3 under pressure +The suppressed TN of NiPSe3 under pressure is due to the variation of the magnetic exchange couplings. +At low pressure, NiPSe3 is a Mott insulator with large resistance and its magnetic moment is around +2μB/Ni2+. Because the AFM transition has an influence on the resistance, TN could be determined from +the kink in the first derivative of the resistance. There is a dramatic decrease in the resistance at the first +structural transition. After the structural transition, TN could be identified from the distinctive change in +the resistance. Figure S3 displays selected temperature dependences of the resistance under various +pressures from 2.0 to 7.6 GPa. The resistance curves are collected from two different measurements, +including run 1 and run 2. The kinks indicate the values of TN that are marked in these plots. TN decreases +from 203.1 K at 2.0 GPa to 139.2 K at 7.9 GPa. With further increasing pressure, the signal of the AFM +transition becomes weaker and cannot be identified from the resistance. As shown in Fig. S4, the five +nearest-neighbor exchange couplings are considered in NiPSe3. Combining the DFT calculations and +Monte Carlo simulations, the values of TN are determined accordingly. The results are listed in Table S2. + +a +10% +b +P (GPa) +LP +HP-I +HP-II +(α) +105 +run1 +() e +105 +104 +10° +0-50krun1 +Resistance +Resistance +10° +0—100K +103 +IMT +150K +102 +1.4 +2.0 +3.8 +5.2 +102 +12.6 +7-200K +101 +7.6 +8.6 +10.6 +16.6 +18.6 +20.8 +22.6 +10 +250 K +100 +27.6 +29.2 +31.2 +34.0 +10° +300K +10-1 +10-1 +ATTA +0 +50 +100 +150 +200 +250 +300 +0 +5 +10 +15 +20 +25 +30 +35 +Temperature (K) +Pressure(GPa) +c +0.101 +p +1.5 +Resistance +0.100 +P = 8.0 GPa +E +1.0 +OT +P = 8.0 GPa +0.099 +0.5 T +1 T +T = 4.7 K +1.5 T +,H(0) = 1.26 T +0.098 +0.0 +2 +4 +6 +8 +0 +2 +4 +6 +Temperature (K) +T, (K) +e +0.114 +f +3.0 +P = 34.0 GPa +Resistance +0.108 +2.0 +10 +0.5T +E +1T +1.5T +工 +P = 34.0 GPa +2T +2.5T +0.102 +3T +3.5T +1.0 +T. = 5.8 K +4T +4.5T +μ,H(0) = 2.80 T +5T +0.096 +0.0 +2 +6 +8 +0 +2 +4 +6 +Temperature (K) +T。 (K) +Figure S3 Temperature dependence of the resistance under pressures of a 2.0, b 3.8, c 5.2, d 7.6, e 2.5, f 3.4, and g 5.5 GPa. The +resistance curves are selected from different measurements (run 1 to 4). TN is determined by the inflection points marked in the +plots. + +Figure S4 Schematics of the Ni ions in the a LP phase and b HP-I phase. + +Table S2 DFT calculated magnetic exchange couplings J in units of meV, electronic band gap, TN, magnetic moment (M) and magnetic +anisotropy energy (MAE) of NiPSe3 under pressure. +Pressure(GP +a) +J1 +J2 +J3 +J4 +J5 +Gap(eV +) +TN(K +) +M(μ +B/Ni2+) +MAE(meV/Ni2 ++) +0 +7.90 +-2.89 +1.47 +-1.03 +-7.41 +0.994 +189.87 +1.264 +0.991 +10.4 +-8.66 +53.9 +1.71 +2.64 +3.72 +0.004 +142.40 +0.866 +-0.074 +12.3 +-7.49 +4.42 +-2.52 +2.97 +3.35 +-0.168 +122.76 +0.772 +-0.091 + +As listed in Table S2, the interlayer magnetic exchange coupling J5 changes from AFM to FM with +pressure increasing. Thus, we establish a 1×1×2 out-of-plane supercell to deliberate the magnetic ground +state. As shown in Fig.S5, the zigzag-out magnetic structure is more stable than the zigzag-in structure +when the pressure on NiPSe3 is less than 6 GPa. However, zigzag-out has higher energy than zigzag-in +when the pressure is larger than 6 GPa. When the pressure is above 15 GPa, NiPSe3 favors a non-magnetic +phase. + +a +b +dR/dT (arb.unit) +(arb.unit) +-40 +dR/dT +P = 2.0 GPa (run 1) +P = 3.8 GPa (run 1) +-80 +T, = 203.1 K +T, = 201.5 K +-12 +C +d +1.2 +0.124 +Resistance (2) +() +Resistance +0.122 +0.8 +0.120 +P = 5.2 GPa (run 1) +P = 7.6 GPa (run 1) +0.4 +T, = 179.4 K +T. = 139.2 K +0.118 +e +f +dR/dT (arb.unit) +-20 +(arb.unit) +dR/dT +10 +P = 2.5 GPa (run 2) +P = 3.4 GPa (run 2) +1: +T,= 203.3 K +T,= 203.9 K +-60 +155 +g +50 +100 +150 +200 +250 +300 +Temperature (K) +Resistance( +1.7 +P = 5.5 GPa (run 2) +1.6 +T = 177.3 K +50 +100 +150 +200 +250 +300 +Temperature (K)a +b +Figure S5 Pressure dependence of the energy difference between the zigzag-out and zigzag-in AFM orders of NiPSe3. Here, the +energies of the zigzag-out AFM order are taken as reference. + + +10 +△E (meV/atom) +0 +-10 +一 zigzag-out AFM +一zigzag-in AFM +-20 +-30 +1 +1 +0 +2 +4 +6 +8 +10 +12 +14 +Pressure (GPa) \ No newline at end of file diff --git a/gdE3T4oBgHgl3EQfIAlQ/content/tmp_files/load_file.txt b/gdE3T4oBgHgl3EQfIAlQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..039c7903e71b6c67fbbffb91a19bf9a439a0a541 --- /dev/null +++ b/gdE3T4oBgHgl3EQfIAlQ/content/tmp_files/load_file.txt @@ -0,0 +1,974 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf,len=973 +page_content='Coexistence of zigzag antiferromagnetic order and superconductivity in compressed NiPSe3 Hualei Sun1, Liang Qiu1, Yifeng Han3, Enkui Yi1, Junlong Li4, Mengwu Huo1, Chaoxin Huang1, Hui Liu1, Manrong Li3, Weiliang Wang2, Dao-Xin Yao1, Benjamin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Frandsen5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Bing Shen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Yusheng Hou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='and Meng Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='† 1Center for Neutron Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong 510275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 2 School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong Province Key Laboratory of Display Material and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong 510275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 3 Key Laboratory of Bioinorganic and Synthetic Chemistry of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' School of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong 510275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 4Beijing Synchrotron Radiation Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 5Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Brigham Young University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Provo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Utah 84602,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' USA # houysh@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='cn shenbing@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='cn † wangmeng5@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='cn Abstract NiPSe3 is regarded as a bandwidth-controlled Mott insulator, distinct from the widely studied Mott insulating magnetic graphene MPSe3 (M = Mn and Fe) family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' By employing high-pressure synchrotron X-ray diffraction, we observe two structural transitions as a function of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' With the help of first- principles calculations, we discover the antiferromagnetic (AFM) moment directions of NiPSe3 switch from out-of-plane to in-plane and the honeycomb layers slide relative to each other at the first structural transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The in-plane AFM order persists until the second structural transition, whereupon the two- dimensional (2D) structure assumes a more three-dimensional (3D) character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' A bandwidth-controlled Mott insulator-metal transition (IMT) occurs between the two structural transitions at Pc≈8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa, concomitant with the emergence of superconductivity with Tc≈4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The superconductivity in NiPSe3 emerging in the 2D monoclinic phase coexists with the in-plane AFM order and continues into the 3D trigonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Our electronic structure calculations reveal that the Mott IMT and superconductivity in NiPSe3 are both closely related to the enhanced Se2- 4p and Ni2+ 3d electronic hybridizations under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' From these results, we construct the temperature-pressure electronic phase diagram of NiPSe3, revealing rich physics and many similarities with copper oxide and iron-based superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Introduction The MPSe3 family, where M is a transition metal, has attracted extensive attention for their unique magnetic properties and potential for spintronic device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The long-range AFM order in MPSe3 can be maintained even down to the monolayer scale, such that these materials have been described as “magnetic graphene”1,2,3,4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' This is possible because the magnetocrystalline anisotropic energy of magnetism counteracts the tendency of thermal fluctuations to destroy 2D magnetic order6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The magnetocrystalline anisotropic energy gap results from distortions of the MSe6 octahedra and the hexagonal honeycomb structure7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The anisotropic magnetism also leads to different types of magnetic interactions and diverse magnetic ordered ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' For example, MnPSe3 exhibits a Néel type AFM with spins pointing parallel to the van der Waals (vdW) plane9, while FePSe3 possesses zig-zag type AFM with spins perpendicular to the vdW plane10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' For MPSe3 (M = Mn, Fe, and Ni) under atmospheric pressure, the MSe6 octahedra have a nearly regular coordination structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The transition metals Mn, Fe, and Ni are all in the high spin magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure can induce distortions of the MSe6 octahedra and is expected to tune the crystal field, possibly resulting in spin state transitions of the 3d metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' MnPSe3 and FePSe3 both have partially filled eg and t2g states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The pressure-induced spin state transitions, called spin crossover transitions, are indeed found coincident with a dramatic decrease of the ionic radius of Mn2+ and Fe2+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Concomitantly, the d-d overlap of the two t2g orbitals between the nearest two Mn2+ or Fe2+ ions causes a Mn or Fe dimer to form9,10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Thus, the pressure-induced spin state transition and the formation of metallic bonds in MnPSe3 and FePSe3 are accompanied by a structural transition12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' In addition, superconductivity (SC) was found in the non-magnetic state of FePSe3, where the spin of Fe2+ is S=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Previous calculations predicted the lattice structures of NiPSe3 to undergo different behaviors during the IMT under pressure, due to the different occupation states of the eg and t2g orbitals of Ni ions12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Therefore, it is of highly interesting to explore the properties of NiPSe3 under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' However, high pressure studies on NiPSe3 are absent due to the lack of available single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Here, we report comprehensive high-pressure studies on NiPSe3 single crystals up to 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa utilizing synchrotron X-ray diffraction (XRD), electrical resistance measurements, and first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' NiPSe3 undergoes two structural transitions at pressures of ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 and ~15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The first structural transition corresponds to a sliding of the honeycomb layers accompanied by a reorientation of the moments in the AFM zigzag order from out-of-plane to in-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The second one is a transition from the monoclinic symmetry high-pressure I (HP-I) phase to the nonmagnetic trigonal symmetry HP-II phase, coincident with a 2D to 3D structural transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The in-plane AFM order is suppressed gradually in the 2D monoclinic HP-I phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' An IMT occurs at ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa between the two structural transitions, consistent with a bandwidth-controlled Mott transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Superconductivity appears immediately following the IMT, coexisting with the zigzag AFM order in the HP-I phase and persisting into the nonmagnetic HP-II phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Results Pressure-induced structural transitions in NiPSe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The Ni2+ ions in NiPSe3 form a 2D honeycomb layered lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Each Ni2+ ion is octahedrally coordinated by six nearest-neighbor Se atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The honeycomb sublattices stack along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Figure 1a shows the high-pressure powder XRD patterns taken at room temperature up to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Two distinct phase transitions can be identified in this pressure range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The first transition occurs at ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa with a new peak appearing at ~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We call this the HP- I phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The peak at ~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7° in the low pressure (LP) phase is suppressed with further increasing pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The second transition occurs around 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa, as evidenced by a new peak emerging at ~14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The peak at ~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6° from the HP-I phase disappears quickly under additional pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rietveld refinements performed with TOPAS-Academic 14 are shown for representative pressures in the HP-II, HP-I, and LP phases in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 1b, 1c, and 1d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Figure 2 displays the refined structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The detailed structural parameters are listed in Supplementary Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The structural transition at ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa is an isomorphic structural transition, where the honeycomb layers shift relative to each other in a sliding motion of ~a/3 along the a-axis, resulting in the β angle of the monoclinic unit cell contracting to nearly 90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Such a large sliding of the honeycomb sublattices is possible due to the weak interlayer vdW interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The unit cell remains in the monoclinic space group C2/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The second structural transition to a trigonal symmetry (space group P-31m) at ~15 GPa is non- isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' There is an obvious reduction of the interlayer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' At 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa, the distance between the two nearest P ions is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='224(3) Å, which is much less than the distance between two vdW-coupled P ions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 Å) and quite close to the phosphate dimer distance (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='21 Å)15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' This collapse along the c axis therefore corresponds to a transition from the quasi-2D layered structure at lower pressure to a genuinely 3D structure at higher pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' A similar structural transition and 2D-3D crossover in NiPSe3 were recently reported16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 1 a High-pressure XRD patterns from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The X-ray wavelength is λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6199 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Two structural transitions occur, one between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 GPa, and the other between 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' b-d Rietveld refinements at 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 GPa, corresponding to the HP-II, HP-I, and LP phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 2 Schematics of the structural phases of NiPSe3 under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (a, d) Refined structure in the LP phase at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 GPa, displayed along a viewing axis perpendicular to and parallel to the vdW planes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The different orientations are drawn to the same scale with respect to the lattice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (b, e) Equivalent figures for the HP-I structure at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (c, f) Equivalent figures for the HP-II structure at 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure-induced IMT and superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We now investigate the possibility that changes in electrical transport properties of NiPSe3 accompany the observed structural transitions17–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Figure 3a shows the temperature dependence of the resistance for a single crystal of NiPSe3 measured at high a P (GPa) Observed P = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa Calculated Difference 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 HP-II phase Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='unit) Observed P = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 GPa 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 Calculated Difference 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 HP-I phase Observed P= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 GPa Calculated Difference 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 LP phase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 10 12 14 16 18 20 22 24 26 10 12 14 16 1820222426 20 (degree) 20 (degree)a CORO b Q b b b Ni P Se C apressures up to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We observe a clear IMT under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The resistance as a function of pressure for selected temperatures is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' At all measured temperatures, we observe an abrupt decrease of the resistance at the pressure corresponding to the isomorphic structural transition from the LP phase to the HP-I phase, with NiPSe3 becoming completely metallic at ~ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa, which is between the two structural transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The electrical transport measurements have been repeated on several samples (see Supplementary Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S2a and S2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' In conjunction with this IMT, we observed a significant drop in resistance below 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='9 K for an applied pressure of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We will show that this corresponds to a SC transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The transition temperature extracted from the resistance curves initially increases as a function of pressure across the HP-I and HP- II phases, reaches a maximum of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='9 K around 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa, and then remains constant or decreases slightly with higher pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' To determine whether the drop in resistance represents a SC transition, we measured the resistance in an applied magnetic field in the HP-I phase at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 3e and 3f) and the HP-II phase at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 3g and 3h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The drop in resistance is clearly suppressed to lower temperature with increasing magnetic field, revealing a SC transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We note that the resistance remains nonzero at low temperature, but this is commonly observed for pressure-induced superconductivity, possibly due to lattice distortions or inhomogeneous pressure21,22,23,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The Ginzburg- Landau formula 𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='𝐻"#(𝑇) = 𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='𝐻"#(0) (1 − + $ $!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', # is adopted to fit the upper critical field of the superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The fitted values of 𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='𝐻"# are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='91 T at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='95 T at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa, lower than the Pauli limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We see that both the transition temperature and the upper critical field are enhanced in the HP-II phase (see Supplementary Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S2c-S2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 3 Electrical transport measurements under pressure and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' a Temperature dependence of the resistance for single-crystal NiPSe3 at pressures between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The vertical axis is on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The color of each curve indicates the corresponding pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' b Pressure dependence of the resistance at various temperatures up to 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The vertical axis is on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The grey, blue, and yellow backgrounds indicate the LP, HP-I, and HP-II phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The dashed line indicates the IMT at ~ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' c A zoomed-in plot of the resistance as a function of temperature from 2 to 8 K at pressures between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' d SC transition temperature Tc extracted from the measurements of two single crystals at various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' e, g Magnetic field dependence of Tc at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' f, h Ginzburg-Landau fits to the experimentally determined Tc as a function of magnetic field at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The values of the upper critical field 𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='𝐻"#(0) and Tc in the absence of an applied magnetic field are labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' DFT calculations of electronic structure and magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' To obtain the comprehensive electronic structure of NiPSe3 under different pressures, we first investigate the band structure using density a 105 b e LP HP-I HP-II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 10* P (GPa) run2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0550 F 104 (u) 50K run2 Resistance (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 103 103 Resistance Resistance 100K 150K P = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa E 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 102 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='9 IMT 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 200K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0525 OT 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 P = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa 101 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 10 250K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 T 300K 1 T T, = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 K 10° 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 TCT μ,H(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='91T 10-1 10-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0500 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 0 50 100 150 200 250 300 0 5 10 20 25 30 2 4 6 8 0 2 4 6 Temperature (K) Pressure (GPa) Temperature (K) T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='12 d 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 LP HP-I1 HP-II g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='165 h 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 P (GPa) run1 () 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='160 P = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa E 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 Resistance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='11 Resistance ( 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='155 1T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 P = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 2T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 T, run1 3T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T T, = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='150 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 T run2 4T 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='10 μ,H(0) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='95 T 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='145, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='05 2 4 6 8 0 5 10 15 20 25 30 35 2 4 6 8 0 2 4 6 Temperature (K) Pressure(GP) Temperature (K) T (K)functional theory (DFT) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 4a, NiPSe3 remains insulating with an indirect gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='316 eV when the pressure is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' As the pressure increases to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 GPa, there are electronic pockets between Γ and Y but holes between B and E in our DFT calculated band structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Such features indicate that NiPSe3 has become a metal, consistent with the experimentally observed pressure-induced IMT at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' When the pressure is further increased to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa, many bands cross the Fermi level (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 4c), revealing that NiPSe3 is a good metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' From the projected density of state (PDOS) as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 4d-4f, we can see that the 3d states of Ni2+ ions are mainly located below the Fermi level, irrespective of the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' It is worth noting that the PDOS from Ni2+ and Se2- ions significantly increase near the Fermi level as the pressure increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' This is understandable because the pressure shortens the bond lengths between Ni2+ and Se2- ions and thus enhances the p-d hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Therefore, the IMT and superconductivity in NiPSe3 are closely related to the enhanced p-d hybridization between Ni2+ and Se2- ions under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Previous calculations also show that the partially filled eg orbital of Ni2+ is at the Fermi level and contributes to the metallic electrons25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 4 DFT-calculated electronic structure of NiPSe3 under different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' a-c Band structure with spin–orbit coupling (SOC) included at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3, and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' d-f PDOS of Ni 3d, Se 4p, and P 3p orbitals at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3, and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' To understand the influence of pressure on the magnetic order and the Néel temperature TN in NiPSe3, we investigate the magnetic exchange couplings based on the following spin model: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Considering the layered honeycomb structure of NiPSe3, the five nearest neighbor (NN) Heisenberg exchange couplings (parameterized by Jij) are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (1), A is the single-ion magnetic anisotropy parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The values of TN under different pressures could be identified from both resistance and calculations (see Supplementary Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S3 and S4, and Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The calculated TN at atmospheric pressure is 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='9 K, close to 206 K determined by neutron scattering26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' As the pressure increases, TN obtained from the resistance curves decreases from 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='1 K at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa to 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 K at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' DFT calculations and Monte Carlo simulations indicate that the magnetic structures are different in the LP and HP-I phases, despite the isomorphic nature of the structural phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' For the LP phase, the magnetic moment directions are perpendicular to the honeycomb layers and exhibit an ( ) ( ) 2 = 1 z ij i j i ij i H J A S × + å å S S a b c 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 GPa Ni GPa S (eV) (eV) 0 Energy Energy 0 Z r Y A B D E C Z Y A B D E C M K r A L H A d 6 e 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 GPa 6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3GPa 6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa PDOS (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=') (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=') (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=') 4 4 Ni-d Se-p PDOS PDOS P-p 2 2 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 3 2 1 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 2 1 0 2 3 Energy (eV) Energy (eV) Energy (eV)AFM arrangement within the plane and a FM arrangement between planes (designated as zigzag-out, see the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The gap is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='99 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' For the HP-I phase, the magnetic moment directions are within the honeycomb layers, displaying an AFM arrangement both within and between layers (designated as zigzag-in, see the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lastly, our DFT calculations show that the HP-II phase is non-magnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure causes a significant decrease of the interlayer distance, resulting in further enhancement of the metallization and the disappearance of the magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' This is confirmed by high pressure Raman scattering in NiPS3, which showed a complete suppression of TN at the second structural transition27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 5 Phase diagram of NiPSe3 under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The insets illustrate the magnetic structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' zigzag-out for the LP phase and zigzag-in for the HP-I phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' TN for the AFM order and Tc for the superconductivity are extracted from resistance measurements and DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Discussion Bandwidth controlled insulator-metal transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The IMT occurring at ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa in NiPSe3 is separated from the structural transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' This is distinct from the d4-d7 transition metal complexes, where both the t2g and eg orbitals of the transition metals are partially filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' For FePSe3 and MnPSe3, the IMT, spin crossover, and structural transition appear simultaneously due to the interorbital eg-t2g hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' However, Ni2+ (3d8) has fully filled t2g orbitals and partially filled eg orbitals, producing no ionic radius collapse under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure induces the sliding of the vdW layers before the IMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' It was suggested that NiPS3 exhibits negative charge-transfer character with a d9 L configuration where L indicates a ligand (chalcogen) hole 28–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The IMT can be ascribed to progressive enhancement of the p-d hybridization between Ni2+ and Se2- ions and increase of the Ni-d9 configuration (S=1/2) under pressure12,13, thus qualifying it as a bandwidth controlled IMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The 2D monoclinic structure with vdW layers observed in the LP and HP-I phases transforms into a 3D trigonal structure in the HP-II phase, in low-spin S=0 Ni ions with itinerant electrons are favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Coexistence of superconductivity and magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Our theoretical analysis reveals the LP phase to be a Mott insulator with out-of-plane zigzag AFM order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' With increasing pressure, the structure transitions to the HP-I phase characterized by the honeycomb layers sliding relative to each other, and the magnetic moments reorient to point in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Superconductivity emerges from the zigzag AFM parent phase in the HP-I structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The magnetic order in the SC phase is dominated by the S=1/2 AFM 300 LP HP-I HP-1I C2/m C2/m 250 P/31m IMT (K) 200 5 × Tc run1 Temperature 5 × Tc run2 150 T run1 T run2 100 T calculation 50 AFM SC 0 0 5 10 15 20 25 30 Pressure (GPa)order, due to charge transfer from Se under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The magnetism and superconductivity coexist and compete reminiscent of copper oxide and iron-based superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' In the HP-II phase, the S=0 state is favored, yielding a nonmagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' In summary, our work reveals that pressure induces structural and magnetic transitions in NiPSe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' A bandwidth-controlled insulator-metal transition is observed, accompanied by the emergence of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Interestingly, the LP phase and HP-I phase have different zigzag antiferromagnetic orders with magnetic moments perpendicular and parallel to the honeycomb layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure induces a significant reduction of the interlayer distance, resulting in further enhancement of the metallization and the disappearance of the magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The HP-II phase shows 3D character with a nonmagnetic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The variation of the structural dimension and the cooperation between spin and lattice degrees of freedom make NiPSe3 an interesting compound to explore intertwined superconductivity and magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Methods Material synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' NiPSe3 powders were synthesized by heating the stoichiometric Ni, P and Se powders at 600 ℃ in sealed quartz ampoule for two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' High quality single crystals of NiPSe3 were grown with pure NiPSe3 powders mixed with NaCl/AlCl3 using the chemical vapor transport method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The starting materials were sealed in a quartz ampoule and placed in a two-zone furnace for one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The NiPSe3 single crystals were acquired by dissolving the flux by water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The crystals were naturally cleaved along the (001) surface with shiny gray appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The crystals were irregular hexagons with sides of ~1 mm in length and ~10-50 μm in thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' High-pressure XRD measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' High-pressure synchrotron radiation XRD patterns of NiPSe3 were collected at 300 K with an X-ray wavelength of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6199 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' A symmetric diamond anvil cell (DAC) with a pair of 300-μm-diameter culets was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' A sample chamber with a diameter of 110 μm was laser- drilled in a pre-indented steel gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The NiPSe3 powders were compressed into a pellet with a 60-μm diameter and 20-μm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The pellet was loaded into the middle of the sample chamber and silicone oil was used as a pressure transmitting medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' A ruby sphere was also loaded into the sample chamber and pressure was determined by measuring the shift of its fluorescence wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The data were initially integrated using Dioptas31 (with a CeO2 calibration) and the subsequent Rietveld refinements were processed using TOPAS-Academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='14 High-pressure electrical property measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Magnetic and electrical measurements were taken on a physical property measurement system (PPMS, Quantum Design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' High-pressure electrical transport measurements of NiPSe3 single crystals were carried out using a miniature DAC made from a Be–Cu alloy on a PPMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Diamond anvils with a 300-μm culet were used, and the corresponding sample chamber (with a diameter of 110-μm) was made in an insulating gasket achieved by cubic boron nitride and epoxy mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' NaCl powders were employed as the pressure-transmitting medium, providing a quasi- hydrostatic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The pressure was also calibrated by measuring the shift of the fluorescence wavelength of the ruby sphere, which was loaded in the sample chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The standard four-probe technique was adopted for these measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' First-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' DFT calculations are performed using the Vienna ab initio Simulation Package (VASP) at the level of the generalized gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='32,33 We adopted the projector augmented wave pseudopotentials and a plane-wave cutoff energy of 500 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='34 The experimentally measured lattice constants were used in our calculations and the positions of all atoms were fully relaxed until the force on each atom was less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='01 eV/Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We used U = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 eV for atmospheric pressure but U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 eV for higher pressures to describe the correlation among 3d electrons of Ni2+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The values of TN were obtained through parallel tempering Monte Carlo (MC) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='35,36 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Opportunities and challenges of 2D magnetic van der Waals materials : magnetic graphene ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Matter 28, 301001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Gong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Two-dimensional magnetic crystals and emergent heterostructure devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 363, eaav4450 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Critical behavior of the quasi-two-dimensional semiconducting ferromagnet CrSiTe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 6, 33873 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer NiPS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 10, 345 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Huang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Nature 546, 270–273 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Seyler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Ligand-field helical luminescence in a 2D ferromagnetic insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 14, 277–281 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Tan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Probing the Magnetic Ordering of Antiferromagnetic MnPS3 by Raman Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 10, 3087–3093 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Calder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al Magnetic exchange interactions in the van der Waals layered antiferromagnet MnPSe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' B 103, 024414 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Kawakami, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Spin transition in a four-coordinate iron oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 1, 371–376 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Struzhkin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Badro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Shu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Hemley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Mao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure-Induced High-Spin to Low-Spin Transition in FeS Evidenced by X-Ray Emission Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 82, 3284–3287 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Giant Pressure-Driven Lattice Collapse Coupled with Intermetallic Bonding and Spin-State Transition in Manganese Chalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chemie - Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 55, 10350– 10353 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure-Driven Cooperative Spin-Crossover, Large-Volume Collapse, and Semiconductor-to-Metal Transition in Manganese(II) Honeycomb Lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 138, 15751 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Emergent superconductivity in an iron-based honeycomb lattice initiated by pressure-driven spin-crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 9, 1914 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Coelho, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' TOPAS and TOPAS-Academic : an optimization program integrating computer algebra and crystallographic objects written in C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Crystallogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 51, 210 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Mann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Structure Calculations II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='Hartree-Fock Wavefunctions and Radial Expectation Values: Hydrogen to Lawrencium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al Dimensional Crossover Tuned by Pressure in Layered Magnetic NiPS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China Physics, Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 64, 297011 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Magnetism variation of the compressed antiferromagnetic topological insulator EuSn2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China Physics, Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 64, 118211 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Exchange field enhanced upper critical field of the superconductivity in compressed antiferromagnetic EuTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Cai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure-induced superconductivity and structural transition in ferromagnetic CrSiTe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' B 102, 144525 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Evidence for charge and spin density wave in single crystals of La3Ni2O7 and La3Ni2O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China Physics, Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 66, 217411 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Monoclinic EuSn2As2: A Novel High-Pressure Network Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 126, 155701 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Debessai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Matsuoka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Hamlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Schilling, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure-Induced Superconducting State of Europium Metal at Low Temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 102, 197002 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pfleiderer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Coexistence of superconductivity and ferromagnetism in the d-band metal ZrZn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Nature 412, 58–61 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Observation of two superconducting domes under pressure in tetragonal FeS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' npj Quantum Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 2, 49 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Haule, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Vanderbilt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Mott Metal-Insulator Transitions in Pressurized Layered Trichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 123, 236401 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Le Flem, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Brec, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Ouvard, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Louisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Segransan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Magnetic interactions in the layer compounds MPX3 (M = Mn, Fe, Ni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' X = S, Se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Solids 43, 455–461 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Charge-Spin Correlation in van der Waals Antiferromagnet NiPS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 120, 136402 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Takubo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Unusual superexchange pathways in an NiS2 triangular lattice with negative charge-transfer energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 99, 037203 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Patel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Hole doping in a negative charge transfer insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 5, s42005 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Bisogni, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Ground-state oxygen holes and the metal-insulator transition in the negative charge-transfer rare-earth nickelates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 7, 13017 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Prescher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Prakapenka, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' DIOPTAS : a program for reduction of two-dimensional X- ray diffraction data and data exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' High Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 7959, 223–230 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Furthmuller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' B 54, 169–186 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Perdew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Ernzerhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Generalized gradient approximation made simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 77, 3865–3868 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Joubert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' From ultrasoft pseudopotentials to the projector augmented-wave method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 59, 1758–1775 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Lou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' PASP: Property analysis and simulation package for materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 154, 114103 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Hukushima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' & Nemoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Exchange Monte Carlo Method and Application to Spin Glass Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Journal of the Physical Society of Japan 65, 1604–1608 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Acknowledgements Work at Sun Yat-Sen University was supported by the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 12174454, 12104518, U213010013, 11974432, 92165204), Guangdong Basic and Applied Basic Research Funds (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 2021B1515120015, 2022A1515012643, 2022A1515010035, 2022B1212010008), Guangzhou Basic and Applied Basic Research Funds (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 202201011123, 202201011118, 202201011798), National Key Research and Development Program of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 2019YFA0705702, 2022YFA1402802, 2018YFA0306001), Shenzhen International Quantum Academy (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' SIQA202102), the Fundamental Research Funds for the Central Universities, Sun Yat-sen University (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 22QNTD3004), DFT calculations are performed on Tianhe-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' We appreciate the support of BSRF, IHEP, CAS for high pressure XRD measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Supplementary Materials for “Coexistence of zigzag antiferromagnetic order and superconductivity in compressed NiPSe3” Hualei Sun1, Liang Qiu1, Yifeng Han3, Enkui Yi1, Junlong Li4, Mengwu Huo1, Chaoxin Huang1, Hui Liu1, Manrong Li3, Weiliang Wang2, Dao-Xin Yao1, Benjamin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Frandsen5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Bing Shen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Yusheng Hou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='and Meng Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='† 1Center for Neutron Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong 510275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 2 School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong Province Key Laboratory of Display Material and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong 510275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 3 Key Laboratory of Bioinorganic and Synthetic Chemistry of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' School of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Guangdong 510275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 4Beijing Synchrotron Radiation Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' China 5Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Brigham Young University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Provo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Utah 84602,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' USA # houysh@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='cn shenbing@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='cn † wangmeng5@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='cn Properties of the single crystals of NiPSe3 at atmospheric pressure An X-ray diffraction (XRD) pattern of a single crystal of NiPSe3 was measured at ambient pressure, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The XRD pattern shows (00L) diffraction peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The obtained lattice parameter c is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8663(3) Å, which is consistent with the previous reports, confirming the high quality of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Figure S1 XRD pattern of a single crystal of NiPSe3 with the corresponding Miller indices (00L) at atmospheric pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The wavelength is λ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='54 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The inset shows an image of a typical single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Structural transitions of NiPSe3 under pressure The in situ high pressure XRD patterns of both the LP and HP-I phases can be well indexed by the monoclinic C2/m space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' During the isomorphic structural transition from LP to HP-I, there is an obvious sliding of the honeycomb layers relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The transition occurring at about 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa is a non-isomorphic structural transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' At this transition, there is a large decrease of the interlayer (002) 2000 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='unit) (001) 1000 (004) (005) (006) 0 0 10 20 30 40 50 60 70 80 90 100 20 (degree)spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The XRD patterns of the HP-II phase can be well indexed by the trigonal P-31m space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The related structural parameters are listed in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Table S1 Refined lattice parameters, atomic coordinates, and Wyckoff positions (WP) of NiPSe3 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3, and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The LP phase at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 GPa, space group: C2/m a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='052(1), b = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='440(1), and c = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='705(1) Å, α =90˚, β = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='45(1)°, γ =90˚, Rwp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='04%, Rp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='42% atom x y z Occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' WP Ni 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='331(1) 0 1 4g P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='102(7) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='161(8) 1 4i Se 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='759(2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='261(3) 1 4i Se 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='251(5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='173(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='247(2) 1 8j The HP-I phase at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 GPa, space group: C2/m a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='898(1), b = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='348(1), and c = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='303(1) Å, α = 90˚, β = 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='38(1)°, γ = 90˚, Rwp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='12%, Rp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='71% atom x y z Occ WP Ni 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3038(5) 0 1 4g P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='04(1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='18(1) 1 4i Se 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='166(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='181(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='271(2) 1 4i Se 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='632(3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='309(3) 1 8j The HP-II phase at 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 GPa, space group: P-31m a = b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='873(2) and c = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='274(4) Å, α=90˚, β = 90°, γ= 120°, Rwp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='12%, Rp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='71% atom x y z Occ WP Ni 1/3 2/3 0 1 2c P 0 0 0 1 2e Se 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='327(3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='330(2) 1 6k High-pressure resistance and Ginzburg-Landau fitting of the 𝝁𝟎𝑯𝒄𝟐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The electrical resistance as a function of pressure and temperature for single crystals of NiPSe3 is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S2a and S2b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The resistance decreases dramatically at the first structural transition and further decreases at ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The simple Ginzburg-Landau formula 𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='𝐻"#(𝑇) = 𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='𝐻"#(0) (1 − + $ $!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=', # is adopted to fit the upper critical field of the superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The fitted values of 𝜇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='𝐻"# are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='26 T at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='80 T at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa, lower than the Pauli limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Figure S2 High-pressure resistance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' a Temperature dependence of the resistance under various pressures up to 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The resistance is shown on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' b Pressure dependence of the resistance at various temperatures up to 300 K shown on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The grey, blue and yellow backgrounds indicate the LP, HP-I, and HP-II phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The vertical dashed line indicates the IMT at ~ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' c and e Magnetic field dependence of the SC transition at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' d and f Ginzburg-Landau formula fits to the experimentally determined Tc as a function of magnetic field at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The AFM transition in NiPSe3 under pressure The suppressed TN of NiPSe3 under pressure is due to the variation of the magnetic exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' At low pressure, NiPSe3 is a Mott insulator with large resistance and its magnetic moment is around 2μB/Ni2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Because the AFM transition has an influence on the resistance, TN could be determined from the kink in the first derivative of the resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' There is a dramatic decrease in the resistance at the first structural transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' After the structural transition, TN could be identified from the distinctive change in the resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Figure S3 displays selected temperature dependences of the resistance under various pressures from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The resistance curves are collected from two different measurements, including run 1 and run 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The kinks indicate the values of TN that are marked in these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' TN decreases from 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='1 K at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa to 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 K at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='9 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' With further increasing pressure, the signal of the AFM transition becomes weaker and cannot be identified from the resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' S4, the five nearest-neighbor exchange couplings are considered in NiPSe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Combining the DFT calculations and Monte Carlo simulations, the values of TN are determined accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The results are listed in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' a 10% b P (GPa) LP HP-I HP-II (α) 105 run1 () e 105 104 10° 0-50krun1 Resistance Resistance 10° 0—100K 103 IMT 150K 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 102 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 7-200K 101 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 10 250 K 100 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 10° 300K 10-1 10-1 ATTA 0 50 100 150 200 250 300 0 5 10 15 20 25 30 35 Temperature (K) Pressure(GPa) c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='101 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 Resistance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='100 P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 OT P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 T 1 T T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 T ,H(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='26 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 2 4 6 8 0 2 4 6 Temperature (K) T, (K) e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='114 f 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 P = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa Resistance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='108 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T E 1T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 工 P = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa 2T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='102 3T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 K 4T 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5T μ,H(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='80 T 5T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 2 6 8 0 2 4 6 Temperature (K) T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' (K) Figure S3 Temperature dependence of the resistance under pressures of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0, b 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8, c 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2, d 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6, e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5, f 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4, and g 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' The resistance curves are selected from different measurements (run 1 to 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' TN is determined by the inflection points marked in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Figure S4 Schematics of the Ni ions in the a LP phase and b HP-I phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Table S2 DFT calculated magnetic exchange couplings J in units of meV, electronic band gap, TN, magnetic moment (M) and magnetic anisotropy energy (MAE) of NiPSe3 under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Pressure(GP a) J1 J2 J3 J4 J5 Gap(eV ) TN(K ) M(μ B/Ni2+) MAE(meV/Ni2 +) 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='994 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='991 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='074 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='168 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='091 As listed in Table S2, the interlayer magnetic exchange coupling J5 changes from AFM to FM with pressure increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Thus, we establish a 1×1×2 out-of-plane supercell to deliberate the magnetic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='S5, the zigzag-out magnetic structure is more stable than the zigzag-in structure when the pressure on NiPSe3 is less than 6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' However, zigzag-out has higher energy than zigzag-in when the pressure is larger than 6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' When the pressure is above 15 GPa, NiPSe3 favors a non-magnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' a b dR/dT (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='unit) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='unit) 40 dR/dT P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='0 GPa (run 1) P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 GPa (run 1) 80 T, = 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='1 K T, = 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 K 12 C d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='124 Resistance (2) () Resistance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='120 P = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 GPa (run 1) P = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 GPa (run 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 T, = 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 K T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' = 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='2 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='118 e f dR/dT (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='unit) 20 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='unit) dR/dT 10 P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 GPa (run 2) P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='4 GPa (run 2) 1: T,= 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 K T,= 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='9 K 60 155 g 50 100 150 200 250 300 Temperature (K) Resistance( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='7 P = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='5 GPa (run 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='6 T = 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content='3 K 50 100 150 200 250 300 Temperature (K)a b Figure S5 Pressure dependence of the energy difference between the zigzag-out and zigzag-in AFM orders of NiPSe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' Here, the energies of the zigzag-out AFM order are taken as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE3T4oBgHgl3EQfIAlQ/content/2301.04329v1.pdf'} +page_content=' 10 △E (meV/atom) 0 10 一 zigzag-out AFM 一zigzag-in AFM 20 30 1 1 0 2 4 6 8 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+Stony Brook, NY, United States of America +nengkun.yu@cs.stonybrook.edu +ABSTRACT +In this paper, we propose a quantum algorithm for majority voting. +We prove that the time complexity of quantum majority voting +could be quadratically smaller than its classical algorithms, which +is supported by our experimental studies. +KEYWORDS +Voting Rules, Majority Voting, Quantum Computation +ACM Reference Format: +Ao Liu, Lirong Xia and Nengkun Yu. 2023. Accelerating Majority Voting +by Quantum Computation. In Proc. of the 22nd International Conference on +Autonomous Agents and Multiagent Systems (AAMAS 2023), London, United +Kingdom, May 29 – June 2, 2023, IFAAMAS, 9 pages. +1 +INTRODUCTION +Voting is a widely used methodology to make collective decisions +in a wide range of applications [24], such as political elections, +recommender systems [7], crowdsourcing [17], and blockchain +governance [1, 9], just to name a few. While many voting rules, such +as plurality and single transferable vote (STV), can be computed in +polynomial time, in large-scale, high-frequency decision-making +scenarios, it is desirable that the winner can be computed in a +short amount of time [25]. Like in database systems, a polynomial +or even linear runtime may be too slow already. As many such +scenarios are low-stakes, randomized algorithms and small “errors” +are acceptable. +One natural solution is to randomly sample a subset of votes +(with or without replacement), and then compute the winner of the +sampled votes. Can we do better? +Quantum computation appears to be a promising approach, as +it has successfully accelerated many computational tasks such as +search [10], optimization [11], and machine learning [2–4]. How- +ever, we are not aware of a previous work on accelerating voting +using quantum computation. Thus, the following problem remains +open +Can voting be accelerated by quantum computation? +In this paper, we take a first step to address this question by fo- +cusing on the simple majority voting (for two-candidate), where +the winner is the candidate with more votes. Majority voting has +many desired properties (e.g., fairness, decisiveness, and monotonic- +ity) and is widely used in group decision-makings [8]. We target +to use quantum computation to accelerate the quantum majority +voting when a small probability of “errors” are allowed. Also see +Section 4 for a detailed explanation of when quantum computation +accelerates. +Proc. of the 22nd International Conference on Autonomous Agents and Multiagent Sys- +tems (AAMAS 2023), A. Ricci, W. Yeoh, N. Agmon, B. An (eds.), May 29 – June 2, 2023, +London, United Kingdom. © 2023 International Foundation for Autonomous Agents +and Multiagent Systems (www.ifaamas.org). All rights reserved. +Our contributions are three-fold. Firstly, we propose the quantum +majority voting algorithm (Algorithm 1). Secondly, we theoretically +prove that our proposed quantum majority voting algorithm is +quadratically faster than its classical algorithm (see Table 1 for a +more detailed comparison). Thirdly, we experimentally verified our +theoretical results (Section 6). +time complexity +space complexity +Quantum (Thm. 4.1) +Θ +�𝑛·log(1/𝜖) +MoV +� +𝑂 +� +log( 𝑛·log(1/𝜖) +MoV +) +� +Classical (Thm. 5.1) +Θ +�𝑛2·log(1/𝜖) +MoV2 +� +Θ +� +log( 𝑛2·log(1/𝜖) +MoV2 +) +� +Table 1: Summary for the theoretical results, where MoV +means margin of victory (See Section 2 for its formal def- +inition). All algorithms in the table assume the algorithm +output the winner with no less than 1 − 𝜖 probability. +Related works and discussions. To the best of our knowledge, +none of the literature has used quantum computation to accelerate +voting. Vaccaro et al. [20] firstly introduces the idea of quantum +communication or quantum computation to voting. The quantum +voting algorithm in Vaccaro et al. [20] provides theoretically guar- +anteed security (against colluding attacks [15]). Xue and Zhang [26] +improved the result in Vaccaro et al. [20] by proposing a simpler vot- +ing protocol but with stronger security guarantees. Khabiboulline +et al. [14] proposed an “all-in-one” quantum voting protocol, which +focuses on achieving anonymity without losing security guaran- +tees. However, all of the above approaches require Ω(𝑛) quantum +communication cost, which means their proposed method does +accelerate voting. +2 +PRELIMINARY +Majority voting. In (two-candidate) majority voting, 𝑛 > 1 voters +cast their votes for one of the two candidates. We use 𝑛1 to denote +the number of votes for the first candidate. Similarly, 𝑛2 = 𝑛 − 𝑛1 +is the number of votes for the second candidate. If 𝑛1 > 𝑛2 (or +𝑛2 > 𝑛1), the first (or the second) candidate will be announced +as the winner. If 𝑛1 = 𝑛2, the voting rule will break the tie by +outputting a candidate according to the “tie-breaking rule”, which +usually outputs a candidate uniformly at random. Margin of vic- +tory (MOV) describes the smallest number 𝑘 such that 𝑘 voters can +change the winner by voting differently [22]. In majority voting, it +is easy to check that MoV = ⌈|𝑛/2 − 𝑛1|⌉ = ⌈|𝑛/2 − 𝑛2|⌉. +Basic quantum computation.1 Quantum bit (or qubit in short) +is the counterpart of classical bit, which takes a deterministic binary +from {0, 1}. Qubit, on the other hand, is represented by a linear com- +bination of {|0⟩, |1⟩}, which are counterparts to {0, 1}, respectively. +1This paper adopts the same notation system as Nielsen and Chuang [18], which is a +textbook about quantum computation. +arXiv:2301.02995v1 [cs.CY] 8 Jan 2023 + +That is, every qubit |𝜓⟩ is written as +|𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩, +where 𝛼 and 𝛽 are complex numbers and are usually called ampli- +tudes. If we measure the qubit, there is |𝛼|2 probability to get 0 and +|𝛽|2 probability to get 1. Naturally, we always have |𝛼|2 + |𝛽|2 = 1 +because the probabilities should sum to 1. Qubits sometimes are +written as vectors to simplify notations. Formally, +�𝛼 +𝛽 +� +≜ 𝛼|0⟩ + 𝛽|1⟩. +𝑡 > 1 qubits are presented as a 2𝑡-dimensional vector, where the 𝑗- +th component of the vector (denoted as 𝛼𝑗) represents the amplitude +of |𝑗1 · · · 𝑗𝑡⟩ (or |𝑗⟩), where 𝑗1 · · · 𝑗𝑡 is the binary representation of +𝑗. Similar to the 1-qubit case, the probability of observing 𝑗1, · · · , 𝑗𝑡 +from those 𝑡 qubit equals to |𝛼𝑗 |2. +A quantum operation (quantum gate) 𝑄 on 𝑡 qubits is denoted +by a 2𝑡 × 2𝑡 unitary matrix, which means the matrix’s inverse is its +Hermitian conjugate. Applying a quantum operation 𝑄 on quantum +state |𝜓⟩ is denoted by +𝑄|𝜓⟩ ≜ 𝑸(2𝑡 ×2𝑡 ) �𝜓(2𝑡 ), +where the the quantum operator 𝑸(2𝑡 ×2𝑡 ) is a 2𝑡 ×2𝑡 unitary matrix +and the quantum state �𝜓(2𝑡 ) is a 2𝑡 dimensional column vector. +Quantum circuit of some useful quantum operators.2 Quan- +tum circuits run from the left-hand side to the right-hand side. For +example, the following circuit means applying Hadamard gate 𝐻 +on a quantum state |𝜓⟩. +|𝜓⟩ +𝐻 +where 𝑯 = 1 +√ +2 +�1 +1 +1 +−1 +� +. +The quantum circuit notion +|𝜓⟩ +0/1 +𝑏 +denotes measuring quantum state |𝜓⟩ with 0/1 base (𝑏 denotes +the result of measurement). Naturally, the complexity of quantum +measurement and Hadamard gate are both Θ(1). +Quantum oracle [5, 12, 21] is a widely-used operator to encode +binary functions or binary information. Given 𝑡 qubits and a binary +function 𝑓 : {0, · · · , 2𝑡 − 1} ↦→ {0, 1}, quantum oracle (based on +function 𝑓 (·)) applies a phase shift of −1 = 𝑒𝜋𝑖 if 𝑓 (𝑥) = 1 and does +nothing otherwise. We can query oracle many times and regard the +number of queries as the cost [10]. Formally, +� 𝑂𝑓 |𝑥⟩ = |𝑥⟩ +if 𝑓 (𝑥) = 1 +𝑂𝑓 |𝑥⟩ = −|𝑥⟩ +otherwise +. +Suppose we have a quantum gate 𝐺 on 𝑡 qubits. The following +operation is called controlled-𝐺. +𝐺 += +�𝑰(2𝑡 ×2𝑡 ) +0(2𝑡 ×2𝑡 ) +0(2𝑡 ×2𝑡 ) +𝑮(2𝑡 ×2𝑡 ) +� +, +2All quantum circuits of this paper are drawn using the Quantikz package [13] for +LATEX. +where 𝑰 denotes the identity matrix, and 0 denotes the zeros matrix. +To simplify notations, we also write +𝐺𝑎 += +. . . +. . . +𝐺 +𝐺 +(repeat 𝑎 times). +3 +QUANTUM MAJORITY VOTING +Voting through quantum computers. In classical voting, the +votes usually are sent to an “aggregator”, who is responsible for +aggregating the votes and announcing the winner. Our quantum +majority voting follows a similar procedure, where the “aggregator” +constructs a quantum oracle based on the votes and uses our pro- +posed quantum majority voting algorithm (Algorithm 1) to decide +the winner. +Construct the quantum oracle. Since there are two candidates +in majority voting, the votes can be treated as binary data. In this +paper, 1 represents a vote to the first candidate and 0 represents +a vote to the second candidate. As a quantum oracle with 𝑡 qubits +has the ability to encode 2𝑡 binary bits, we use 𝑡 = 2⌈log𝑛⌉ qubits to +encode the votes from𝑛 voters.3 The rest +� +2⌈log𝑛⌉ − 𝑛 +� +bits are filled +in 0 and 1 half-by-half. Formally, the function 𝑓 : {0, · · · , 2𝑡 − 1} ↦→ +{0, 1} to construct the quantum oracle is defined as +𝑓 (𝑥) = + + +1 +if candidate 𝑥 votes the first candidate +or 𝑥 > 𝑛 − 1 − ⌊(2𝑡 − 𝑛)/2⌋ +0 +otherwise +, +where the voters are numbered from 0 to (𝑛 − 1). We let the ad- +justed number of votes for the first candidate (the number of 1’s) +as +◦𝑛1 ≜ # {𝑥 : 𝑓 (𝑥) = 1} = 𝑛1 + +� +2𝑡 −𝑛 +2 +� +. +Algorithm 1: Quantum Majority Voting +1: Inputs: 𝑛 voters’ votes 𝑉0, · · · ,𝑉𝑛−1, the number of qubits +𝑠 ≥ 2, and the number of iterations 𝐾 +2: Initialization: Set 𝐾1 = 0 and construct the quantum oracle +𝑂𝑓 based on 𝑉0, · · · ,𝑉𝑛−1 +3: Construct Grover operator 𝐺 using 𝑂𝑓 according to Figure 1 +4: for 𝑘 ∈ {1, · · · , 𝐾} do +5: +Apply the Grover operator to the quantum counting circuit +(Figure 3) with 𝑠 qubits in Register 1 +6: +if the binary decimal 0.𝑏1 · · ·𝑏𝑠 ≥ 0.01 then +7: +𝐾1 = 𝐾1 + 1 +8: +end if +9: end for +10: if 𝐾1 ≥ 𝐾/2 then +11: +Announce the first candidate as the winner +12: else +13: +Announce the second candidate as the winner +14: end if +3In all discussions of this paper, logarithm function log(·) uses 2 as its base, and ln(·) +uses Euler’s number 𝑒 as its base. + +Formal definition of quantum majority voting. We formally +define quantum majority voting in Algorithm 1. Basically, Algo- +rithm 1 repeats the quantum counting algorithm by 𝐾 rounds. In +each round, quantum counting estimates the number of votes for +each candidate and makes a prediction about the winner. Then, +the “aggregator” announces the candidate who wins in a larger +number of rounds as the winner of the majority voting. Usually, the +number of rounds 𝐾 is set as an odd number to avoid ties. Next, we +will introduce the functionality and implementation of each step of +Algorithm 1 in detail. +... +... +𝑡 qubits +𝑂𝑓 +𝐻 +𝑄𝑃𝑆 +𝐻 +𝐻 +𝐻 +Figure 1: The circuit for Grover operator. +Prepare Grover operator. The Grover operation is constructed +by the quantum circuit in Figure 1, where 𝑡 = ⌈log𝑛⌉ denotes the +minimum number of bits to represent 𝑛. The quantum operator +𝑄𝑃𝑆 is called quantum phase shifting, which provides a phase shift +of −1 on every state except |0⟩. Mathematically, +|0⟩ +𝑄𝑃𝑆 +−→ |0⟩ +and +|𝑥⟩ +𝑄𝑃𝑆 +−→ −|𝑥⟩ for any 𝑥 ∈ 1, · · · , 2𝑡 − 1. +Here, |𝑥⟩ represents the 𝑥-th base state of the 𝑡 qubits. The high- +level idea of Grover operator’s functionality is shown in Figure 2, +where |𝜓⟩ is the input of Grover operators in quantum counting, +and {|𝛼⟩, |𝛽⟩} is a pair of orthogonal bases. The formal definition +of |𝜓⟩, |𝛼⟩, and |𝛽⟩ can be found in Appendix A. Under the |𝛼⟩ |𝛽⟩ +base, the quantum oracle 𝑂𝑓 reflects |𝜓⟩ over |𝛼⟩, while the rest +parts of 𝐺 reflects 𝑂𝑓 |𝜓⟩ over |𝜓⟩. The angle between the output +state 𝐺|𝜓⟩ and initial state |𝜓⟩ +𝜃 = 2 arcsin +�√︁ ◦𝑛1 · 2−𝑡 +� +, +which includes the information about +◦𝑛1. Since function arcsin(√𝑥) +grows quadratically faster than linear functions when 𝑥 is small, we +expect that an estimation about arcsin(√𝑥) could be quadratically +more accurate than directly estimate 𝑥. +Quantum counting. The quantum circuit of quantum counting +is shown in Figure 3, where 𝑄𝐹𝑇 † denotes the quantum reverse +Fourier transformation (its time complexity is Θ(𝑠2)). At a high level, +quantum counting estimates angle 𝜃 in Grover operator (plotted +in Figure 2). Mathematically, the output of quantum counting ˆ𝜑 = +0.𝑏1 · · ·𝑏𝑠 is an estimation of 𝜑 ≜ 𝜃/(2𝜋) = arcsin +�√︁ ◦𝑛1 · 2−𝑡 +� +/𝜋. +Here, 0.𝑏1 · · ·𝑏𝑠 is a binary decimal. For example, 0.011 represents +(2−2 + 2−3) = 3/8. Next, we present a useful error bound for quan- +tum counting. +G| ۧ +𝝍 +| ۧ +𝝍 +Of | ۧ +𝝍 +| ۧ +𝜷 +| ۧ +𝜶 +𝜃 +𝜃/2 +𝜃/2 +Figure 2: An illustration of Grover operator’s functionality +(Figure 6.3 in Nielsen and Chuang [18]). +Lemma 3.1 (Error bound for qantum counting, Ineqality +(5.34) in Nielsen and Chuang [18]). Using the notations above, for +quantum counting, +Pr[| ˆ𝜑 − 𝜑| ≥ 𝛿] ≤ +1 +2(𝛿 · 2𝑠 − 1) , +where 𝑠 is the number of qubits in Register 1 of quantum counting’s +circuit (Figure 3). +According to the design of 𝑂𝑓 , we need to decide whether +◦𝑛1 is +larger than or smaller than 2𝑡−1. We also note that +◦𝑛1 ≥ 2𝑡−1 ⇐⇒ 𝜑 ≥ 1 +4. +Thus, predicting the winner of majority voting reduces to the prob- +lem of predicting the relationship between 𝜑 and 1 +4 (or 0.01 in +binary decimal). +4 +THEORETICAL ANALYSIS OF FAST +QUANTUM MAJORITY VOTING +In Section 3, we proposed quantum majority voting (Algorithm 1) +and explained why it works at a high level. In this section, we +provide theoretical guarantees about the accuracy (probability of +outputting the correct winner), time complexity, and space com- +plexity of quantum majority voting. +When quantum majority voting (may) accelerate? We first +think about the cases where classical algorithms (e.g., Algorithm 2, +which randomly sample a subset of votes and use the subset to pre- +dict the winner) do not need to be improved or cannot be improved. +When the margin of victory MoV = Θ(𝑛), classical algorithms +are already very fast according to the Chernoff bound, which says +the classical algorithms’ error rate can be exponentially small in +terms of complexity. +Another case is when MoV is very small (e.g., MoV = Θ(1)) +where classical algorithms’ performance is close to the optimal. In + +|0⟩ +. . . +0/1 +𝑏1 +|0⟩ +. . . +0/1 +𝑏2 +... +... +... +... +|0⟩ +. . . +0/1 +𝑏𝑠 +|0⟩ +. . . +... +... +. . . +|0⟩ +. . . +Register 1 +𝑠 qubits +𝐻 +𝑄𝐹𝑇 † +𝐻 +𝐻 +Register 2 +𝑡 qubits +𝐻 +𝐺20 +𝐺21 +𝐺2𝑠−1 +trash +𝐻 +Figure 3: The circuit for quantum counting algorithm. +this case, any algorithms have to look into each vote to decide the +winner. Since the complexity of counting every vote is Θ(𝑛), there +is not a lot of space for the classical algorithms to be improved. In +order to improve readability, we let “tie” (MoV ≤ 1) be a special +case of MoV = Θ(1). +Throughout this paper, we assume that the margin of victory +MoV = Θ(𝑛𝑐), where 𝑐 ∈ (0, 1) is a constant. For example, in one of +the settings in our experimental verification (Figure 4), the number +of voters 𝑛 ≈ 106 and MoV = √𝑛. In this example, the winner only +got ∼0.2% more votes than the loser. +The theoretical guarantee of quantum majority voting. In +Theorem 4.1, we provide the theoretical guarantee of Algorithm 1’s +performance under the above-discussed conditions where quantum +algorithms may accelerate. +Theorem 4.1 (Theoretical guarantee of qantum major- +ity voting). For arbitrary constant 𝜖 ∈ (0, 1), quantum major- +ity voting (Algorithm 1) has the following three properties when +𝑠 = max {2, 𝑡 − ⌊log(MoV − 1)⌋ + 4}, and 𝐾 = ⌈12.3 · ln(1/𝜖)⌉, +1. It outputs the correct voting outcome with at least 1 −𝜖 probability. +2. It’s time complexity is Θ +�𝑛·log(1/𝜖) +MoV +� +. +3. It’s space complexity is Θ +� +log( 𝑛·log(1/𝜖) +MoV +) +� +. +Theorem 4.1 describes the required parameter 𝑠, time complex- +ity, and space complexity of quantum majority voting to achieve +arbitrary accuracy (probability of outputting the correct winner). +Theorem 4.1 is proved by combining Chernoff bound and Theo- +rem 4.2, which shows the performance of quantum majority voting +when the number of iterations 𝐾 = 1. +Proof of Theorem 4.1. According to Theorem 4.2, we know +that each iteration has at least 𝑝 = 32−3𝜋 +32−2𝜋 ≈ 87.8% probability to out- +put the correct winner if setting𝑠 = max {2, 𝑡 − ⌊log(MoV − 1)⌋ + 4}. +W.L.O.G., we assume that 𝑛1 ≥ 𝑛2. Then, we apply Chernoff bound +and have, +1 − 𝜖 = Pr[correct winner] +≥ 1 − Pr[𝐾1 ≥ 𝐾/2] +≥ 1 − exp +� +−(1 − 1 +2𝑝 )2 · 𝐾 · 𝑝/2 +� +, +(1) +which is equivalent with +𝐾 ≥ 2𝑝 · ln(1/𝜖) +(𝑝 − 1/2)2 ≈ 12.3 · ln(1/𝜖). +Then, Theorem 4.1 follows by directly apply Theorem 4.2. +□ +Theorem 4.2 (Theoretical guarantee of qantum majority +voting when 𝐾 = 1). For arbitrary constant 𝜖 ∈ (0, 1), quantum +majority voting (Algorithm 1) has the following three properties when +𝐾 = 1 and 𝑠 = max +� +2, +� +𝑡 − log(MoV − 1) + log( 𝜋 +2𝜖 + 𝜋) +�� +. +1. It outputs the correct voting outcome with at least 1 −𝜖 probability. +2. It’s time complexity is Θ +� +𝑛 +𝜖·MoV +� +. +3. It’s space complexity is Θ +� +log( +𝑛 +𝜖·MoV) +� +. +Comparing Theorem 4.2 with Theorem 4.1, we know that Algo- +rithm 1 reduces the 1/𝜖 term in (either time or space) complexity to +log(1/𝜖) by setting 𝐾 = ⌈12.3 · ln(1/𝜖)⌉. The proof of Theorem 4.2 +directly follows by setting parameter +𝑠 = +� +𝑡 − log(MoV − 1) + log( 𝜋 +2𝜖 + 𝜋) +� +for Lemma 4.3, which proves the accuracy, time complexity, and +space complexity for arbitrary parameter 𝑠 ≥ 2 when 𝐾 = 1. +Lemma 4.3. For any parameter 𝑠 ≥ 2, quantum majority voting +(Algorithm 1) with the number of iterations 𝐾 = 1 has the following +three properties, + +1. It outputs the correct voting outcome with at least 1− 𝜋 +2 · +1 +(MoV−1) ·2𝑠−𝑡 −𝜋 +probability. +2. It’s time complexity is Θ (2𝑠). +3. It’s space complexity is Θ (log(𝑛) + 𝑠). +Proof. Accuracy. Firstly, we prove the case that 𝑛1 ≤ 𝑛2 (or +equivalently, +◦𝑛1 ≤ 2𝑡−1). We recall the reasoning behind the quan- +tum counting algorithm +𝜑 = +arcsin +�√︁ ◦𝑛1 · 2−𝑡 +� +𝜋 +. +(2) +Next, we provide an upper bound for 𝜑. Since +◦𝑛1 = 𝑛1 + ⌈ 2𝑡 −𝑛 +2 ⌉, we +know that +◦𝑛1 = 𝑛1 + +�2𝑡 − 𝑛 +2 +� += +�𝑛 +2 +� +− MoV + +�2𝑡 − 𝑛 +2 +� +≤ 2𝑡−1 − MoV + 1. +(3) +To simplify notations, we let NMoVQ = (MoV − 1) · 2−𝑡 to denote +the “normalized” MoV for quantum majority voting. Combining (2) +and (3), we have, +𝜑 ≤ +arcsin +�√︁1/2 − NMoVQ +� +𝜋 +. +We define +𝑔(𝑥) = +arcsin +�√︁ +1/2 − 𝑥 +� +𝜋 +. +By standard calculus, we know that +𝑔(0) = 1 +4 +𝑔′(𝑥)|𝑥=0 = − 1 +𝜋 +𝑔′′(𝑥)|𝑥=0 = 0 +𝑔′′(𝑥)|𝑥>0 < 0. +Thus, 𝑔(𝑥) is a concave function when 𝑥 ≥ 0. Then, we have, +𝜑 ≤ +arcsin +�√︁ +1/2 − 𝑥 +� +𝜋 +≤ 1 +4 − +NMoVQ +𝜋 +. +(4) +Then we have that +Pr [incorrect winner] +≤ Pr[0.𝑏1 · · ·𝑏𝑠 ≥ 0.01] += Pr[0.𝑏1 · · ·𝑏𝑠 − 𝜑 ≥ 0.01 − 𝜑] +≤ Pr[|0.𝑏1 · · ·𝑏𝑠 − 𝜑| ≥ 0.01 − 𝜑], +where 0.01 represent 1/4. Next, we apply the error bound for quan- +tum counting (Lemma 3.1). +Pr [incorrect winner] +≤ Pr[|0.𝑏1 · · ·𝑏𝑠 − 𝜑| ≥ 0.01 − 𝜑] +≤ +1 +2[(0.01 − 𝜑) · 2𝑠 − 1] . +Note that the binary decimal 0.01 represent 1 +4. Then, we combine +the above inequality with inequality (4) and have +Pr [incorrect winner] +≤ +1 +2[(0.01 − 𝜑) · 2𝑠 − 1] +≤ +1 +2 +� NMoVQ +𝜋 +· 2𝑠 − 1 +� += 𝜋 +2 · +1 +NMoVQ · 2𝑠 − 𝜋 += 𝜋 +2 · +1 +(MoV − 1) · 2𝑠−𝑡 − 𝜋 . +Then, we know that +Pr [correct winner] += 1 − Pr [incorrect winner] +≥ 1 − 𝜋 +2 · +1 +(MoV − 1) · 2𝑠−𝑡 − 𝜋 . +(5) +By now, we have proved the case that 𝑛1 ≤ 𝑛2. Since 𝑔(𝑥) is rota- +tionally symmetric over (0, 1/4). Then, the 𝑛1 ≥ 𝑛2 case directly +follows by symmetry. +Time complexity. In the quantum circuit of quantum counting +(Figure 3), the controlled-𝐺 gate is called �𝑠−1 +𝑖=0 2𝑖 = 2𝑠 − 1 times. +Since the quantum Fourier transformation requires Θ(𝑠2) time, +then, the time complexity is +Hadamard +���� +Θ(1) ++ +Controlled-𝐺 +�������������� +Θ(2𝑠 − 1) + +𝑄𝐹𝑇 † +���� +Θ(𝑠2) + +measurement +���� +Θ(1) += Θ(2𝑠). +Then, the time complexity part of Theorem 4.3 follows. +Space Complexity. We note that Register 1 contains 𝑠 qubits and +Register 2 contains 𝑡 = 2⌈log(𝑛)⌉ = Θ(log(𝑛)) qubits. Then. the +space complexity of the algorithm is 𝑠 + 𝑡 = Θ (log(𝑛) + 𝑠). +□ +5 +COMPARING QUANTUM & CLASSICAL +MAJORITY VOTING +In this section, we compare quantum majority voting (Algorithm 1) +with classical majority voting (Algorithm 2). The classical algorithm +is designed according to the idea of sampling (either with or without +replacement). At the high level, it uses the 𝑇 randomly sampled +votes to estimate the winner. In the next theorem, we will present +the accuracy, time complexity, and space complexity of classical +majority voting under the conditions where quantum algorithms +may accelerate (discussed in Section 4). +Theorem 5.1. For any 𝜖 ∈ (0, 1), Algorithm 2 (with or without +replacement) with the number of samples 𝑇 = Θ +�𝑛2·log(1/𝜖) +MoV2 +� +has +the following three properties, +1. It outputs the correct winner with 1 − 𝜖 probability. +2. Its time complexity is Θ +�𝑛2·log(1/𝜖) +MoV2 +� +. +3. Its space complexity is Θ +� +log( 𝑛2·log(1/𝜖) +MoV2 +) +� +. +Theorem 5.1 describes the required parameter 𝑇, time complex- +ity, and space complexity of classical majority voting to achieve + +Algorithm 2: Classical Majority Voting +1: Inputs: 𝑛 voters’ votes 𝑉1, · · · ,𝑉𝑛, and the number of samples +𝑇 ≤ 𝑛 +2: Randomly sample 𝑇 votes with or without replacement +3: Count the number of sampled votes for each candidate +4: if the first candidate gets more votes then +5: +Announce the first candidate as the winner +6: else +7: +Announce the second candidate as the winner +8: end if +arbitrary accuracy (probability of outputting the correct winner). +The proof of Theorem 5.1 directly follows by setting parameter +𝑇 = Θ +�𝑛2·log(1/𝜖) +MoV2 +� +in Lemma 5.2, which proves the accuracy, time +complexity, and space complexity of classical majority voting for +arbitrary 𝑇 parameter. +Lemma 5.2. Classical majority voting (Algorithm 2, either sample +with or without replacement) has the following three asymptotic +properties, +1. It output the correct winner with 1−exp +� +−Θ +� +MoV2·𝑇 +𝑛2 +�� +probability. +2. Its time complexity is Θ(𝑇). +3. Its space complexity is Θ(log(𝑇)). +Proof. W.L.O.G., we assume that 𝑛1 < 𝑛2. Then, the probability +such that a vote for the first candidate got sampled +𝑝 ≜ Pr [first candidate’s vote sampled] = 1 +2 − NMoVQ, +where NMoVC ≜ 2MoV−(𝑛 mod 2) +2𝑛 +denotes the “normalized” MoV +for the classical algorithm. Next, we will derive an asymptotic upper +and lower bound for the accuracy of Algorithm 2. +Asymptotic Upper Bound. If the votes are sampled with replace- +ment, we know that the number of votes for the first candidate +follows binomial distribution B(𝑇, 𝑝), which converges to Gaussian +distribution N (𝑛𝑝,𝑛𝑝(1 − 𝑝)) when 𝑇 → ∞. +If the votes are sampled without replacement, we know that +the number of votes for the first candidate follows hypergeometric +distribution H (𝑇,𝑛,𝑇/2,𝑛𝑝), which converges to Gaussian distribu- +tion N (𝑛𝑝,𝑛𝑝(1 − 𝑝)) when 𝑇 = 𝑐 · 𝑛 and 𝑛 → ∞, where 𝑐 ∈ (0, 1) +is a constant. We note that the normal distribution of sampling +without replacement is the same normal distribution as sampling +with replacement. +Mathematically, for both sampling with or without replacement, +we have, +lim +𝑇→∞ Pr [incorrect output] +≥ +lim +𝑇→∞ Pr +� +𝑇1 > 𝑇 +2 +� += Φ +��� +� +NMoVC · +√ +𝑇 +√︃ +1/4 − NMoV2 +C +��� +� +, +(6) +where 𝑇1 is the number of sampled votes for the first candidate. +Function Φ(·) denotes the cumulative distribution function (CDF) +of the standard normal distribution N (0, 1). According to Cook [6], +we have the following lower bound for the CDF of standard normal +distribution, +Φ(𝑥) > +1 +√ +2𝜋 +· +𝑥 +𝑥2 + 1 · 𝑒−𝑥2/2. +(7) +Combining (6) and (7), we have, +lim +𝑇→∞ Pr [incorrect output] +> +1 +√ +2𝜋 +· +NMoVC· +√ +𝑇 +√︃ +1/4−NMoV2 +C +� +NMoVC· +√ +𝑇 +√︃ +1/4−NMoV2 +C +�2 ++ 1 +· exp +� +− +NMoV2 +C · 𝑇 +1/2 − 2NMoV2 +C +� +. +(8) +Case 1. When 𝑇 = 𝑂(𝑛2/MoV2), we have that +NMoV2 +C · 𝑇 = +� 2MoV − (𝑛 mod 2) +2𝑛 +�2 +· 𝑇 = 𝑂(1). +Since MoV = 𝑜(𝑛), we have, +1 +4 − NMoV2 +C = Θ(1) +and +√︂ +1 +2 − 2NMoV2 +C = Θ(1). +Thus, we have +NMoVC · +√ +𝑇 +√︃ +1/4 − NMoV2 +C += Θ(NMoVC · +√ +𝑇) +and +exp +� +− +NMoV2 +C · 𝑇 +1/2 − 2NMoV2 +C +� += Θ(1). +(9) +By combining (9) and (8), we know that +lim +𝑇→∞ Pr [correct output] += 1 − lim +𝑇→∞ Pr [incorrect output] += 1 − 𝑂 +� +1 +NMoVC · +√ +𝑇 +� += 1 − 𝑂 +� +𝑛 +MoV · +√ +𝑇 +� += 1 − Ω(1) += 𝑂(1). +Case 2. When 𝑇 = 𝜔(𝑛2/MoV2), we have that +NMoVC · +√ +𝑇 +√︃ +1/4 − NMoV2 +C += 𝜔(1). +Combining the above equation with (8), we have that +lim +𝑇→∞ Pr [correct output] += 1 − lim +𝑇→∞ Pr [incorrect output] += 1 − exp +� +−Ω +� +NMoV2 +C · 𝑇 +�� += 1 − exp +� +−Ω +� MoV2 · 𝑇 +𝑛2 +�� +. + +Since 1 − exp +� +−Ω +� +MoV2·𝑇 +𝑛2 +�� += 𝑂(1) when 𝑇 = 𝑂(𝑛2/MoV2), we +can combine Case 1 and Case 2 as +lim +𝑇→∞ Pr [correct output] = 1 − exp +� +−Ω +� MoV2 · 𝑇 +𝑛2 +�� +. +Asymptotic Lower Bound. Similar to the upper bound, we use the +relationship between normal distribution and binomial distribution +(or hypergeometric distribution) and have +lim +𝑇→∞ Pr [incorrect output] +≤ +lim +𝑇→∞ Pr +� +𝑇1 ≥ 𝑇 +2 +� += Φ +��� +� +NMoVC · +√ +𝑇 +√︃ +1/4 − NMoV2 +C +��� +� +, +According to Cook [6], we have the following upper bound for the +CDF of standard normal distribution, +Φ(𝑥) < +1 +√ +2𝜋 +· 1 +𝑥 · 𝑒−𝑥2/2. +Then, we repeat a similar process as the asymptotic upper bound +and have the following bound. +lim +𝑇→∞ Pr [correct output] = 1 − exp +� +−𝑂 +� MoV2 · 𝑇 +𝑛2 +�� +. +Then, the accuracy part of Lemma 5.2 follows by combining the +asymptotic upper and lower bounds. +Time complexity. Algorithm 2 draw 𝑇 samples and count them, +which cost Θ(𝑇) time in total. +Space complexity. The counting step of Algorithm 2 needs to +store an integer 𝑇1, which is the number of votes to the first candi- +date. For either average-case analysis or worst-case analysis, storing +𝑇1 requires Θ(log(𝑇)) bits. +□ +6 +EXPERIMENTAL VERIFICATION +Basic settings. We numerically compare the proposed quantum +majority voting (Algorithm 1) with classical majority voting (Algo- +rithm 2, sample with replacement). We set the number of samples +𝑇 in Algorithm 2 to be 𝐾 · 2𝑠, where 𝑠 and 𝐾 are the parameters of +Algorithm 1. By doing this, the complexity of both algorithms is +Θ(𝐾 · 2𝑠). We set the number of voters 𝑛 = 220 ≈ 106, which is at a +similar order of magnitude as the number of voters in each state of +US. For example, the number of registered voters in New Hampshire +is 1,009,004≈106 [19]. In Figure 4. we set MoV = √𝑛 = 210 ≈ 103. +Or equivalently, one candidate got 219 + 210 = 525, 312 votes while +the other candidate got 219 − 210 = 523, 264 votes. Figure 5 sets +MoV = 2√𝑛 = 211 ≈ 2 × 103 +Implementation details. For quantum majority voting, we di- +rectly calculate the probability of outputting the correct winner +Pr[correct] through equation (5.26) in Nielsen and Chuang [18], +which is the output distribution of quantum counting algorithm. +The classical majority algorithm’s Pr[correct] is calculated from +the distribution of 𝑇1 (the number of sampled votes for the first +candidate, follows binomial distribution). Our experiment also plots +a lower bound for quantum majority voting (Inequality (5) for 𝐾 = 1 +and Inequality (1) for 𝐾 > 1) and an asymptotic bound for classical +majority voting (Inequality (6)). As mentioned above, none of our +experimental results rely on random sampling. Thus, the curves in +Figure 4 and Figure 5 have no randomness (thus has no error bar on +it). All experiments of this paper are implemented through MAT- +LAB 2022b and run on a Windows 11 desktop with AMD Ryzen 9 +5900X CPU and 32GB RAM. +Observations. The first observation is, no matter which setting +for MoV, the quantum majority voting has better accuracy than +classical majority voting. Especially, for the case that MoV = 211, +𝐾 = 1, and 𝑠 = 13, the quantum majority voting outputs the correct +winner almost for certain. However, the classical algorithm only has +∼63% probability to output the correct winner. It is not surprising +that the accuracy of quantum and classical majority voting both +increases when the increase of time complexity. We also observed +that the lower bound of 𝐾 = 69 and 𝐾 = 139 for quantum majority +voting are looser than the 𝐾 = 1 case. We believe that this behavior +is caused by the Chernoff bound, which is not asymptotically tight, +used in Inequality (1). We also note that providing an asymptotically +tight tail bound for binomial distribution is too far from the main +topic of this paper. +7 +CONCLUSIONS AND FUTURE WORKS +In conclusion, we took the first step in using quantum computation +to accelerate voting. We found that majority voting can be accel- +erated quadratically using quantum computation. Our proposed +quantum computation has the potential to improve the efficiency +of voting in large-scale and/or high-frequency decision-making +scenarios. A simple extension of this paper is applying quantum +computation on biased majority voting, where the threshold of +winning is not half-by-half. It would also be interesting to apply +quantum computation to accelerate other widely used voting rules. +For example, Borda, STV, Copeland, Ranked Pairs, or even general- +ized scoring rules [16, 23], which contain most of the widely-used +voting rules in real-world elections. +A +APPENDIX: ADDITIONAL INFORMATION +ABOUT QUANTUM MAJORITY VOTING +According to (6.4) in Nielsen and Chuang [18], Hadamard gate +changes 𝑡 qubits of |0⟩ to an equal superposition state (equal proba- +bility of observing any outcome under quantum measurements). +|𝜓⟩ = +1 +2𝑡/2 · +2𝑡 −1 +∑︁ +𝑥=0 +|𝑥⟩. +Letting 𝑓 : {0, · · · , 2𝑡 − 1} ↦→ {0, 1} be the binary function to +construct the quantum oracle, the orthogonal bases |𝛼⟩ and |𝛽⟩ are +defined as, +|𝛼⟩ ≜ +1 +√︁ +2𝑡 − +◦𝑛1 +· +∑︁ +𝑥:𝑓 (𝑥)=0 +|𝑥⟩ +and +|𝛽⟩ ≜ +1 +√︁ ◦𝑛1 +· +∑︁ +𝑥:𝑓 (𝑥)=1 +|𝑥⟩. + +13 +15 +17 +19 +log2 (K 2 s) +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Pr[correct] +K = 1 +Quantum +Quantum Lower +Classical +Classical Asymptotic +19 +21 +23 +25 +log2 (K 2 s) +0.92 +0.94 +0.96 +0.98 +1 +Pr[correct] +K = 69 +Quantum +Quantum Lower +Classical +Classical Asymptotic +20 +22 +24 +26 +log2 (K 2 s) +0.98 +0.985 +0.99 +0.995 +1 +Pr[correct] +K = 139 +Quantum +Quantum Lower +Classical +Classical Asymptotic +13 +15 +17 +19 +log2 (K 2 s) +0.6 +0.7 +0.8 +0.9 +1 +Pr[correct] +K = 1 +Quantum +Quantum Lower +Classical +Classical Asymptotic +19 +21 +23 +25 +log2 (K 2 s) +0.998 +0.9985 +0.999 +0.9995 +1 +Pr[correct] +K = 69 +Quantum +Quantum Lower +Classical +Classical Asymptotic +20 +22 +24 +26 +log2 (K 2 s) +0.99998 +0.999985 +0.99999 +0.999995 +1 +Pr[correct] +K = 139 +Quantum +Quantum Lower +Classical +Classical Asymptotic +Figure 4: Compare quantum majority voting (red squares) with classic majority voting (blue circles) when MoV = 210. In both +curves, we set 𝑠 = 13, 14, 15, 16, 17, 18, 19 for the seven points from left to the right respectively. The red hashed curve “Quantum +Lower” represents our lower bound for quantum majority voting. The blue dashed curve “Classical Asymptotic” represents +our asymptotic bound for classic majority voting. The horizontal axis can be seen as the logarithm of the algorithms’ time +complexity. +13 +15 +17 +19 +log2 (K 2 s) +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Pr[correct] +K = 1 +Quantum +Quantum Lower +Classical +Classical Asymptotic +19 +21 +23 +25 +log2 (K 2 s) +0.92 +0.94 +0.96 +0.98 +1 +Pr[correct] +K = 69 +Quantum +Quantum Lower +Classical +Classical Asymptotic +20 +22 +24 +26 +log2 (K 2 s) +0.98 +0.985 +0.99 +0.995 +1 +Pr[correct] +K = 139 +Quantum +Quantum Lower +Classical +Classical Asymptotic +13 +15 +17 +19 +log2 (K 2 s) +0.6 +0.7 +0.8 +0.9 +1 +Pr[correct] +K = 1 +Quantum +Quantum Lower +Classical +Classical Asymptotic +19 +21 +23 +25 +log2 (K 2 s) +0.998 +0.9985 +0.999 +0.9995 +1 +Pr[correct] +K = 69 +Quantum +Quantum Lower +Classical +Classical Asymptotic +20 +22 +24 +26 +log2 (K 2 s) +0.99998 +0.999985 +0.99999 +0.999995 +1 +Pr[correct] +K = 139 +Quantum +Quantum Lower +Classical +Classical Asymptotic +Figure 5: Compare quantum majority voting (red squares) with classic majority voting (blue circles) when MoV = 211. In both +curves, we set 𝑠 = 13, 14, 15, 16, 17, 18, 19 for the seven points from left to the right respectively. The red hashed curve “Quantum +Lower” represents our lower bound for quantum majority voting. The blue dashed curve “Classical Asymptotic” represents +our asymptotic bound for classic majority voting. The horizontal axis can be seen as the logarithm of the algorithms’ time +complexity. +Under the |𝛼⟩ |𝛽⟩ base, the equal superposition state +|𝜓⟩ = +√︂ +2𝑡 − +◦𝑛1 +2𝑡 +|𝛼⟩ + +√︂ ◦𝑛1 +2𝑡 |𝛽⟩. +Since +𝜃 = 2 arcsin +�√︁ ◦𝑛1 · 2−𝑡 +� +, +we have +|𝜓⟩ = cos +�𝜃 +2 +� +|𝛼⟩ + sin +�𝜃 +2 +� +|𝛽⟩, +𝑂𝑓 |𝜓⟩ = cos +�𝜃 +2 +� +|𝛼⟩ + sin +� +−𝜃 +2 +� +|𝛽⟩, and +𝐺|𝜓⟩ = cos +� 3𝜃 +2 +� +|𝛼⟩ + sin +� 3𝜃 +2 +� +|𝛽⟩. +One can see that the above states match the geometric illustration +in Figure 2. + +REFERENCES +[1] Ben Abramowitz, Edith Elkind, Davide Grossi, Ehud Shapiro, and Nimrod Talmon. +2021. Democratic Forking: Choosing Sides with Social Choice. In Proceedings of +ADT. +[2] Akshay Ajagekar and Fengqi You. 2020. Quantum computing assisted deep learn- +ing for fault detection and diagnosis in industrial process systems. Computers & +Chemical Engineering 143 (2020), 107119. https://doi.org/10.1016/j.compchemeng. +2020.107119 +[3] Akshay Ajagekar and Fengqi You. 2021. Quantum computing based hybrid deep +learning for fault diagnosis in electrical power systems. Applied Energy 303 +(2021), 117628. https://doi.org/10.1016/j.apenergy.2021.117628 +[4] Marcello Benedetti, John Realpe-Gómez, Rupak Biswas, and Alejandro Perdomo- +Ortiz. 2016. Estimation of effective temperatures in quantum annealers for +sampling applications: A case study with possible applications in deep learning. +Phys. Rev. A 94 (Aug 2016), 022308. Issue 2. https://doi.org/10.1103/PhysRevA.94. +022308 +[5] André Berthiaume and Gilles Brassard. 1994. Oracle quantum computing. Journal +of modern optics 41, 12 (1994), 2521–2535. +[6] John D Cook. 2009. Upper and lower bounds for the normal distribution function. +[7] Cynthia Dwork, Ravi Kumar, Moni Naor, and D. Sivakumar. 2001. Rank aggrega- +tion methods for the web. In Proceedings of the 10th World Wide Web Conference. +613–622. +[8] Mark Fey. 2004. May’s theorem with an infinite population. Social Choice and +Welfare 23, 2 (2004), 275–293. +[9] Davide Grossi. 2021. Social Choice Around the Block: On the Computational +Social Choice of Blockchain. In Proceedings of AAMAS. +[10] Lov K. Grover. 1996. +A Fast Quantum Mechanical Algorithm for Database +Search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory +of Computing (Philadelphia, Pennsylvania, USA) (STOC ’96). Association for +Computing Machinery, New York, NY, USA, 212–219. https://doi.org/10.1145/ +237814.237866 +[11] Tad Hogg and Dmitriy Portnov. 2000. Quantum optimization. Information +Sciences 128, 3-4 (2000), 181–197. +[12] Elham Kashefi, Adrian Kent, Vlatko Vedral, and Konrad Banaszek. 2002. Com- +parison of quantum oracles. Physical Review A 65, 5 (2002), 050304. +[13] Alastair Kay. 2018. Tutorial on the quantikz package. +[14] Emil T Khabiboulline, Juspreet Singh Sandhu, Marco Ugo Gambetta, Mikhail D +Lukin, and Johannes Borregaard. 2021. +Efficient Quantum Voting with +Information-Theoretic Security. +[15] Shiguo Lian and Yan Zhang. 2009. Handbook of research on secure multimedia +distribution. +[16] Ao LIU, Yun Lu, Lirong Xia, and Vassilis Zikas. 2020. How Private Are Commonly- +Used Voting Rules?. In Conference on Uncertainty in Artificial Intelligence. PMLR, +629–638. +[17] Andrew Mao, Ariel D. Procaccia, and Yiling Chen. 2013. Better Human Compu- +tation Through Principled Voting. In Proceedings of the National Conference on +Artificial Intelligence (AAAI). Bellevue, WA, USA. +[18] Michael A Nielsen and Isaac Chuang. 2002. Quantum computation and quantum +information. +[19] Independent Voter Project. 2020. New Hampshire Voter Statistics. +[20] Joan Alfina Vaccaro, Joseph Spring, and Anthony Chefles. 2007. Quantum pro- +tocols for anonymous voting and surveying. Physical Review A 75, 1 (2007), +012333. +[21] Wim Van Dam. 1998. Quantum oracle interrogation: Getting all information for +almost half the price. In Proceedings 39th Annual Symposium on Foundations of +Computer Science (Cat. No. 98CB36280). IEEE, 362–367. +[22] Lirong Xia. 2012. Computing the margin of victory for various voting rules. In +Proceedings of the 13th ACM conference on electronic commerce. 982–999. +[23] Lirong Xia. 2013. Generalized scoring rules: a framework that reconciles Borda +and Condorcet. ACM SIGecom Exchanges 12, 1 (2013), 42–48. +[24] Lirong Xia. 2019. Learning and Decision-Making from Rank Data. Morgan & +Claypool Publishers. +[25] Lirong Xia and Weiqiang Zheng. 2022. Beyond the Worst Case: Semi-Random +Complexity Analysis of Winner Determination. +[26] Peng Xue and Xin Zhang. 2017. A simple quantum voting scheme with multi- +qubit entanglement. Scientific reports 7, 1 (2017), 1–4. + diff --git a/idE1T4oBgHgl3EQfNAMs/content/tmp_files/load_file.txt b/idE1T4oBgHgl3EQfNAMs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad65e0ecd89ea13f9f6bf0bc3e07781557737da0 --- /dev/null +++ b/idE1T4oBgHgl3EQfNAMs/content/tmp_files/load_file.txt @@ -0,0 +1,598 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf,len=597 +page_content='Accelerating Majority Voting by Quantum Computation Ao Liu, Lirong Xia Rensselaer Polytechnic Institute Troy, NY, United States of America liua6@rpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='edu,xialirong@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='com Nengkun Yu Stony Brook University Stony Brook, NY, United States of America nengkun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='yu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='edu ABSTRACT In this paper, we propose a quantum algorithm for majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We prove that the time complexity of quantum majority voting could be quadratically smaller than its classical algorithms, which is supported by our experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' KEYWORDS Voting Rules, Majority Voting, Quantum Computation ACM Reference Format: Ao Liu, Lirong Xia and Nengkun Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Accelerating Majority Voting by Quantum Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023), London, United Kingdom, May 29 – June 2, 2023, IFAAMAS, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 1 INTRODUCTION Voting is a widely used methodology to make collective decisions in a wide range of applications [24], such as political elections, recommender systems [7], crowdsourcing [17], and blockchain governance [1, 9], just to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' While many voting rules, such as plurality and single transferable vote (STV), can be computed in polynomial time, in large-scale, high-frequency decision-making scenarios, it is desirable that the winner can be computed in a short amount of time [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Like in database systems, a polynomial or even linear runtime may be too slow already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' As many such scenarios are low-stakes, randomized algorithms and small “errors” are acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' One natural solution is to randomly sample a subset of votes (with or without replacement), and then compute the winner of the sampled votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Can we do better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum computation appears to be a promising approach, as it has successfully accelerated many computational tasks such as search [10], optimization [11], and machine learning [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' How- ever, we are not aware of a previous work on accelerating voting using quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Thus, the following problem remains open Can voting be accelerated by quantum computation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In this paper, we take a first step to address this question by fo- cusing on the simple majority voting (for two-candidate), where the winner is the candidate with more votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Majority voting has many desired properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=', fairness, decisiveness, and monotonic- ity) and is widely used in group decision-makings [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We target to use quantum computation to accelerate the quantum majority voting when a small probability of “errors” are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Also see Section 4 for a detailed explanation of when quantum computation accelerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Sys- tems (AAMAS 2023), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Ricci, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Yeoh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Agmon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' An (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' ), May 29 – June 2, 2023, London, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' © 2023 International Foundation for Autonomous Agents and Multiagent Systems (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='ifaamas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Our contributions are three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Firstly, we propose the quantum majority voting algorithm (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Secondly, we theoretically prove that our proposed quantum majority voting algorithm is quadratically faster than its classical algorithm (see Table 1 for a more detailed comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Thirdly, we experimentally verified our theoretical results (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' time complexity space complexity Quantum (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1) Θ �𝑛·log(1/𝜖) MoV � 𝑂 � log( 𝑛·log(1/𝜖) MoV ) � Classical (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1) Θ �𝑛2·log(1/𝜖) MoV2 � Θ � log( 𝑛2·log(1/𝜖) MoV2 ) � Table 1: Summary for the theoretical results, where MoV means margin of victory (See Section 2 for its formal def- inition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' All algorithms in the table assume the algorithm output the winner with no less than 1 − 𝜖 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Related works and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' To the best of our knowledge, none of the literature has used quantum computation to accelerate voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Vaccaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [20] firstly introduces the idea of quantum communication or quantum computation to voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The quantum voting algorithm in Vaccaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [20] provides theoretically guar- anteed security (against colluding attacks [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Xue and Zhang [26] improved the result in Vaccaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [20] by proposing a simpler vot- ing protocol but with stronger security guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Khabiboulline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [14] proposed an “all-in-one” quantum voting protocol, which focuses on achieving anonymity without losing security guaran- tees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' However, all of the above approaches require Ω(𝑛) quantum communication cost, which means their proposed method does accelerate voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2 PRELIMINARY Majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In (two-candidate) majority voting, 𝑛 > 1 voters cast their votes for one of the two candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We use 𝑛1 to denote the number of votes for the first candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Similarly, 𝑛2 = 𝑛 − 𝑛1 is the number of votes for the second candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' If 𝑛1 > 𝑛2 (or 𝑛2 > 𝑛1), the first (or the second) candidate will be announced as the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' If 𝑛1 = 𝑛2, the voting rule will break the tie by outputting a candidate according to the “tie-breaking rule”, which usually outputs a candidate uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Margin of vic- tory (MOV) describes the smallest number 𝑘 such that 𝑘 voters can change the winner by voting differently [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In majority voting, it is easy to check that MoV = ⌈|𝑛/2 − 𝑛1|⌉ = ⌈|𝑛/2 − 𝑛2|⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Basic quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 Quantum bit (or qubit in short) is the counterpart of classical bit, which takes a deterministic binary from {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Qubit, on the other hand, is represented by a linear com- bination of {|0⟩, |1⟩}, which are counterparts to {0, 1}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 1This paper adopts the same notation system as Nielsen and Chuang [18], which is a textbook about quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='02995v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='CY] 8 Jan 2023 That is, every qubit |𝜓⟩ is written as |𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩, where 𝛼 and 𝛽 are complex numbers and are usually called ampli- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' If we measure the qubit, there is |𝛼|2 probability to get 0 and |𝛽|2 probability to get 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Naturally, we always have |𝛼|2 + |𝛽|2 = 1 because the probabilities should sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Qubits sometimes are written as vectors to simplify notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Formally, �𝛼 𝛽 � ≜ 𝛼|0⟩ + 𝛽|1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 𝑡 > 1 qubits are presented as a 2𝑡-dimensional vector, where the 𝑗- th component of the vector (denoted as 𝛼𝑗) represents the amplitude of |𝑗1 · · · 𝑗𝑡⟩ (or |𝑗⟩), where 𝑗1 · · · 𝑗𝑡 is the binary representation of 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Similar to the 1-qubit case, the probability of observing 𝑗1, · · · , 𝑗𝑡 from those 𝑡 qubit equals to |𝛼𝑗 |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' A quantum operation (quantum gate) 𝑄 on 𝑡 qubits is denoted by a 2𝑡 × 2𝑡 unitary matrix, which means the matrix’s inverse is its Hermitian conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Applying a quantum operation 𝑄 on quantum state |𝜓⟩ is denoted by 𝑄|𝜓⟩ ≜ 𝑸(2𝑡 ×2𝑡 ) �𝜓(2𝑡 ), where the the quantum operator 𝑸(2𝑡 ×2𝑡 ) is a 2𝑡 ×2𝑡 unitary matrix and the quantum state �𝜓(2𝑡 ) is a 2𝑡 dimensional column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum circuit of some useful quantum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2 Quan- tum circuits run from the left-hand side to the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For example, the following circuit means applying Hadamard gate 𝐻 on a quantum state |𝜓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' |𝜓⟩ 𝐻 where 𝑯 = 1 √ 2 �1 1 1 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The quantum circuit notion |𝜓⟩ 0/1 𝑏 denotes measuring quantum state |𝜓⟩ with 0/1 base (𝑏 denotes the result of measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Naturally, the complexity of quantum measurement and Hadamard gate are both Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum oracle [5, 12, 21] is a widely-used operator to encode binary functions or binary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Given 𝑡 qubits and a binary function 𝑓 : {0, · · · , 2𝑡 − 1} ↦→ {0, 1}, quantum oracle (based on function 𝑓 (·)) applies a phase shift of −1 = 𝑒𝜋𝑖 if 𝑓 (𝑥) = 1 and does nothing otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We can query oracle many times and regard the number of queries as the cost [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Formally, � 𝑂𝑓 |𝑥⟩ = |𝑥⟩ if 𝑓 (𝑥) = 1 𝑂𝑓 |𝑥⟩ = −|𝑥⟩ otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Suppose we have a quantum gate 𝐺 on 𝑡 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The following operation is called controlled-𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 𝐺 = �𝑰(2𝑡 ×2𝑡 ) 0(2𝑡 ×2𝑡 ) 0(2𝑡 ×2𝑡 ) 𝑮(2𝑡 ×2𝑡 ) � , 2All quantum circuits of this paper are drawn using the Quantikz package [13] for LATEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' where 𝑰 denotes the identity matrix, and 0 denotes the zeros matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' To simplify notations, we also write 𝐺𝑎 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 𝐺 𝐺 (repeat 𝑎 times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 3 QUANTUM MAJORITY VOTING Voting through quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In classical voting, the votes usually are sent to an “aggregator”, who is responsible for aggregating the votes and announcing the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Our quantum majority voting follows a similar procedure, where the “aggregator” constructs a quantum oracle based on the votes and uses our pro- posed quantum majority voting algorithm (Algorithm 1) to decide the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Construct the quantum oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since there are two candidates in majority voting, the votes can be treated as binary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In this paper, 1 represents a vote to the first candidate and 0 represents a vote to the second candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' As a quantum oracle with 𝑡 qubits has the ability to encode 2𝑡 binary bits, we use 𝑡 = 2⌈log𝑛⌉ qubits to encode the votes from𝑛 voters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3 The rest � 2⌈log𝑛⌉ − 𝑛 � bits are filled in 0 and 1 half-by-half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Formally, the function 𝑓 : {0, · · · , 2𝑡 − 1} ↦→ {0, 1} to construct the quantum oracle is defined as 𝑓 (𝑥) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 1 if candidate 𝑥 votes the first candidate or 𝑥 > 𝑛 − 1 − ⌊(2𝑡 − 𝑛)/2⌋ 0 otherwise , where the voters are numbered from 0 to (𝑛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We let the ad- justed number of votes for the first candidate (the number of 1’s) as 𝑛1 ≜ # {𝑥 : 𝑓 (𝑥) = 1} = 𝑛1 + � 2𝑡 −𝑛 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Algorithm 1: Quantum Majority Voting 1: Inputs: 𝑛 voters’ votes 𝑉0, · · · ,𝑉𝑛−1, the number of qubits 𝑠 ≥ 2, and the number of iterations 𝐾 2: Initialization: Set 𝐾1 = 0 and construct the quantum oracle 𝑂𝑓 based on 𝑉0, · · · ,𝑉𝑛−1 3: Construct Grover operator 𝐺 using 𝑂𝑓 according to Figure 1 4: for 𝑘 ∈ {1, · · · , 𝐾} do 5: Apply the Grover operator to the quantum counting circuit (Figure 3) with 𝑠 qubits in Register 1 6: if the binary decimal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑏1 · · ·𝑏𝑠 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 then 7: 𝐾1 = 𝐾1 + 1 8: end if 9: end for 10: if 𝐾1 ≥ 𝐾/2 then 11: Announce the first candidate as the winner 12: else 13: Announce the second candidate as the winner 14: end if 3In all discussions of this paper, logarithm function log(·) uses 2 as its base, and ln(·) uses Euler’s number 𝑒 as its base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Formal definition of quantum majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We formally define quantum majority voting in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Basically, Algo- rithm 1 repeats the quantum counting algorithm by 𝐾 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In each round, quantum counting estimates the number of votes for each candidate and makes a prediction about the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, the “aggregator” announces the candidate who wins in a larger number of rounds as the winner of the majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Usually, the number of rounds 𝐾 is set as an odd number to avoid ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Next, we will introduce the functionality and implementation of each step of Algorithm 1 in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 𝑡 qubits 𝑂𝑓 𝐻 𝑄𝑃𝑆 𝐻 𝐻 𝐻 Figure 1: The circuit for Grover operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Prepare Grover operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The Grover operation is constructed by the quantum circuit in Figure 1, where 𝑡 = ⌈log𝑛⌉ denotes the minimum number of bits to represent 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The quantum operator 𝑄𝑃𝑆 is called quantum phase shifting, which provides a phase shift of −1 on every state except |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Mathematically, |0⟩ 𝑄𝑃𝑆 −→ |0⟩ and |𝑥⟩ 𝑄𝑃𝑆 −→ −|𝑥⟩ for any 𝑥 ∈ 1, · · · , 2𝑡 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Here, |𝑥⟩ represents the 𝑥-th base state of the 𝑡 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The high- level idea of Grover operator’s functionality is shown in Figure 2, where |𝜓⟩ is the input of Grover operators in quantum counting, and {|𝛼⟩, |𝛽⟩} is a pair of orthogonal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The formal definition of |𝜓⟩, |𝛼⟩, and |𝛽⟩ can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Under the |𝛼⟩ |𝛽⟩ base, the quantum oracle 𝑂𝑓 reflects |𝜓⟩ over |𝛼⟩, while the rest parts of 𝐺 reflects 𝑂𝑓 |𝜓⟩ over |𝜓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The angle between the output state 𝐺|𝜓⟩ and initial state |𝜓⟩ 𝜃 = 2 arcsin �√︁ ◦𝑛1 · 2−𝑡 � , which includes the information about 𝑛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since function arcsin(√𝑥) grows quadratically faster than linear functions when 𝑥 is small, we expect that an estimation about arcsin(√𝑥) could be quadratically more accurate than directly estimate 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The quantum circuit of quantum counting is shown in Figure 3, where 𝑄𝐹𝑇 † denotes the quantum reverse Fourier transformation (its time complexity is Θ(𝑠2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' At a high level, quantum counting estimates angle 𝜃 in Grover operator (plotted in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Mathematically, the output of quantum counting ˆ𝜑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑏1 · · ·𝑏𝑠 is an estimation of 𝜑 ≜ 𝜃/(2𝜋) = arcsin �√︁ ◦𝑛1 · 2−𝑡 � /𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Here, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑏1 · · ·𝑏𝑠 is a binary decimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For example, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='011 represents (2−2 + 2−3) = 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Next, we present a useful error bound for quan- tum counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' G| ۧ 𝝍 | ۧ 𝝍 Of | ۧ 𝝍 | ۧ 𝜷 | ۧ 𝜶 𝜃 𝜃/2 𝜃/2 Figure 2: An illustration of Grover operator’s functionality (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3 in Nielsen and Chuang [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 (Error bound for qantum counting, Ineqality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='34) in Nielsen and Chuang [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Using the notations above, for quantum counting, Pr[| ˆ𝜑 − 𝜑| ≥ 𝛿] ≤ 1 2(𝛿 · 2𝑠 − 1) , where 𝑠 is the number of qubits in Register 1 of quantum counting’s circuit (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' According to the design of 𝑂𝑓 , we need to decide whether 𝑛1 is larger than or smaller than 2𝑡−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We also note that 𝑛1 ≥ 2𝑡−1 ⇐⇒ 𝜑 ≥ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Thus, predicting the winner of majority voting reduces to the prob- lem of predicting the relationship between 𝜑 and 1 4 (or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 in binary decimal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 4 THEORETICAL ANALYSIS OF FAST QUANTUM MAJORITY VOTING In Section 3, we proposed quantum majority voting (Algorithm 1) and explained why it works at a high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In this section, we provide theoretical guarantees about the accuracy (probability of outputting the correct winner), time complexity, and space com- plexity of quantum majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' When quantum majority voting (may) accelerate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We first think about the cases where classical algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=', Algorithm 2, which randomly sample a subset of votes and use the subset to pre- dict the winner) do not need to be improved or cannot be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' When the margin of victory MoV = Θ(𝑛), classical algorithms are already very fast according to the Chernoff bound, which says the classical algorithms’ error rate can be exponentially small in terms of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Another case is when MoV is very small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=', MoV = Θ(1)) where classical algorithms’ performance is close to the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 0/1 𝑏1 |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 0/1 𝑏2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 0/1 𝑏𝑠 |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Register 1 𝑠 qubits 𝐻 𝑄𝐹𝑇 † 𝐻 𝐻 Register 2 𝑡 qubits 𝐻 𝐺20 𝐺21 𝐺2𝑠−1 trash 𝐻 Figure 3: The circuit for quantum counting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' this case, any algorithms have to look into each vote to decide the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since the complexity of counting every vote is Θ(𝑛), there is not a lot of space for the classical algorithms to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In order to improve readability, we let “tie” (MoV ≤ 1) be a special case of MoV = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Throughout this paper, we assume that the margin of victory MoV = Θ(𝑛𝑐), where 𝑐 ∈ (0, 1) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For example, in one of the settings in our experimental verification (Figure 4), the number of voters 𝑛 ≈ 106 and MoV = √𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In this example, the winner only got ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2% more votes than the loser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The theoretical guarantee of quantum majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1, we provide the theoretical guarantee of Algorithm 1’s performance under the above-discussed conditions where quantum algorithms may accelerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 (Theoretical guarantee of qantum major- ity voting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For arbitrary constant 𝜖 ∈ (0, 1), quantum major- ity voting (Algorithm 1) has the following three properties when 𝑠 = max {2, 𝑡 − ⌊log(MoV − 1)⌋ + 4}, and 𝐾 = ⌈12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3 · ln(1/𝜖)⌉, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It outputs the correct voting outcome with at least 1 −𝜖 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It’s time complexity is Θ �𝑛·log(1/𝜖) MoV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It’s space complexity is Θ � log( 𝑛·log(1/𝜖) MoV ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 describes the required parameter 𝑠, time complex- ity, and space complexity of quantum majority voting to achieve arbitrary accuracy (probability of outputting the correct winner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 is proved by combining Chernoff bound and Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2, which shows the performance of quantum majority voting when the number of iterations 𝐾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' According to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2, we know that each iteration has at least 𝑝 = 32−3𝜋 32−2𝜋 ≈ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='8% probability to out- put the correct winner if setting𝑠 = max {2, 𝑡 − ⌊log(MoV − 1)⌋ + 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=', we assume that 𝑛1 ≥ 𝑛2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, we apply Chernoff bound and have, 1 − 𝜖 = Pr[correct winner] ≥ 1 − Pr[𝐾1 ≥ 𝐾/2] ≥ 1 − exp � −(1 − 1 2𝑝 )2 · 𝐾 · 𝑝/2 � , (1) which is equivalent with 𝐾 ≥ 2𝑝 · ln(1/𝜖) (𝑝 − 1/2)2 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3 · ln(1/𝜖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 follows by directly apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2 (Theoretical guarantee of qantum majority voting when 𝐾 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For arbitrary constant 𝜖 ∈ (0, 1), quantum majority voting (Algorithm 1) has the following three properties when 𝐾 = 1 and 𝑠 = max � 2, � 𝑡 − log(MoV − 1) + log( 𝜋 2𝜖 + 𝜋) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It outputs the correct voting outcome with at least 1 −𝜖 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It’s time complexity is Θ � 𝑛 𝜖·MoV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It’s space complexity is Θ � log( 𝑛 𝜖·MoV) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Comparing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2 with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1, we know that Algo- rithm 1 reduces the 1/𝜖 term in (either time or space) complexity to log(1/𝜖) by setting 𝐾 = ⌈12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3 · ln(1/𝜖)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2 directly follows by setting parameter 𝑠 = � 𝑡 − log(MoV − 1) + log( 𝜋 2𝜖 + 𝜋) � for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3, which proves the accuracy, time complexity, and space complexity for arbitrary parameter 𝑠 ≥ 2 when 𝐾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For any parameter 𝑠 ≥ 2, quantum majority voting (Algorithm 1) with the number of iterations 𝐾 = 1 has the following three properties, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It outputs the correct voting outcome with at least 1− 𝜋 2 · 1 (MoV−1) ·2𝑠−𝑡 −𝜋 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It’s time complexity is Θ (2𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It’s space complexity is Θ (log(𝑛) + 𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Firstly, we prove the case that 𝑛1 ≤ 𝑛2 (or equivalently, 𝑛1 ≤ 2𝑡−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We recall the reasoning behind the quan- tum counting algorithm 𝜑 = arcsin �√︁ ◦𝑛1 · 2−𝑡 � 𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' (2) Next, we provide an upper bound for 𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since 𝑛1 = 𝑛1 + ⌈ 2𝑡 −𝑛 2 ⌉, we know that 𝑛1 = 𝑛1 + �2𝑡 − 𝑛 2 � = �𝑛 2 � − MoV + �2𝑡 − 𝑛 2 � ≤ 2𝑡−1 − MoV + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' (3) To simplify notations, we let NMoVQ = (MoV − 1) · 2−𝑡 to denote the “normalized” MoV for quantum majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Combining (2) and (3), we have, 𝜑 ≤ arcsin �√︁1/2 − NMoVQ � 𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We define 𝑔(𝑥) = arcsin �√︁ 1/2 − 𝑥 � 𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' By standard calculus, we know that 𝑔(0) = 1 4 𝑔′(𝑥)|𝑥=0 = − 1 𝜋 𝑔′′(𝑥)|𝑥=0 = 0 𝑔′′(𝑥)|𝑥>0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Thus, 𝑔(𝑥) is a concave function when 𝑥 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, we have, 𝜑 ≤ arcsin �√︁ 1/2 − 𝑥 � 𝜋 ≤ 1 4 − NMoVQ 𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' (4) Then we have that Pr [incorrect winner] ≤ Pr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑏1 · · ·𝑏𝑠 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01] = Pr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑏1 · · ·𝑏𝑠 − 𝜑 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 − 𝜑] ≤ Pr[|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑏1 · · ·𝑏𝑠 − 𝜑| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 − 𝜑], where 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 represent 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Next, we apply the error bound for quan- tum counting (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Pr [incorrect winner] ≤ Pr[|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑏1 · · ·𝑏𝑠 − 𝜑| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 − 𝜑] ≤ 1 2[(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 − 𝜑) · 2𝑠 − 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Note that the binary decimal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 represent 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, we combine the above inequality with inequality (4) and have Pr [incorrect winner] ≤ 1 2[(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='01 − 𝜑) · 2𝑠 − 1] ≤ 1 2 � NMoVQ 𝜋 2𝑠 − 1 � = 𝜋 2 · 1 NMoVQ · 2𝑠 − 𝜋 = 𝜋 2 · 1 (MoV − 1) · 2𝑠−𝑡 − 𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, we know that Pr [correct winner] = 1 − Pr [incorrect winner] ≥ 1 − 𝜋 2 · 1 (MoV − 1) · 2𝑠−𝑡 − 𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' (5) By now, we have proved the case that 𝑛1 ≤ 𝑛2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since 𝑔(𝑥) is rota- tionally symmetric over (0, 1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, the 𝑛1 ≥ 𝑛2 case directly follows by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In the quantum circuit of quantum counting (Figure 3), the controlled-𝐺 gate is called �𝑠−1 𝑖=0 2𝑖 = 2𝑠 − 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since the quantum Fourier transformation requires Θ(𝑠2) time, then, the time complexity is Hadamard ���� Θ(1) + Controlled-𝐺 �������������� Θ(2𝑠 − 1) + 𝑄𝐹𝑇 † ���� Θ(𝑠2) + measurement ���� Θ(1) = Θ(2𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, the time complexity part of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='3 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Space Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We note that Register 1 contains 𝑠 qubits and Register 2 contains 𝑡 = 2⌈log(𝑛)⌉ = Θ(log(𝑛)) qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' the space complexity of the algorithm is 𝑠 + 𝑡 = Θ (log(𝑛) + 𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' □ 5 COMPARING QUANTUM & CLASSICAL MAJORITY VOTING In this section, we compare quantum majority voting (Algorithm 1) with classical majority voting (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The classical algorithm is designed according to the idea of sampling (either with or without replacement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' At the high level, it uses the 𝑇 randomly sampled votes to estimate the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In the next theorem, we will present the accuracy, time complexity, and space complexity of classical majority voting under the conditions where quantum algorithms may accelerate (discussed in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For any 𝜖 ∈ (0, 1), Algorithm 2 (with or without replacement) with the number of samples 𝑇 = Θ �𝑛2·log(1/𝜖) MoV2 � has the following three properties, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It outputs the correct winner with 1 − 𝜖 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Its time complexity is Θ �𝑛2·log(1/𝜖) MoV2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Its space complexity is Θ � log( 𝑛2·log(1/𝜖) MoV2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 describes the required parameter 𝑇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' time complex- ity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' and space complexity of classical majority voting to achieve Algorithm 2: Classical Majority Voting 1: Inputs: 𝑛 voters’ votes 𝑉1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='𝑉𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' and the number of samples 𝑇 ≤ 𝑛 2: Randomly sample 𝑇 votes with or without replacement 3: Count the number of sampled votes for each candidate 4: if the first candidate gets more votes then 5: Announce the first candidate as the winner 6: else 7: Announce the second candidate as the winner 8: end if arbitrary accuracy (probability of outputting the correct winner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1 directly follows by setting parameter 𝑇 = Θ �𝑛2·log(1/𝜖) MoV2 � in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2, which proves the accuracy, time complexity, and space complexity of classical majority voting for arbitrary 𝑇 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Classical majority voting (Algorithm 2, either sample with or without replacement) has the following three asymptotic properties, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It output the correct winner with 1−exp � −Θ � MoV2·𝑇 𝑛2 �� probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Its time complexity is Θ(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Its space complexity is Θ(log(𝑇)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=', we assume that 𝑛1 < 𝑛2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, the probability such that a vote for the first candidate got sampled 𝑝 ≜ Pr [first candidate’s vote sampled] = 1 2 − NMoVQ, where NMoVC ≜ 2MoV−(𝑛 mod 2) 2𝑛 denotes the “normalized” MoV for the classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Next, we will derive an asymptotic upper and lower bound for the accuracy of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Asymptotic Upper Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' If the votes are sampled with replace- ment, we know that the number of votes for the first candidate follows binomial distribution B(𝑇, 𝑝), which converges to Gaussian distribution N (𝑛𝑝,𝑛𝑝(1 − 𝑝)) when 𝑇 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' If the votes are sampled without replacement, we know that the number of votes for the first candidate follows hypergeometric distribution H (𝑇,𝑛,𝑇/2,𝑛𝑝), which converges to Gaussian distribu- tion N (𝑛𝑝,𝑛𝑝(1 − 𝑝)) when 𝑇 = 𝑐 · 𝑛 and 𝑛 → ∞, where 𝑐 ∈ (0, 1) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We note that the normal distribution of sampling without replacement is the same normal distribution as sampling with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Mathematically, for both sampling with or without replacement, we have, lim 𝑇→∞ Pr [incorrect output] ≥ lim 𝑇→∞ Pr � 𝑇1 > 𝑇 2 � = Φ ��� � NMoVC · √ 𝑇 √︃ 1/4 − NMoV2 C ��� � , (6) where 𝑇1 is the number of sampled votes for the first candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Function Φ(·) denotes the cumulative distribution function (CDF) of the standard normal distribution N (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' According to Cook [6], we have the following lower bound for the CDF of standard normal distribution, Φ(𝑥) > 1 √ 2𝜋 𝑥 𝑥2 + 1 · 𝑒−𝑥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' (7) Combining (6) and (7), we have, lim 𝑇→∞ Pr [incorrect output] > 1 √ 2𝜋 NMoVC· √ 𝑇 √︃ 1/4−NMoV2 C � NMoVC· √ 𝑇 √︃ 1/4−NMoV2 C �2 + 1 exp � − NMoV2 C · 𝑇 1/2 − 2NMoV2 C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' (8) Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' When 𝑇 = 𝑂(𝑛2/MoV2), we have that NMoV2 C · 𝑇 = � 2MoV − (𝑛 mod 2) 2𝑛 �2 𝑇 = 𝑂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since MoV = 𝑜(𝑛), we have, 1 4 − NMoV2 C = Θ(1) and √︂ 1 2 − 2NMoV2 C = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Thus, we have NMoVC · √ 𝑇 √︃ 1/4 − NMoV2 C = Θ(NMoVC · √ 𝑇) and exp � − NMoV2 C · 𝑇 1/2 − 2NMoV2 C � = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' (9) By combining (9) and (8), we know that lim 𝑇→∞ Pr [correct output] = 1 − lim 𝑇→∞ Pr [incorrect output] = 1 − 𝑂 � 1 NMoVC · √ 𝑇 � = 1 − 𝑂 � 𝑛 MoV · √ 𝑇 � = 1 − Ω(1) = 𝑂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' When 𝑇 = 𝜔(𝑛2/MoV2), we have that NMoVC · √ 𝑇 √︃ 1/4 − NMoV2 C = 𝜔(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Combining the above equation with (8), we have that lim 𝑇→∞ Pr [correct output] = 1 − lim 𝑇→∞ Pr [incorrect output] = 1 − exp � −Ω � NMoV2 C · 𝑇 �� = 1 − exp � −Ω � MoV2 · 𝑇 𝑛2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since 1 − exp � −Ω � MoV2·𝑇 𝑛2 �� = 𝑂(1) when 𝑇 = 𝑂(𝑛2/MoV2), we can combine Case 1 and Case 2 as lim 𝑇→∞ Pr [correct output] = 1 − exp � −Ω � MoV2 · 𝑇 𝑛2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Asymptotic Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Similar to the upper bound, we use the relationship between normal distribution and binomial distribution (or hypergeometric distribution) and have lim 𝑇→∞ Pr [incorrect output] ≤ lim 𝑇→∞ Pr � 𝑇1 ≥ 𝑇 2 � = Φ ��� � NMoVC · √ 𝑇 √︃ 1/4 − NMoV2 C ��� � , According to Cook [6], we have the following upper bound for the CDF of standard normal distribution, Φ(𝑥) < 1 √ 2𝜋 1 𝑥 · 𝑒−𝑥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, we repeat a similar process as the asymptotic upper bound and have the following bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' lim 𝑇→∞ Pr [correct output] = 1 − exp � −𝑂 � MoV2 · 𝑇 𝑛2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Then, the accuracy part of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2 follows by combining the asymptotic upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Algorithm 2 draw 𝑇 samples and count them, which cost Θ(𝑇) time in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The counting step of Algorithm 2 needs to store an integer 𝑇1, which is the number of votes to the first candi- date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For either average-case analysis or worst-case analysis, storing 𝑇1 requires Θ(log(𝑇)) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' □ 6 EXPERIMENTAL VERIFICATION Basic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We numerically compare the proposed quantum majority voting (Algorithm 1) with classical majority voting (Algo- rithm 2, sample with replacement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We set the number of samples 𝑇 in Algorithm 2 to be 𝐾 · 2𝑠, where 𝑠 and 𝐾 are the parameters of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' By doing this, the complexity of both algorithms is Θ(𝐾 · 2𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We set the number of voters 𝑛 = 220 ≈ 106, which is at a similar order of magnitude as the number of voters in each state of US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For example, the number of registered voters in New Hampshire is 1,009,004≈106 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' we set MoV = √𝑛 = 210 ≈ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Or equivalently, one candidate got 219 + 210 = 525, 312 votes while the other candidate got 219 − 210 = 523, 264 votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Figure 5 sets MoV = 2√𝑛 = 211 ≈ 2 × 103 Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For quantum majority voting, we di- rectly calculate the probability of outputting the correct winner Pr[correct] through equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='26) in Nielsen and Chuang [18], which is the output distribution of quantum counting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The classical majority algorithm’s Pr[correct] is calculated from the distribution of 𝑇1 (the number of sampled votes for the first candidate, follows binomial distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Our experiment also plots a lower bound for quantum majority voting (Inequality (5) for 𝐾 = 1 and Inequality (1) for 𝐾 > 1) and an asymptotic bound for classical majority voting (Inequality (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' As mentioned above, none of our experimental results rely on random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Thus, the curves in Figure 4 and Figure 5 have no randomness (thus has no error bar on it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' All experiments of this paper are implemented through MAT- LAB 2022b and run on a Windows 11 desktop with AMD Ryzen 9 5900X CPU and 32GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The first observation is, no matter which setting for MoV, the quantum majority voting has better accuracy than classical majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Especially, for the case that MoV = 211, 𝐾 = 1, and 𝑠 = 13, the quantum majority voting outputs the correct winner almost for certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' However, the classical algorithm only has ∼63% probability to output the correct winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It is not surprising that the accuracy of quantum and classical majority voting both increases when the increase of time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We also observed that the lower bound of 𝐾 = 69 and 𝐾 = 139 for quantum majority voting are looser than the 𝐾 = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We believe that this behavior is caused by the Chernoff bound, which is not asymptotically tight, used in Inequality (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We also note that providing an asymptotically tight tail bound for binomial distribution is too far from the main topic of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 7 CONCLUSIONS AND FUTURE WORKS In conclusion, we took the first step in using quantum computation to accelerate voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' We found that majority voting can be accel- erated quadratically using quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Our proposed quantum computation has the potential to improve the efficiency of voting in large-scale and/or high-frequency decision-making scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' A simple extension of this paper is applying quantum computation on biased majority voting, where the threshold of winning is not half-by-half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' It would also be interesting to apply quantum computation to accelerate other widely used voting rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' For example, Borda, STV, Copeland, Ranked Pairs, or even general- ized scoring rules [16, 23], which contain most of the widely-used voting rules in real-world elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' A APPENDIX: ADDITIONAL INFORMATION ABOUT QUANTUM MAJORITY VOTING According to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='4) in Nielsen and Chuang [18], Hadamard gate changes 𝑡 qubits of |0⟩ to an equal superposition state (equal proba- bility of observing any outcome under quantum measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' |𝜓⟩ = 1 2𝑡/2 · 2𝑡 −1 ∑︁ 𝑥=0 |𝑥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Letting 𝑓 : {0, · · · , 2𝑡 − 1} ↦→ {0, 1} be the binary function to construct the quantum oracle, the orthogonal bases |𝛼⟩ and |𝛽⟩ are defined as, |𝛼⟩ ≜ 1 √︁ 2𝑡 − 𝑛1 ∑︁ 𝑥:𝑓 (𝑥)=0 |𝑥⟩ and |𝛽⟩ ≜ 1 √︁ ◦𝑛1 ∑︁ 𝑥:𝑓 (𝑥)=1 |𝑥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 13 15 17 19 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9 1 Pr[correct] K = 1 Quantum Quantum Lower Classical Classical Asymptotic 19 21 23 25 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='98 1 Pr[correct] K = 69 Quantum Quantum Lower Classical Classical Asymptotic 20 22 24 26 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='995 1 Pr[correct] K = 139 Quantum Quantum Lower Classical Classical Asymptotic 13 15 17 19 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9 1 Pr[correct] K = 1 Quantum Quantum Lower Classical Classical Asymptotic 19 21 23 25 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9995 1 Pr[correct] K = 69 Quantum Quantum Lower Classical Classical Asymptotic 20 22 24 26 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='99998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='999985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='99999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='999995 1 Pr[correct] K = 139 Quantum Quantum Lower Classical Classical Asymptotic Figure 4: Compare quantum majority voting (red squares) with classic majority voting (blue circles) when MoV = 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In both curves, we set 𝑠 = 13, 14, 15, 16, 17, 18, 19 for the seven points from left to the right respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The red hashed curve “Quantum Lower” represents our lower bound for quantum majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The blue dashed curve “Classical Asymptotic” represents our asymptotic bound for classic majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The horizontal axis can be seen as the logarithm of the algorithms’ time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 13 15 17 19 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9 1 Pr[correct] K = 1 Quantum Quantum Lower Classical Classical Asymptotic 19 21 23 25 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='98 1 Pr[correct] K = 69 Quantum Quantum Lower Classical Classical Asymptotic 20 22 24 26 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='995 1 Pr[correct] K = 139 Quantum Quantum Lower Classical Classical Asymptotic 13 15 17 19 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9 1 Pr[correct] K = 1 Quantum Quantum Lower Classical Classical Asymptotic 19 21 23 25 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='9995 1 Pr[correct] K = 69 Quantum Quantum Lower Classical Classical Asymptotic 20 22 24 26 log2 (K 2 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='99998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='999985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='99999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='999995 1 Pr[correct] K = 139 Quantum Quantum Lower Classical Classical Asymptotic Figure 5: Compare quantum majority voting (red squares) with classic majority voting (blue circles) when MoV = 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In both curves, we set 𝑠 = 13, 14, 15, 16, 17, 18, 19 for the seven points from left to the right respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The red hashed curve “Quantum Lower” represents our lower bound for quantum majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The blue dashed curve “Classical Asymptotic” represents our asymptotic bound for classic majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' The horizontal axis can be seen as the logarithm of the algorithms’ time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Under the |𝛼⟩ |𝛽⟩ base, the equal superposition state |𝜓⟩ = √︂ 2𝑡 − 𝑛1 2𝑡 |𝛼⟩ + √︂ ◦𝑛1 2𝑡 |𝛽⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Since 𝜃 = 2 arcsin �√︁ ◦𝑛1 · 2−𝑡 � , we have |𝜓⟩ = cos �𝜃 2 � |𝛼⟩ + sin �𝜃 2 � |𝛽⟩, 𝑂𝑓 |𝜓⟩ = cos �𝜃 2 � |𝛼⟩ + sin � −𝜃 2 � |𝛽⟩, and 𝐺|𝜓⟩ = cos � 3𝜃 2 � |𝛼⟩ + sin � 3𝜃 2 � |𝛽⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' One can see that the above states match the geometric illustration in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' REFERENCES [1] Ben Abramowitz, Edith Elkind, Davide Grossi, Ehud Shapiro, and Nimrod Talmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Democratic Forking: Choosing Sides with Social Choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proceedings of ADT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [2] Akshay Ajagekar and Fengqi You.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum computing assisted deep learn- ing for fault detection and diagnosis in industrial process systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Computers & Chemical Engineering 143 (2020), 107119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='compchemeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='107119 [3] Akshay Ajagekar and Fengqi You.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum computing based hybrid deep learning for fault diagnosis in electrical power systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Applied Energy 303 (2021), 117628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='apenergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='117628 [4] Marcello Benedetti, John Realpe-Gómez, Rupak Biswas, and Alejandro Perdomo- Ortiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' A 94 (Aug 2016), 022308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Issue 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 022308 [5] André Berthiaume and Gilles Brassard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Oracle quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Journal of modern optics 41, 12 (1994), 2521–2535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [6] John D Cook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Upper and lower bounds for the normal distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [7] Cynthia Dwork, Ravi Kumar, Moni Naor, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Sivakumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Rank aggrega- tion methods for the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proceedings of the 10th World Wide Web Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 613–622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [8] Mark Fey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' May’s theorem with an infinite population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Social Choice and Welfare 23, 2 (2004), 275–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [9] Davide Grossi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Social Choice Around the Block: On the Computational Social Choice of Blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proceedings of AAMAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [10] Lov K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Grover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' A Fast Quantum Mechanical Algorithm for Database Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (Philadelphia, Pennsylvania, USA) (STOC ’96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 212–219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='1145/ 237814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='237866 [11] Tad Hogg and Dmitriy Portnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Information Sciences 128, 3-4 (2000), 181–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [12] Elham Kashefi, Adrian Kent, Vlatko Vedral, and Konrad Banaszek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Com- parison of quantum oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Physical Review A 65, 5 (2002), 050304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [13] Alastair Kay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Tutorial on the quantikz package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [14] Emil T Khabiboulline, Juspreet Singh Sandhu, Marco Ugo Gambetta, Mikhail D Lukin, and Johannes Borregaard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Efficient Quantum Voting with Information-Theoretic Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [15] Shiguo Lian and Yan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Handbook of research on secure multimedia distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [16] Ao LIU, Yun Lu, Lirong Xia, and Vassilis Zikas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' How Private Are Commonly- Used Voting Rules?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content='. In Conference on Uncertainty in Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' PMLR, 629–638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [17] Andrew Mao, Ariel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Procaccia, and Yiling Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Better Human Compu- tation Through Principled Voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proceedings of the National Conference on Artificial Intelligence (AAAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Bellevue, WA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [18] Michael A Nielsen and Isaac Chuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum computation and quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [19] Independent Voter Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' New Hampshire Voter Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [20] Joan Alfina Vaccaro, Joseph Spring, and Anthony Chefles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum pro- tocols for anonymous voting and surveying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Physical Review A 75, 1 (2007), 012333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [21] Wim Van Dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Quantum oracle interrogation: Getting all information for almost half the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 98CB36280).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' IEEE, 362–367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [22] Lirong Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Computing the margin of victory for various voting rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' In Proceedings of the 13th ACM conference on electronic commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 982–999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [23] Lirong Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Generalized scoring rules: a framework that reconciles Borda and Condorcet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' ACM SIGecom Exchanges 12, 1 (2013), 42–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [24] Lirong Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Learning and Decision-Making from Rank Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Morgan & Claypool Publishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [25] Lirong Xia and Weiqiang Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Beyond the Worst Case: Semi-Random Complexity Analysis of Winner Determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' [26] Peng Xue and Xin Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' A simple quantum voting scheme with multi- qubit entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} +page_content=' Scientific reports 7, 1 (2017), 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfNAMs/content/2301.02995v1.pdf'} diff --git a/jdFJT4oBgHgl3EQfXixy/content/tmp_files/2301.11522v1.pdf.txt b/jdFJT4oBgHgl3EQfXixy/content/tmp_files/2301.11522v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f7ae3f92741603d361d0ed516181ed8dbd4ef08 --- /dev/null +++ b/jdFJT4oBgHgl3EQfXixy/content/tmp_files/2301.11522v1.pdf.txt @@ -0,0 +1,673 @@ +A Comparison of Tiny-nerf versus Spatial +Representations for 3d Reconstruction +Saulo Abraham Gante[0000−0001−6012−4003], Juan Irving +Vasquez[0000−0001−8427−9333], Marco Antonio Valencia[0000−0003−3990−0463], and +Mauricio Olgu´ın Carbajal[0000−0002−2296−8536] +Instituto Polit´ecnico Nacional (IPN), Centro de Innovaci´on y Desarrollo Tecnol´ogico +en C´omputo (CIDETEC), Ciudad de M´exico, M´exico. sganted1500@ipn.mx +Abstract. Neural rendering has emerged as a powerful paradigm for +synthesizing images, offering many benefits over classical rendering by +using neural networks to reconstruct surfaces, represent shapes, and syn- +thesize novel views, either for objects or scenes. In this neural rendering, +the environment is encoded into a neural network. We believe that these +new representations can be used to codify the scene for a mobile robot. +Therefore, in this work, we perform a comparison between a trending +neural rendering, called tiny-NeRF, and other volume representations +that are commonly used as maps in robotics, such as voxel maps, point +clouds, and triangular meshes. The target is to know the advantages and +disadvantages of neural representations in the robotics context. The com- +parison is made in terms of spatial complexity and processing time to ob- +tain a model. Experiments show that tiny-NeRF requires three times less +memory space compared to other representations. In terms of processing +time, tiny-NeRF takes about six times more to compute the model. +Keywords: Neural Rendering · NeRF · 3D Reconstruction · Mapping. +1 +Introduction +The recent and continuous advances in neural rendering have shown numerous +applications and became a new field of study in the graphics community. Some +of these efforts are in the implicit functions which represent shapes in three +dimensions (3D) [4,15]. The tool for creating the neural representations is a +multi-layer perceptron (MLP). This MLP works as a general implicit function +approximator. On the other hand, there are plenty of methods to reconstruct +surfaces, and represent shapes and volumes in 3D space. Some examples are +meshes [3], point clouds [1], voxel maps [7], and octrees [5]. The latter offers +diverse capabilities to reconstruct or create 3D models and those have diverse +applications in robotics and artificial vision [18,10,17,8]. +In this paper, we perform a comparison between neural representations, point +clouds, meshes, and voxel maps in terms of memory space and processing time +required to obtain a model. The main objective is to show, clarify or put some +important considerations for future works related to object reconstruction. +arXiv:2301.11522v1 [cs.AI] 27 Jan 2023 + +2 +Gante et al. +The MLP employed follows the architecture proposed in tiny-NeRF by the +authors of Neural Radiance Fields (NeRF) [6]. We design a grid search-based +experimentation. For the tiny-NeRF, the independent variables are i) learning +rate ii) encoding functions and iii) seed; the resulting grid search has 36 experi- +mental units. In addition, the experiment employs the same capturing positions +to create the neural model, point cloud, and voxel map. The experiment shows +that neural representations require 3 times less memory to store a model but on +the other hand they takes about 6 times more to compute a model concerning +the other representations. +The rest of the paper is structured as follows. Section 2 introduces the re- +quired concepts. Section 3 presents the related work and advances of volume, +surface and neural representations. In Section 4, we present the methodology +used to perform the experiments carry out in Section 5, where results are also +reported. Finally, the conclusions and the future work are given in Section 6. +2 +Preeliminaries +In this section, we define certain concepts required to understand the topics +tackled in this paper. +A commonly used data representation is the point cloud, that is a represen- +tation of body shapes and is made by points mapped in the 3D space which are +usually produced by sensors or scanners. The point cloud could be processed +in order to create more accurate representations. Meshes are representations of +shapes formed by a set of nodes and connections between them, one of the ad- +vantages is that it has a range of different resolutions which means that it could +be as accurate as wanted but the more resolution those have the more computa- +tion it needs to complete the representation. Another 3D shape representation +is the voxel, which is a cube of unitary distance, and the union of a set creates +a voxel map representation. +The voxel map represents shapes of objects or it is possible to represent +the volume/solid by using a voxel carving technique. Those representations are +commonly used to reconstruct shapes, objects and maps. +According to [16] there is no definition for neural rendering and suggests a +definition for Neural Rendering as: “Deep image or video generation approaches +that enable explicit or implicit control of scene properties such as illumination, +camera parameters, pose, geometry, appearance, and semantic structure.” +A recent proposed neural rendering technique is NeRF [6], it became one of +the most popular and extensively used to render objects in 3D space due to its +capabilities to create novel views in the reconstructed scene. +NeRF [6] is an approach for creating novel view synthesis, it uses a set of +input views to optimize a continuous volumetric scene function, as a result, this +optimization produces a novel view of a complex scene. Its input is a 5D vector +function, which contains the 3D space location(x,y,z) and 2D viewing direction +(θ, φ) and the output is an emitted color: Red, Green, Blue (r,g,b) and volume +density (σ). NeRF uses the concept of encoding functions where the purpose of + +Title Suppressed Due to Excessive Length +3 +these functions is to map the input into a higher dimensional space where the +MLP can more easily approximate higher frequency functions. +To generate a NeRF from a specific viewpoint, first, a set of rays are marched +through the scene, the data generated is fed into the neural network and produce +a set of RGBσ values then the data is structured into a 2D image. +3 +Related Work +The 3D reconstruction of spaces and objects is not a new research topic and has +many approaches which reconstruct scenes employing different techniques, the +accuracy of the volumetric representations relies on the resolution employed to +map, and the more resolution is wanted the more computation is needed which +means more time is needed to achieve a good result. +Volume representations in 3D space have many methods to represent syn- +thetically objects, like meshes [3], point clouds [1], voxel maps [7], and octree +[5]. Those offer diverse capabilities to reconstruct or create 3D models and those +have diverse applications in robotics and artificial vision [18,10,17,8]. Despite +the popularity that they have, resolution of the representations is one of it cons. +Also, the memory consumption between them is variable and usually requires +memory in the order of Megabytes (MB). +Neural rendering has gained popularity since it employs a multi-layer per- +ceptron (MLP) to achieve these tasks [11,12]. In [15,4] are presented different +techniques to represent shapes and volumes in 3D space. The proposed meth- +ods concentrate its effort in creating those representations and compare it with +state-of-the-art. On the other hand, in [14] they propose an approach for volume +compression and compare it with voxel maps. A Simultaneous Localization And +Mapping system is proposed by [13], they compare it with truncated signed dis- +tance function (TSDF) method, both approaches do a comparison in terms of +memory consumption, and stand out a good performance. +We believe that time taken in the process of the reconstructions is an impor- +tant variable to take in consideration, and that the consulted approaches do not +report those differences in terms of time. +4 +Methodology +We want to experiment with neural representations, exploring the advantages +that those have over existent representations used for 3D space reconstruction. +We do a comparison between neural representations proposed by the authors of +NeRF [6], and three different spatial representations used to model objects, such +as meshes, voxel maps, and point clouds. We use a dataset that contains 106 (see +Fig. 3) pairs of sensor poses, and using those poses, we extract the required data +in order to create the proposed representations (Figure 1). The main reason to +use one dataset is that we want to give the algorithms the same point of view +for a fair comparison in terms of data that could be extracted given the poses +in the data set. + +4 +Gante et al. +Registration +Voxel carving +Tiny-NeRF +training +Dataset +Tiny-NeRF +neural +representation +Point Cloud +Voxel map +Fig. 1: General diagram of the experiments. Given a dataset which contains po- +sitions in 3D space and images, the data required is extracted with a simulator, +as a result the three representations to be compared are obtained. +4.1 +Dataset +Given that we propose a comparison with synthetic data, we use a simulator +to render images of an object (simulating a camera inside the simulated world). +See Figure 2. Then, a world is needed to set up, configure it with a ground, and +place the mesh of the object of interest in it. From the synthetic datasets used +in NeRF [6], we apply transformation matrices as in Equation (1) +Fig. 2: World setup. An object is placed in the simulated world. +T = +�R p +0 1 +� +, +(1) + +Title Suppressed Due to Excessive Length +5 +where R indicates the rotation matrix, whose values represent the rotations over +the three axes, and the p indicates the position vector, whose values contain the +position of a body in a 3D space (x, y, z). Please see Figure 3. +Fig. 3: Capturing poses. In this figure it is shown the 106 poses used, the frame +of reference is the described by OpenGL [19] for synthetic cameras. +Having those positions in 3D space the required datasets are extracted, that +is RGB images, ray casting points, and depth data. All in order to create the +proposed reconstructions. +4.2 +Point cloud and voxel map +Open3D library [22] allows us to visualize objects and create representations. +For the point cloud, it is created by the use of ray-tracing which emits synthetic +rays in simulation when those touches or intersect with a surface return a value, +having this is possible to calculate in R3 and map them into points in space, +creating a point cloud. This process is repeated every capture, then the resulting +points are concatenated and filtered to reduce possible noise created by captures. +We create a voxel model using the technique of voxel carving, using a pinhole +camera and homogeneous transformation matrix is possible to create a voxel +dense given the resulting images and a silhouette to employ a carve silhouette +method provided by Open3D, resulting in a voxel model. +4.3 +Tiny-NeRF +As explained above, NeRF [6] receives as input a set of data that express location +and viewing direction where the output is an emitted color and a volume density. +Tiny-NeRF is a simplified version of NeRF, which is an MLP conformed by 6 + +6 +Gante et al. +fully-connected ReLU layers each with 256 filter size, one fully-connected ReLU +layer with a filter size of 64 then an output layer that expresses the emitted +RGBσ at a certain position with a four filter size layer. The process starts by +getting rays according to the pose, then the returned rays become useful to +map 3D points which are going to be fed into Tiny-NeRF input, the output of +the model is used to compute opacities and RGB data, finally the weights are +calculated and the process is repeated. +5 +Experiments +We evaluate the Tiny-NeRF describing a grid search where certain variables are +changed over the experiments. Using the same position captures we perform +reconstruction with voxels and a point cloud. All the data was synthetic and +obtained using Open3D. +Our experiments run in Python and the libraries employed are Pytorch [9] for +the MLP or neural representations, Pybullet [2] and Open3D [22].The hardware +employed for those experiments is the CPU/GPU provided by Colab which al- +lows us to use a Graphic card: Tesla P100-PCIE-16GB with 16GB of GPU-RAM, +25.46 GB of RAM, and 166.83 GB in Hard disc drive. +5.1 +Tiny-NeRF training +Tiny-NeRF is a simplified version of NeRF, which is an MLP conformed by six +fully-connected ReLU layers each with a 256 filter size, one fully-connected layer +with a filter size of 64 then an output that expresses the RGBσ values. The grid +search proposed to vary over three variables and the values are: +– Seed: 2057, 5680 and 7461. +– Learning rate: 5x10−3, 5x10−3 and 5x10−3 +– Encoding functions : 6, 9, 10 and 12. +For the Neural Networks (NN) training a commonly used metric is Loss +since it evaluates how bad predicts on an example, the Peak Signal-to-Noise +Ratio (PSNR) is used to measure the ratio between a signal and the noise which +affects the representation of this signal; in this case, the PSNR is used to measure +how well the Tiny-NeRF does a representation compared to the original images. +On the other hand, to measure time the unit employed is seconds (s) and to +measure space in memory we utilize MB. +To obtain the data set we employed Pybullet [2] simulator which let us set +simulated worlds, set objects in it (Figure 2) and create pinhole cameras to +extract or create synthetic images, among other things. Once the object is set +in the world, it is possible to create a synthetic camera given its position, target +position, field of view (FOV), near and far plane distance, weight and height of +the image. The positions are given by the data capturing positions, the FOV is +17.70◦, the weight and height are equal to 100. Resulting in images like the ones +in Figure 4. + +Title Suppressed Due to Excessive Length +7 +(a) Frontal point of view. +(b) Lateral point of view. +Fig. 4: Images of the object of study extracted using Pybullet[2] +The experiments with the Tiny-NeRF are iterated for five thousand epochs +each and the variables are modified at each experiment, the number of exper- +iments is 36. The results of this experiment are shown in Table 1. The entire +experiment took about 65,100 seconds which means that approximately every +experiment took 1,808.33 seconds to be completed. To summarize the informa- +tion in Table 1, the average was calculated (Table 2) in order to easily extract +which parameters perform better results with the MLP. +Analysis of experiments showed that nine coding functions help the Tiny- +NeRF to accurately (Figure 5) create a neural representation of the object, +and the learning rate helped to achieve good performance in fewer epochs. Ad- +ditionally, the neural representations took 1,808.33 seconds to complete an +experiment, and the memory space to store a representation is 1.5 MB. +Additionally to PSNR, we perform evaluations over two more metrics SSIM +and LPIPS [20,21] which are commonly used to measure distances over images, +looking for a measure of how well the Tiny-NeRF is rendering views. Comparing +images like the ones in Fig. 6 the metrics proposed gave as a result 0.8481 and +0.0565, respectively. Those results affirm that the representations are good in +quality but it could improved. +5.2 +Comparison of Tiny-NeRF versus spatial representations +Once Tiny-NeRF has been trained and the representations were created, we +compared the time taken to do a representation. To measure the time, it was +printed every time a process started and finished the difference between those +shows the time taken. The space in memory is measured by the file space in +memory that is required to store the representations. + +8 +Gante et al. +ID Factor 1 Factor 2 +Factor 3 +Metric 1 +Metric 2 +Seed +Learning Rate Coding +func- +tions +Loss +PSNR (dB) +1 +2057 +5 × 10−3 +6 +0.5463 +2.6257 +2 +2057 +5 × 10−3 +9 +0.0030 +25.2288 +3 +2057 +5 × 10−3 +10 +0.0493 +13.0715 +4 +2057 +5 × 10−3 +12 +0.5463 +2.6257 +5 +2057 +5 × 10−4 +6 +0.5463 +2.6257 +6 +2057 +5 × 10−4 +9 +0.0025 +26.0206 +7 +2057 +5 × 10−4 +10 +0.5463 +2.6257 +8 +2057 +5 × 10−4 +12 +0.5463 +2.6257 +9 +2057 +5 × 10−5 +6 +0.5463 +2.6257 +10 2057 +5 × 10−5 +9 +0.0026 +25.8503 +11 2057 +5 × 10−5 +10 +0.5463 +2.6257 +12 2057 +5 × 10−5 +12 +0.5463 +2.6257 +13 7461 +5 × 10−3 +6 +0.0921 +10.3574 +14 7461 +5 × 10−3 +9 +0.0032 +24.9485 +15 7461 +5 × 10−3 +10 +0.5463 +2.6257 +16 7461 +5 × 10−3 +12 +0.5463 +2.6257 +17 7461 +5 × 10−4 +6 +0.5463 +2.6257 +18 7461 +5 × 10−4 +9 +0.0026 +25.8503 +19 7461 +5 × 10−4 +10 +0.5463 +2.6257 +20 7461 +5 × 10−4 +12 +0.5463 +2.6257 +21 7461 +5 × 10−5 +6 +0.5463 +2.6257 +22 7461 +5 × 10−5 +9 +0.0025 +26.0206 +23 7461 +5 × 10−5 +10 +0.5463 +2.6257 +24 7461 +5 × 10−5 +12 +0.5463 +2.6257 +25 5680 +5 × 10−3 +6 +0.5463 +2.6257 +26 5680 +5 × 10−3 +9 +0.0032 +24.9485 +27 5680 +5 × 10−3 +10 +0.5463 +2.6257 +28 5680 +5 × 10−3 +12 +0.0033 +24.8149 +29 5680 +5 × 10−4 +6 +0.5463 +2.6257 +30 5680 +5 × 10−4 +9 +0.0027 +25.6864 +31 5680 +5 × 10−4 +10 +0.5463 +2.6257 +32 5680 +5 × 10−4 +12 +0.0024 +26.1979 +33 5680 +5 × 10−5 +6 +0.5463 +2.6257 +34 5680 +5 × 10−5 +9 +0.0027 +25.6864 +35 5680 +5 × 10−5 +10 +0.5463 +2.6257 +36 5680 +5 × 10−5 +12 +0.0027 +25.6864 +Table 1: Grid search. ID express an identification number, the variable values +employed for each experiment with Tiny-NeRF and the results expressed in +terms of Loss and PSNR. +The data employed to reconstruct was obtained by capturing in the positions +of the data set, mentioned above, once the captures are done the process of data +was done employing Open3D [22]. + +Title Suppressed Due to Excessive Length +9 +LR +Functions +Loss +PSNR (dB) +5 × 10−3 +6 +0.3949 +4.0351 +5 × 10−3 +9 +0.0031 +25.0863 +5 × 10−3 +10 +0.3806 +4.1953 +5 × 10−3 +12 +0.3653 +4.3735 +5 × 10−4 +6 +0.5463 +2.6256 +5 × 10−4 +9 +0.0026 +25.8502 +5 × 10−4 +10 +0.5463 +2.6256 +5 × 10−4 +12 +0.365 +4.3770 +5 × 10−5 +6 +0.5463 +2.6256 +5 × 10−5 +9 +0.0026 +25.8502 +5 × 10−5 +10 +0.5463 +2.6256 +5 × 10−5 +12 +0.3651 +4.3758 +Table 2: Average of the results in grid search. +(a) Frontal point of view. +(b) Lateral point of view. +Fig. 5: Neural representations created with the Tiny-NeRF. +The point cloud (Figure 7 (a)) was obtained by mapping the points resultant +of a ray-tracing operation into XYZ or 3D space, those points are concatenated +and finally filtrated to avoid noise in the reconstruction. The experiment took +about 2 seconds and the memory space needed is 12 MB. +The resultant voxel map (Figure 7 (b)) was created with the voxel carving +method which not only reconstructs the surface of an object, it creates a voxel +map that is a cube of certain dimensions and according to the visualized data, +the algorithm carves the shape into it, creating a solid voxel representation. The +experiment took about 166 seconds and the memory space needed is 21.2 MB. +The performed experiments results showed that implicit or neural represen- +tation requires at least 3 times less memory compared with other representations +(Table 3). In terms of time to process a representation, point clouds and voxel + +P10 +Gante et al. +(a) Simulation captures. +(b) Neural representations. +Fig. 6: Qualitative results. We extracted different posses and visualizations using +Tiny-NeRF. +(a) Point cloud reconstruction. +(b) Voxel reconstruction. +Fig. 7: Volumetric reconstructions. +maps build the representations in about 6 times less time than the implicit rep- +resentations (Table 4). +6 +Conclusions and Future work +This paper tackles the trend research topic, neural rendering, which has many +advances in graphics generation. We compare a simplified version of NeRF with +different volume representations commonly used in robotics and vision recon- +struction tasks, all compared in terms of memory space and time to build rep- +resentations. First, we experimented with Tiny-NeRF that computes the colors +over a certain position with a viewing direction; the experiments were conducted +by a grid search looking to perform good representations of an object. In addi- + +Title Suppressed Due to Excessive Length +11 +Representation +Size (MB) +Meshes +4.5 +Point cloud +12.0 +Voxelization +21.2 +Implicit representation +1.5 +Table 3: Comparative of memory size. +Representation +Time (s) +Meshes +28800 +Point cloud +2 +Voxelization +166 +Implicit representation +1008 +Table 4: Comparative of time taken to perform a representation. +tion, we perform a reconstruction using voxels and point clouds. The compari- +son, in terms of memory space and time, shows that the Tiny-NeRF architecture +(MLP) requires less memory but takes more time to build a representation. On +the other hand, this experiment showed that the neural representation relies on +the fine-tuning of the variables implied in the training of the MLP. We believe +that the results of the experiments can offer some relevant information or con- +siderations to take when a reconstruction task is needed. In a future work we +will experiment with more objects and with a mobile manipulator robot. +References +1. Berger, M., Tagliasacchi, A., Seversky, L., Alliez, P., Levine, J., Sharf, A., Silva, +C.: State of the Art in Surface Reconstruction from Point Clouds. In: Eurographics +2014 - State of the Art Reports (2014) +2. Coumans, E., Bai, Y.: Pybullet, a python module for physics simulation for games, +robotics and machine learning. http://pybullet.org (2016–2021 note = Accessed: +2022-05-26) +3. Delingette, H.: General object reconstruction based on simplex meshes. Interna- +tional journal of computer vision (1999) +4. Genova, K., Cole, F., Sud, A., Sarna, A., Funkhouser, T.: Local deep implicit +functions for 3d shape. Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition (CVPR) (June 2020) +5. Hornung, A., Wurm, K.M., Bennewitz, M., Stachniss, C., Burgard, W.: OctoMap: +An efficient probabilistic 3D mapping framework based on octrees. Autonomous +Robots (2013) +6. Mildenhall, B., Srinivasan, P.P., Tancik, M., Barron, J.T., Ramamoorthi, R., Ng, +R.: Nerf: Representing scenes as neural radiance fields for view synthesis. In: ECCV +(2020) +7. Muglikar, M., Zhang, Z., Scaramuzza, D.: Voxel map for visual slam. In: 2020 +IEEE International Conference on Robotics and Automation (ICRA). pp. 4181– +4187. IEEE (2020) + +12 +Gante et al. +8. Mur-Artal, R., Tard´os, J.D.: Orb-slam2: An open-source slam system for monoc- +ular, stereo, and rgb-d cameras. IEEE transactions on robotics 33(5), 1255–1262 +(2017) +9. Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., +Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., +Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S.: +Pytorch: An imperative style, high-performance deep learning library. In: Wallach, +H., Larochelle, H., Beygelzimer, A., d'Alch´e-Buc, F., Fox, E., Garnett, R. (eds.) +Advances in Neural Information Processing Systems 32, pp. 8024–8035. Curran +Associates, Inc. (2019) +10. Respall, V.M., Devitt, D., Fedorenko, R., Klimchik, A.: Fast sampling-based next- +best-view exploration algorithm for a mav. In: 2021 IEEE International Conference +on Robotics and Automation (ICRA). pp. 89–95. IEEE (2021) +11. Saito, S., , Huang, Z., Natsume, R., Morishima, S., Kanazawa, A., Li, H.: Pifu: +Pixel-aligned implicit function for high-resolution clothed human digitization. +arXiv preprint arXiv:1905.05172 (2019) +12. Sitzmann, V., Martel, J.N.P., Bergman, A.W., Lindell, D.B., Wetzstein, G.: Im- +plicit neural representations with periodic activation functions (2020) +13. Sucar, E., Liu, S., Ortiz, J., Davison, A.: iMAP: Implicit mapping and positioning +in real-time. In: Proceedings of the International Conference on Computer Vision +(ICCV) (2021) +14. Tang, D., Singh, S., Chou, P.A., H¨ane, C., Dou, M., Fanello, S.R., Taylor, J., +Davidson, P.L., Guleryuz, O.G., Zhang, Y., Izadi, S., Tagliasacchi, A., Bouaziz, S., +Keskin, C.: Deep implicit volume compression. CoRR abs/2005.08877 (2020) +15. Tang, J., Lei, J., Xu, D., Ma, F., Jia, K., Zhang, L.: Sign-agnostic conet: Learn- +ing implicit surface reconstructions by sign-agnostic optimization of convolutional +occupancy networks. CoRR (2021) +16. Tewari, A., Fried, O., Thies, J., Sitzmann, V., Lombardi, S., Sunkavalli, K., Martin- +Brualla, R., Simon, T., Saragih, J.M., Nießner, M., Pandey, R., Fanello, S.R., +Wetzstein, G., Zhu, J.Y., Theobalt, C., Agrawala, M., Shechtman, E., Goldman, +D.B., Zollh¨ofer, M.: State of the art on neural rendering. Comput. Graph. Forum +39, 701–727 (2020) +17. Uyanik, C., Secil, S., Ozkan, M., Dutagaci, H., Turgut, K., Parlaktuna, O.: Spgs: A +new method for autonomous 3d reconstruction of unknown objects by an industrial +robot. In: Annual Conference Towards Autonomous Robotic Systems. pp. 15–27. +Springer (2018) +18. Vasquez-Gomez, J.I., Sucar, L.E., Murrieta-Cid, R.: View planning for 3d object +reconstruction with a mobile manipulator robot. In: 2014 IEEE/RSJ International +Conference on Intelligent Robots and Systems (2014) +19. de Vries, J.: Learn opengl: Camera. https://learnopengl.com/Getting-started/ +Camera, accessed: 2022-05-26 +20. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: +from error visibility to structural similarity. IEEE transactions on image processing +13(4), 600–612 (2004) +21. Zhang, R., Isola, P., Efros, A.A., Shechtman, E., Wang, O.: The unreasonable +effectiveness of deep features as a perceptual metric. In: CVPR (2018) +22. Zhou, Q.Y., Park, J., Koltun, V.: Open3D: A modern library for 3D data process- +ing. arXiv:1801.09847 (2018) + diff --git a/jdFJT4oBgHgl3EQfXixy/content/tmp_files/load_file.txt b/jdFJT4oBgHgl3EQfXixy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b2443e0afb9a1dcb70de6da35d0d7a07968ba7f --- /dev/null +++ b/jdFJT4oBgHgl3EQfXixy/content/tmp_files/load_file.txt @@ -0,0 +1,504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf,len=503 +page_content='A Comparison of Tiny-nerf versus Spatial Representations for 3d Reconstruction Saulo Abraham Gante[0000−0001−6012−4003], Juan Irving Vasquez[0000−0001−8427−9333], Marco Antonio Valencia[0000−0003−3990−0463], and Mauricio Olgu´ın Carbajal[0000−0002−2296−8536] Instituto Polit´ecnico Nacional (IPN), Centro de Innovaci´on y Desarrollo Tecnol´ogico en C´omputo (CIDETEC), Ciudad de M´exico, M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' sganted1500@ipn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='mx Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Neural rendering has emerged as a powerful paradigm for synthesizing images, offering many benefits over classical rendering by using neural networks to reconstruct surfaces, represent shapes, and syn- thesize novel views, either for objects or scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In this neural rendering, the environment is encoded into a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We believe that these new representations can be used to codify the scene for a mobile robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Therefore, in this work, we perform a comparison between a trending neural rendering, called tiny-NeRF, and other volume representations that are commonly used as maps in robotics, such as voxel maps, point clouds, and triangular meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The target is to know the advantages and disadvantages of neural representations in the robotics context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The com- parison is made in terms of spatial complexity and processing time to ob- tain a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Experiments show that tiny-NeRF requires three times less memory space compared to other representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In terms of processing time, tiny-NeRF takes about six times more to compute the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Keywords: Neural Rendering · NeRF · 3D Reconstruction · Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 1 Introduction The recent and continuous advances in neural rendering have shown numerous applications and became a new field of study in the graphics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Some of these efforts are in the implicit functions which represent shapes in three dimensions (3D) [4,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The tool for creating the neural representations is a multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' This MLP works as a general implicit function approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' On the other hand, there are plenty of methods to reconstruct surfaces, and represent shapes and volumes in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Some examples are meshes [3], point clouds [1], voxel maps [7], and octrees [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The latter offers diverse capabilities to reconstruct or create 3D models and those have diverse applications in robotics and artificial vision [18,10,17,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In this paper, we perform a comparison between neural representations, point clouds, meshes, and voxel maps in terms of memory space and processing time required to obtain a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The main objective is to show, clarify or put some important considerations for future works related to object reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='11522v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='AI] 27 Jan 2023 2 Gante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The MLP employed follows the architecture proposed in tiny-NeRF by the authors of Neural Radiance Fields (NeRF) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We design a grid search-based experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' For the tiny-NeRF, the independent variables are i) learning rate ii) encoding functions and iii) seed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' the resulting grid search has 36 experi- mental units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In addition, the experiment employs the same capturing positions to create the neural model, point cloud, and voxel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The experiment shows that neural representations require 3 times less memory to store a model but on the other hand they takes about 6 times more to compute a model concerning the other representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Section 2 introduces the re- quired concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Section 3 presents the related work and advances of volume, surface and neural representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In Section 4, we present the methodology used to perform the experiments carry out in Section 5, where results are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Finally, the conclusions and the future work are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 2 Preeliminaries In this section, we define certain concepts required to understand the topics tackled in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' A commonly used data representation is the point cloud, that is a represen- tation of body shapes and is made by points mapped in the 3D space which are usually produced by sensors or scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The point cloud could be processed in order to create more accurate representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Meshes are representations of shapes formed by a set of nodes and connections between them, one of the ad- vantages is that it has a range of different resolutions which means that it could be as accurate as wanted but the more resolution those have the more computa- tion it needs to complete the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Another 3D shape representation is the voxel, which is a cube of unitary distance, and the union of a set creates a voxel map representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The voxel map represents shapes of objects or it is possible to represent the volume/solid by using a voxel carving technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Those representations are commonly used to reconstruct shapes, objects and maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' According to [16] there is no definition for neural rendering and suggests a definition for Neural Rendering as: “Deep image or video generation approaches that enable explicit or implicit control of scene properties such as illumination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' camera parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' pose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' geometry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' appearance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' and semantic structure.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' A recent proposed neural rendering technique is NeRF [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' it became one of the most popular and extensively used to render objects in 3D space due to its capabilities to create novel views in the reconstructed scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' NeRF [6] is an approach for creating novel view synthesis, it uses a set of input views to optimize a continuous volumetric scene function, as a result, this optimization produces a novel view of a complex scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Its input is a 5D vector function, which contains the 3D space location(x,y,z) and 2D viewing direction (θ, φ) and the output is an emitted color: Red, Green, Blue (r,g,b) and volume density (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' NeRF uses the concept of encoding functions where the purpose of Title Suppressed Due to Excessive Length 3 these functions is to map the input into a higher dimensional space where the MLP can more easily approximate higher frequency functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' To generate a NeRF from a specific viewpoint, first, a set of rays are marched through the scene, the data generated is fed into the neural network and produce a set of RGBσ values then the data is structured into a 2D image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 3 Related Work The 3D reconstruction of spaces and objects is not a new research topic and has many approaches which reconstruct scenes employing different techniques, the accuracy of the volumetric representations relies on the resolution employed to map, and the more resolution is wanted the more computation is needed which means more time is needed to achieve a good result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Volume representations in 3D space have many methods to represent syn- thetically objects, like meshes [3], point clouds [1], voxel maps [7], and octree [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Those offer diverse capabilities to reconstruct or create 3D models and those have diverse applications in robotics and artificial vision [18,10,17,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Despite the popularity that they have, resolution of the representations is one of it cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Also, the memory consumption between them is variable and usually requires memory in the order of Megabytes (MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Neural rendering has gained popularity since it employs a multi-layer per- ceptron (MLP) to achieve these tasks [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In [15,4] are presented different techniques to represent shapes and volumes in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The proposed meth- ods concentrate its effort in creating those representations and compare it with state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' On the other hand, in [14] they propose an approach for volume compression and compare it with voxel maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' A Simultaneous Localization And Mapping system is proposed by [13], they compare it with truncated signed dis- tance function (TSDF) method, both approaches do a comparison in terms of memory consumption, and stand out a good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We believe that time taken in the process of the reconstructions is an impor- tant variable to take in consideration, and that the consulted approaches do not report those differences in terms of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 4 Methodology We want to experiment with neural representations, exploring the advantages that those have over existent representations used for 3D space reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We do a comparison between neural representations proposed by the authors of NeRF [6], and three different spatial representations used to model objects, such as meshes, voxel maps, and point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We use a dataset that contains 106 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 3) pairs of sensor poses, and using those poses, we extract the required data in order to create the proposed representations (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The main reason to use one dataset is that we want to give the algorithms the same point of view for a fair comparison in terms of data that could be extracted given the poses in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 4 Gante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Registration Voxel carving Tiny-NeRF training Dataset Tiny-NeRF neural representation Point Cloud Voxel map Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 1: General diagram of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Given a dataset which contains po- sitions in 3D space and images, the data required is extracted with a simulator, as a result the three representations to be compared are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='1 Dataset Given that we propose a comparison with synthetic data, we use a simulator to render images of an object (simulating a camera inside the simulated world).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Then, a world is needed to set up, configure it with a ground, and place the mesh of the object of interest in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' From the synthetic datasets used in NeRF [6], we apply transformation matrices as in Equation (1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 2: World setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' An object is placed in the simulated world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' T = �R p 0 1 � , (1) Title Suppressed Due to Excessive Length 5 where R indicates the rotation matrix, whose values represent the rotations over the three axes, and the p indicates the position vector, whose values contain the position of a body in a 3D space (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Please see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 3: Capturing poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In this figure it is shown the 106 poses used, the frame of reference is the described by OpenGL [19] for synthetic cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Having those positions in 3D space the required datasets are extracted, that is RGB images, ray casting points, and depth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' All in order to create the proposed reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='2 Point cloud and voxel map Open3D library [22] allows us to visualize objects and create representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' For the point cloud, it is created by the use of ray-tracing which emits synthetic rays in simulation when those touches or intersect with a surface return a value, having this is possible to calculate in R3 and map them into points in space, creating a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' This process is repeated every capture, then the resulting points are concatenated and filtered to reduce possible noise created by captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We create a voxel model using the technique of voxel carving, using a pinhole camera and homogeneous transformation matrix is possible to create a voxel dense given the resulting images and a silhouette to employ a carve silhouette method provided by Open3D, resulting in a voxel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3 Tiny-NeRF As explained above, NeRF [6] receives as input a set of data that express location and viewing direction where the output is an emitted color and a volume density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Tiny-NeRF is a simplified version of NeRF, which is an MLP conformed by 6 6 Gante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' fully-connected ReLU layers each with 256 filter size, one fully-connected ReLU layer with a filter size of 64 then an output layer that expresses the emitted RGBσ at a certain position with a four filter size layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The process starts by getting rays according to the pose, then the returned rays become useful to map 3D points which are going to be fed into Tiny-NeRF input, the output of the model is used to compute opacities and RGB data, finally the weights are calculated and the process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 5 Experiments We evaluate the Tiny-NeRF describing a grid search where certain variables are changed over the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Using the same position captures we perform reconstruction with voxels and a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' All the data was synthetic and obtained using Open3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Our experiments run in Python and the libraries employed are Pytorch [9] for the MLP or neural representations, Pybullet [2] and Open3D [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='The hardware employed for those experiments is the CPU/GPU provided by Colab which al- lows us to use a Graphic card: Tesla P100-PCIE-16GB with 16GB of GPU-RAM, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='46 GB of RAM, and 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='83 GB in Hard disc drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='1 Tiny-NeRF training Tiny-NeRF is a simplified version of NeRF, which is an MLP conformed by six fully-connected ReLU layers each with a 256 filter size, one fully-connected layer with a filter size of 64 then an output that expresses the RGBσ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The grid search proposed to vary over three variables and the values are: – Seed: 2057, 5680 and 7461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' – Learning rate: 5x10−3, 5x10−3 and 5x10−3 – Encoding functions : 6, 9, 10 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' For the Neural Networks (NN) training a commonly used metric is Loss since it evaluates how bad predicts on an example, the Peak Signal-to-Noise Ratio (PSNR) is used to measure the ratio between a signal and the noise which affects the representation of this signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' in this case, the PSNR is used to measure how well the Tiny-NeRF does a representation compared to the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' On the other hand, to measure time the unit employed is seconds (s) and to measure space in memory we utilize MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' To obtain the data set we employed Pybullet [2] simulator which let us set simulated worlds, set objects in it (Figure 2) and create pinhole cameras to extract or create synthetic images, among other things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Once the object is set in the world, it is possible to create a synthetic camera given its position, target position, field of view (FOV), near and far plane distance, weight and height of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The positions are given by the data capturing positions, the FOV is 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='70◦, the weight and height are equal to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Resulting in images like the ones in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 7 (a) Frontal point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (b) Lateral point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 4: Images of the object of study extracted using Pybullet[2] The experiments with the Tiny-NeRF are iterated for five thousand epochs each and the variables are modified at each experiment, the number of exper- iments is 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The results of this experiment are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The entire experiment took about 65,100 seconds which means that approximately every experiment took 1,808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='33 seconds to be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' To summarize the informa- tion in Table 1, the average was calculated (Table 2) in order to easily extract which parameters perform better results with the MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Analysis of experiments showed that nine coding functions help the Tiny- NeRF to accurately (Figure 5) create a neural representation of the object, and the learning rate helped to achieve good performance in fewer epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Ad- ditionally, the neural representations took 1,808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='33 seconds to complete an experiment, and the memory space to store a representation is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Additionally to PSNR, we perform evaluations over two more metrics SSIM and LPIPS [20,21] which are commonly used to measure distances over images, looking for a measure of how well the Tiny-NeRF is rendering views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Comparing images like the ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 6 the metrics proposed gave as a result 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='8481 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0565, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Those results affirm that the representations are good in quality but it could improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='2 Comparison of Tiny-NeRF versus spatial representations Once Tiny-NeRF has been trained and the representations were created, we compared the time taken to do a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' To measure the time, it was printed every time a process started and finished the difference between those shows the time taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The space in memory is measured by the file space in memory that is required to store the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 8 Gante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' ID Factor 1 Factor 2 Factor 3 Metric 1 Metric 2 Seed Learning Rate Coding func- tions Loss PSNR (dB) 1 2057 5 × 10−3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 2 2057 5 × 10−3 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0030 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='2288 3 2057 5 × 10−3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0493 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0715 4 2057 5 × 10−3 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 5 2057 5 × 10−4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 6 2057 5 × 10−4 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0025 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0206 7 2057 5 × 10−4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 8 2057 5 × 10−4 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 9 2057 5 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 10 2057 5 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0026 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='8503 11 2057 5 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 12 2057 5 × 10−5 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 13 7461 5 × 10−3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0921 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3574 14 7461 5 × 10−3 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0032 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='9485 15 7461 5 × 10−3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 16 7461 5 × 10−3 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 17 7461 5 × 10−4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 18 7461 5 × 10−4 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0026 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='8503 19 7461 5 × 10−4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 20 7461 5 × 10−4 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 21 7461 5 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 22 7461 5 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0025 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0206 23 7461 5 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 24 7461 5 × 10−5 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 25 5680 5 × 10−3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 26 5680 5 × 10−3 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0032 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='9485 27 5680 5 × 10−3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 28 5680 5 × 10−3 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0033 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='8149 29 5680 5 × 10−4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 30 5680 5 × 10−4 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0027 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6864 31 5680 5 × 10−4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 32 5680 5 × 10−4 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0024 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='1979 33 5680 5 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 34 5680 5 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0027 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6864 35 5680 5 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6257 36 5680 5 × 10−5 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0027 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6864 Table 1: Grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' ID express an identification number, the variable values employed for each experiment with Tiny-NeRF and the results expressed in terms of Loss and PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The data employed to reconstruct was obtained by capturing in the positions of the data set, mentioned above, once the captures are done the process of data was done employing Open3D [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 9 LR Functions Loss PSNR (dB) 5 × 10−3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3949 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0351 5 × 10−3 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0031 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0863 5 × 10−3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3806 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='1953 5 × 10−3 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3653 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3735 5 × 10−4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6256 5 × 10−4 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0026 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='8502 5 × 10−4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6256 5 × 10−4 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='365 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3770 5 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6256 5 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0026 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='8502 5 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='6256 5 × 10−5 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3651 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='3758 Table 2: Average of the results in grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (a) Frontal point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (b) Lateral point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 5: Neural representations created with the Tiny-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The point cloud (Figure 7 (a)) was obtained by mapping the points resultant of a ray-tracing operation into XYZ or 3D space, those points are concatenated and finally filtrated to avoid noise in the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The experiment took about 2 seconds and the memory space needed is 12 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The resultant voxel map (Figure 7 (b)) was created with the voxel carving method which not only reconstructs the surface of an object, it creates a voxel map that is a cube of certain dimensions and according to the visualized data, the algorithm carves the shape into it, creating a solid voxel representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The experiment took about 166 seconds and the memory space needed is 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='2 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The performed experiments results showed that implicit or neural represen- tation requires at least 3 times less memory compared with other representations (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In terms of time to process a representation, point clouds and voxel P10 Gante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (a) Simulation captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (b) Neural representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 6: Qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We extracted different posses and visualizations using Tiny-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (a) Point cloud reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (b) Voxel reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 7: Volumetric reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' maps build the representations in about 6 times less time than the implicit rep- resentations (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 6 Conclusions and Future work This paper tackles the trend research topic, neural rendering, which has many advances in graphics generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We compare a simplified version of NeRF with different volume representations commonly used in robotics and vision recon- struction tasks, all compared in terms of memory space and time to build rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' First, we experimented with Tiny-NeRF that computes the colors over a certain position with a viewing direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' the experiments were conducted by a grid search looking to perform good representations of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In addi- Title Suppressed Due to Excessive Length 11 Representation Size (MB) Meshes 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5 Point cloud 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='0 Voxelization 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='2 Implicit representation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='5 Table 3: Comparative of memory size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Representation Time (s) Meshes 28800 Point cloud 2 Voxelization 166 Implicit representation 1008 Table 4: Comparative of time taken to perform a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' tion, we perform a reconstruction using voxels and point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' The compari- son, in terms of memory space and time, shows that the Tiny-NeRF architecture (MLP) requires less memory but takes more time to build a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' On the other hand, this experiment showed that the neural representation relies on the fine-tuning of the variables implied in the training of the MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' We believe that the results of the experiments can offer some relevant information or con- siderations to take when a reconstruction task is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In a future work we will experiment with more objects and with a mobile manipulator robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Berger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Tagliasacchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Seversky, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Alliez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Levine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Sharf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Silva, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': State of the Art in Surface Reconstruction from Point Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: Eurographics 2014 - State of the Art Reports (2014) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Coumans, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Bai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Pybullet, a python module for physics simulation for games, robotics and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' http://pybullet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='org (2016–2021 note = Accessed: 2022-05-26) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Delingette, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': General object reconstruction based on simplex meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Interna- tional journal of computer vision (1999) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Genova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Cole, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Sud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Sarna, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Funkhouser, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Local deep implicit functions for 3d shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (June 2020) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Hornung, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Wurm, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Bennewitz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Stachniss, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Burgard, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': OctoMap: An efficient probabilistic 3D mapping framework based on octrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Autonomous Robots (2013) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Mildenhall, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Srinivasan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Tancik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Barron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Ramamoorthi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Ng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Nerf: Representing scenes as neural radiance fields for view synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: ECCV (2020) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Muglikar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Scaramuzza, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Voxel map for visual slam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: 2020 IEEE International Conference on Robotics and Automation (ICRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 4181– 4187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' IEEE (2020) 12 Gante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Mur-Artal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Tard´os, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' : Orb-slam2: An open-source slam system for monoc- ular, stereo, and rgb-d cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' IEEE transactions on robotics 33(5), 1255–1262 (2017) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Gross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Massa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Lerer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Bradbury, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Chanan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Killeen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Gimelshein, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Antiga, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Desmaison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Kopf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Yang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', DeVito, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Raison, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Tejani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Chilamkurthy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Steiner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Fang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Chintala, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Pytorch: An imperative style, high-performance deep learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Beygelzimer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=", d'Alch´e-Buc, F." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Fox, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Garnett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=') Advances in Neural Information Processing Systems 32, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 8024–8035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' (2019) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Respall, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Devitt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Fedorenko, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Klimchik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Fast sampling-based next- best-view exploration algorithm for a mav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: 2021 IEEE International Conference on Robotics and Automation (ICRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 89–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' IEEE (2021) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Saito, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', , Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Natsume, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Morishima, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Kanazawa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='05172 (2019) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Sitzmann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Martel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Bergman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Lindell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Wetzstein, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Im- plicit neural representations with periodic activation functions (2020) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Sucar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Ortiz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Davison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': iMAP: Implicit mapping and positioning in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: Proceedings of the International Conference on Computer Vision (ICCV) (2021) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Tang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Chou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', H¨ane, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Dou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Fanello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Davidson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Guleryuz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Izadi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Tagliasacchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Bouaziz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Keskin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Deep implicit volume compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' CoRR abs/2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='08877 (2020) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Lei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Ma, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Jia, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Sign-agnostic conet: Learn- ing implicit surface reconstructions by sign-agnostic optimization of convolutional occupancy networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' CoRR (2021) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Tewari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Fried, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Thies, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Sitzmann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Lombardi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Sunkavalli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Martin- Brualla, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Simon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Saragih, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Nießner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Pandey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Fanello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Wetzstein, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Theobalt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Agrawala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Shechtman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Goldman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Zollh¨ofer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': State of the art on neural rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Forum 39, 701–727 (2020) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Uyanik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Secil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Ozkan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Dutagaci, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Turgut, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Parlaktuna, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Spgs: A new method for autonomous 3d reconstruction of unknown objects by an industrial robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: Annual Conference Towards Autonomous Robotic Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' 15–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Springer (2018) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Vasquez-Gomez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Sucar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Murrieta-Cid, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': View planning for 3d object reconstruction with a mobile manipulator robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (2014) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' de Vries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Learn opengl: Camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' https://learnopengl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='com/Getting-started/ Camera, accessed: 2022-05-26 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Bovik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Sheikh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Simoncelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' : Image quality assessment: from error visibility to structural similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' IEEE transactions on image processing 13(4), 600–612 (2004) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Isola, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Efros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Shechtman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Wang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': The unreasonable effectiveness of deep features as a perceptual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' In: CVPR (2018) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=', Koltun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=': Open3D: A modern library for 3D data process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content=' arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} +page_content='09847 (2018)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFJT4oBgHgl3EQfXixy/content/2301.11522v1.pdf'} diff --git a/k9E2T4oBgHgl3EQfdwdn/content/tmp_files/2301.03909v1.pdf.txt b/k9E2T4oBgHgl3EQfdwdn/content/tmp_files/2301.03909v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..16969a197b8ab46b5c77040aab5d404a6c12b5fa --- /dev/null +++ b/k9E2T4oBgHgl3EQfdwdn/content/tmp_files/2301.03909v1.pdf.txt @@ -0,0 +1,1949 @@ +Metrological detection of purely-non-Gaussian entanglement +David Barral,1, ∗ Mathieu Isoard,1 Giacomo Sorelli,1, 2 Manuel Gessner,3 Nicolas Treps,1 and Mattia Walschaers1, † +1Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS, ENS-PSL Research University, +Coll`ege de France, 4 place Jussieu, F-75252 Paris, France +2Fraunhofer IOSB, Ettlingen, Fraunhofer Institute of Optronics, +System Technologies and Image Exploitation, Gutleuthausstr. 1, 76275 Ettlingen, Germany +3Departamento de F´ısica Te´orica, IFIC, Universidad de Valencia-CSIC, +C/ Dr. +Moliner 50, Burjassot, Valencia 46100, Spain +(Dated: January 10, 2023) +Entanglement and non-Gaussianity are physical resources essential for a large number of quantum- +optics protocols. Non-Gaussian entanglement is indispensable for quantum-computing advantage +and outperforms its Gaussian counterparts in a number of quantum-information protocols. The +characterization of non-Gaussian entanglement is a critical matter as it is in general highly de- +manding in terms of resources. We propose a simple protocol based on the Fisher information for +witnessing entanglement in an important class of non-Gaussian entangled states: photon-subtracted +states. We demonstrate that our protocol is relevant for the detection of purely-non-Gaussian en- +tanglement and that it is experimentally feasible through homodyne detection. +I. +INTRODUCTION +Entanglement is considered one of the most striking +breakthroughs of the 20th century science. The gedanken +experiment proposed by Einstein, Podolsky and Rosen in +1935 [1] pointed out the notion of inseparability of a state +composed by two quantum particles spatially distanced +with maximally correlated momenta and maximally anti- +correlated positions. Nowadays, entanglement stands as +a physical resource underpinning most of current develop- +ment in quantum technologies [2]. The efficient detection +and measurement of entanglement is a very active area of +quantum physics [3], being far from simple especially for +continuous variable (CV) systems which involve physical +quantities with a continuous spectrum of values [4]. +Multimode squeezed states of light are the cornerstone +of CV quantum networks [5, 6]. +They exhibit Gaus- +sian statistics and their entanglement properties are com- +pletely specified by their covariance matrix. Criteria and +witnesses for this Gaussian entanglement have been pro- +posed and tested for decades [7, 8, 10]. However, Gaus- +sian entanglement can always be undone with passive lin- +ear optics, a phenomenon generally refereed to as passive +separabiliy [11]. It was recently found that one requires +states that are not passively separable as a resource for +a quantum computational advantage [12]. +Because all +Gaussian states are passively separable, we can always +find mode bases in which the covariance matrix of the +state will not show any direct signature of entanglement. +Yet, if the state is not passively separable, even the modes +in those bases must be entangled. Because this entangle- +ment is fully hidden in the non-Gaussian features of the +state, we will here refer to it as non-Gaussian entangle- +ment [13, 14]. The goal of this work is to find a practical +way to detect this type of non-Gaussian entanglement. +∗ david.barral@lkb.ens.fr +† mattia.walschaers@lkb.upmc.fr +In order to characterize non-Gaussian entanglement +a number of criteria based on high-order moments and +on uncertainty relations of different classes of operators +have been proposed [15–19]. +Nevertheless, these crite- +ria are far from being feasible with current experimen- +tal methods. Other more experimentally-friendly crite- +ria are based on the Shannon entropy and the fidelity +of teleportation in quantum channels [20, 21]. Here we +tackle the problem from an operational point of view: +non-Gaussian quantum correlations are also known to im- +prove metrological sensitivity, the performance of quan- +tum key distribution and quantum teleportation proto- +cols [22–24]. The advantage of relying on the improve- +ment of quantum protocols is two-fold. On the one hand +the detected entanglement is useful by design, and, on +the other hand, the witness comes with a natural imple- +mentation: executing the protocol. In this Article, we +will focus specifically on metrological protocols, where +quantum estimation tools have been devised to witness +entanglement [25–27]. These witnesses are based on the +fact that metrological sensitivity determines precision of +measurements and this sensitivity is limited for separable +states. This can be used to detect entanglement. Two +powerful assets of these sensitivity-based witnesses are i) +they do not make assumptions about the quantum state +–Gaussianity, purity, etc., and ii) they contain informa- +tion about all high-order moments. +We adapt the approach of refs. +[26, 27] to the ex- +perimental context and limitations of CV quantum op- +tics and propose a general protocol based solely on ho- +modyne detection, using both the variance of the mea- +surement outcomes and the joint measurement statistics. +Our protocol is efficient in terms of resources as the pa- +rameter estimation is done in postprocessing using solely +the data collected by homodyne detection. We show its +relevance analyzing an important class of non-Gaussian +entangled states: photon-subtracted states. We demon- +strate that our protocol is pertinent for the detection of +non-Gaussian entanglement and that it is experimentally +arXiv:2301.03909v1 [quant-ph] 10 Jan 2023 + +2 +feasible. +The article is organized as follows: We first present +our protocol to detect entanglement through homodyne +detection and postprocessing of the joint probability dis- +tribution based on the metrological witness introduced +in [26, 27] in Section II. We then present in Section III +the probe states we will use to test our non-Gaussian +entanglement witness. In Section IV we analyze which +parameter is best suited to measure entanglement in +our metrological protocol and calculate entanglement in +an ideal case. +In Section IV we study a realistic case +taking into account unbalanced input squeezing, losses +and discretization of the measurement outcomes. We fi- +nally discuss possible experimental implementations of +our scheme, their limitations and feasibility in Section +VI and we present our conclusions in Section VII. +II. +ENTANGLEMENT DETECTION VIA +LOCAL HOMODYNE DETECTION AND +POSTPROCESSING +We consider here the following problem: two experi- +menters, Alice and Bob, who share an optical quantum +state ˆρAB, want to elucidate if their shared state is entan- +gled or not, while minimizing the amount of experimental +resources. If the input state is Gaussian, they just need to +measure the variances of linear combinations of optical- +field quadratures and apply second-order moment-based +criteria like for instance those of Duan et al., Simon +or Giovanetti et al. +[7–9]. +This can be easily imple- +mented experimentally using homodyne detection. How- +ever, the larger class of non-Gaussian states do not always +present entanglement that can be revealed by second- +order moment-based criteria. Particularly, the majority +of entanglement criteria for quantum states with purely +non-Gaussian correlations are based on either carrying +out full quantum-state tomography [28] or measuring +high-order moment correlations, protocols which are very +demanding experimentally. +Here, we apply a metrological protocol to detect entan- +glement. Alice and Bob share information in order to es- +timate jointly a parameter θ generated by a Hamiltonian +ˆH = ˆHA + ˆHB that acts locally on both subsystems such +that ˆρθ +AB = e−iθ ˆ +H ˆρABeiθ ˆ +H (see Figure 1). The metrolog- +ical protocol consists of measuring the Fisher information +(FI) defined as +F(P(ξA, ξB|θ)) = +� +R2 P(ξA, ξB|θ) +�∂L(ξA, ξB|θ) +∂θ +�2 +d2ξ, +(1) +where d2ξ = dξA dξB, L(ξA, ξB|θ) = log(P(ξA, ξB|θ)) +represents the logarithmic likelihood related to the prob- +ability density P(ξA, ξB|θ). +The latter quantity repre- +sents the conditional probability to obtain a set of lo- +cal measurement outcomes (ξA, ξB) given the parame- +ter θ. The probability P(ξA, ξB|θ) can be rewritten as +Tr[ˆρθ +AB ˆΠξ], where ˆΠξ = |ξA, ξB⟩⟨ξA, ξB| is a positive- +operator valued measure (POVM) such that +� ˆΠξd2ξ = +1. +In our case, as illustrated in Figure 1, the observ- +ables will correspond to local homodyne measurements +ˆξA = cos φAˆxA+sin φAˆpA and ˆξB = cos φBˆxB+sin φB ˆpB, +where φA, φB are two angles, and ˆxA, ˆxB, ˆpA, ˆpB are the +amplitude and phase quadratures defined from the anni- +hilation operators as +ˆaI = ˆxI + iˆpI +2 +, +I ∈ {A, B}. +(2) +The quadrature operators thus satisfy the commutation +rules [ˆxI, ˆpJ] = 2iδIJ, I, J ∈ {A, B}. +Then, if ˆρAB is separable, the FI of Equation (1) for a +state ˆρθ +AB generated by ˆH is upper bounded by [26, 27] +F(P(ξA, ξB|θ)) ≤ 4Var[ˆρA, ˆHA] + 4Var[ˆρB, ˆHB], +(3) +where ˆρA/B are the reduced density matrices for systems +A and B, respectively. Because this is a necessary condi- +tion for separability, its violation is a sufficient criterion +for entanglement. +Therefore, we can introduce the following metrological +witness of entanglement +E = F(P(ξA, ξB|θ)) +− 4(Var[ˆρA, ˆHA] + Var[ˆρB, ˆHB]) > 0. +(4) +This inequality can reveal entanglement but not its origin +–Gaussian or non-Gaussian. From now on we will refer to +non-Gaussian entanglement as the entanglement that is +not detected by Gaussian entanglement witnesses based +on second order moments –the covariance matrix– such +as Duan et al., Simon and Giovanetti et al. criteria [7–9] +or optimized witnesses such as presented in [29]. +The real interest of the witness (4) is that it also holds +for any state, pure or mixed, and its major asset is the +practicability of its computation. Homodyne measure- +ments in each mode with a common phase reference allow +us to access experimentally i) the joint probability distri- +bution P(ξA, ξB|θ), and thus the FI, and ii) the variances +associated to the local generators, enabling to test the en- +tanglement witness given by Equation (4). Moreover, in +some cases (see Section IV) the parameter-dependence +of the joint probability distribution P(ξA, ξB|θ) can be +generated in postprocessing applying appropriate trans- +formations directly to the joint probability distribution +as P(ξA, ξB|θ) = P(Uθ(ξA), Uθ(ξB)), with Uθ(ξA/B) the +transformation related to the Hamiltonian ˆHA/B in the +quadrature space ξA/B [31, 32]. This important feature +avoids to apply impractical inline transformations to the +state simplifying greatly the detection of entanglement. +The entanglement witness (4) can be maximized by +choosing an optimal measurement observable. It is well +known in quantum metrology that the ultimate precision +on the parameter θ is limited by the quantum Fisher In- +formation FQ (QFI), which represents the sensitivity of +the full quantum state ρAB to small perturbations gener- +ated by ˆH. As a consequence, the FI is bounded by the + +3 +FIG. 1. +Sketch of the proposed metrological protocol for +entanglement detection. +Alice and Bob share a quantum +state ˆρAB. +They jointly estimate a parameter θ generated +by two local Hamiltonians ˆHA/B. Using two homodyne de- +tectors with a common phase reference, Alice and Bob can re- +trieve the parameter-dependent joint probability distribution +P(xA, xB|θ), and thus the Fisher information related to this +parameter estimation, and the local variances of the Hamil- +tonians ˆHA/B. With this in hand, Alice and Bob can jointly +compute the metrological witness of entanglement of Equa- +tion (4). +QFI as +FQ[ˆρAB, ˆH] = max +ˆΠ +F(Tr[ˆρθ +AB ˆΠ]), +(5) +which means that the entanglement witness (4) is max- +imized when the FI saturates the QFI, i.e., when the +measurement observable is optimized [30]. Note that we +restrict ourselves to a POVM ˆΠξ corresponding to homo- +dyne measurements. Thus, the FI related to this mea- +surement observable does not saturate the QFI for every +generator ˆH. +For pure states we can easily obtain the QFI from the +variance of the generator ˆH of the parameter θ as +FQ[ρAB, ˆH] = 4Var[ˆρAB, ˆH]. +Applying this identity into Equation (4) we obtain the +following simple condition for entanglement +EQ ≡ max +ˆΠ +E = 8 Cov[ρAB; ˆHA, ˆHB] > 0. +(6) +This inequality for pure states should not come as an +absolute surprise. After all, any correlation that is seen +in a bipartite pure state is a signature of entanglement. +III. +APPLICATION TO +PHOTON-SUBTRACTED STATES +The protocol described in the previous section is valid +for any CV system, regardless of the nature of the +state under consideration, as long as one has access +to the probability distributions of each subsystem. +In +this section we introduce the states that we will use +as a probe of our non-Gaussian entanglement criterion, +namely, photon-subtracted states. In particular we will +analyze bipartite states without Gaussian correlations in +order to focus on their non-Gaussian features. +We consider two-mode photon subtracted states. This +class of states has been demonstrated in optical systems +using different degrees of freedom, such as polarization or +frequency modes [28, 33]. In Section VI we will explain +in detail different experimental methods for their pro- +duction. Let us consider two independent single-mode +squeezed states respectively related to Alice and Bob +|Ψ0⟩ = ˆSA(rA, θA) ˆSB(rB, θB)|00⟩, +(7) +where ˆSI(rI, θI) = exp{(−rI/2)(ˆa2 +Ie−2iθI − ˆa†2 +I e2iθI)} is +the single-mode squeezing operator, and rI ∈ R+ and +θI ∈ R are respectively the squeezing parameter and +squeezing phase for each mode I = A, B. The amount of +squeezing in decibels is given by sI = 10 log10(e−2rI). +In what follows we analyze two cases: in-phase squeez- +ing (θB += θA) and in-quadrature squeezing (θB += +θA + π/2). Without loss of generality we set θA = 0. +We include the information about the squeezing phase +by extending the domain of the squeezing parameter to +rI ∈ R such that ˆSI(rI, θI) ≡ ˆSI(rI). +Thus, Equa- +tion (7) corresponds to a Gaussian state and all its +information is encoded in the covariance matrix V0 = +diag(e−2rA, e2rA, e−2rB, e2rB), written with respect to the +vector of amplitude and phase quadratures in each mode +⃗ξ = (xA, pA, xB, pB)T . +Note that V0 does not present +off-diagonal terms, thus the input Gaussian state is fully +separable. +Next, we perform a delocalized subtraction of one pho- +ton on this state. This operation produces a superposi- +tion of two-mode squeezed vacuum and squeezed single- +photon states that one can show that up to normaliza- +tions is [34] +|Ψ⟩ ∝ (cos(φ)ˆaA + sin(φ)ˆaB)|Ψ0⟩ = +ˆSA(rA) ˆSB(rB)(cos(φ) sinh(rA)|10⟩ + sinh(rB) sin(φ)|01⟩), +where the parameter φ controls the probability of sub- +traction in each mode and we have considered in-phase +subtraction. A sketch of this operation is shown in Fig- +ure 2. The wavefunction of this state in the amplitude +quadratures of the optical field is given by +Ψ(xA,xB) ≡ ⟨xA, xB|Ψ⟩ ∝ e− +e2rA x2 +A+e2rB x2 +B +4 +× ((e2rA − 1) cos (φ)xA + (e2rB − 1) sin (φ)xB). +(8) +Examples of joint probability distributions P(xA, xB) = +|Ψ(xA, xB)|2 for a photon subtracted state given by +Equation (8) with φ = π/4 and rA = rB += 0.2, +rA = −rB = 0.2, are respectively shown in Figure 3 +a) and b). + +ALICE +i0H +P(CA,CBO) +pAB +LO +Var[βA, HA] +Var[pB, HB] +BOB4 +FIG. 2. +Sketch of an optical setup for delocalized single- +photon subtraction. +Alice and Bob prepare two squeezed +states in given optical modes. A small fraction of each mode +power is diverted to a common beam splitter with a trans- +mittivity controlled by a parameter φ. An event measured by +a single-photon detector heralds the subtraction of a photon +delocalized between the two modes. +The local Hamiltonians ˆHA/B are in general polyno- +mials of amplitude x and phase p local quadratures. The +separability bound related to their variances can be cal- +culated using wavefunctions via +⟨ˆxn +i ˆpm +j ⟩ = +(9) +(−2i)m +�� +R +xn +i Ψ(xA, xB)∗ ∂mΨ(xA, xB) +∂xm +j +dxAdxB, +where the functional relation ˆpj → −2i∂/∂xj is used. +The entanglement present in these states is not grasped +by Gaussian entanglement witnesses: this can be gener- +ally understood from the covariance matrix of a photon- +subtracted state. Ref. [11] shows that this covariance +matrix can generally be written as +V = V0 + 2(V0 − 1)P(V0 − 1) +Tr[(V0 − 1)P] +, +(10) +where V0 is the initial Gaussian state’s covariance matrix +and P is a matrix that projects on the phase space axes +associated with the mode of photon subtraction. In our +present case, we find that +P = +� +� +� +� +cos2(φ) +0 +1 +2 sin(2φ) +0 +0 +cos2(φ) +0 +1 +2 sin(2φ) +1 +2 sin(2φ) +0 +sin2(φ) +0 +0 +1 +2 sin(2φ) +0 +sin2(φ) +� +� +� +� . +Thus, we see in Equation (10) that on the level of the co- +variance matrix the photon subtraction only adds Gaus- +sian noise. +This implies that no additional entangle- +ment can be witnessed by purely looking at the covari- +ance matrix [29]. As a consequence, since we set V0 = +diag(e−2rA, e2rA, e−2rB, e2rB), we find that V should not +display any entanglement. +-� -� -� +� +� +� +� +-� +-� +-� +� +� +� +� +�� +�� +�) +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +-� -� -� +� +� +� +� +-� +-� +-� +� +� +� +� +�� +�� +�) +0.02 +0.04 +0.06 +0.08 +0.10 +FIG. 3. Contour plots of the joint probability distribution for +a photon subtracted state given by Equation (8) with φ = π/4 +and a) rA = rB = 0.2, b) rA = −rB = 0.2 (|sA/B| = 1.74 dB). +IV. +IDEAL DETECTION OF NON-GAUSSIAN +ENTANGLEMENT +In order to decide which Hamiltonian ˆH is best suited +to witness entanglement we can calculate theoretically +EQ through Equation (6). +This can guide us decid- +ing which parameter is best suited to detect entangle- +ment of a given quantum state in a realistic scenario. +Below, we use the estimation of parameters related to +the four single-mode Gaussian gates in CV quantum op- +tics. +Namely: displacement, phase-shift, shearing and +squeezing. We analyze in which cases the joint estima- +tion of these parameters reveals the entanglement of the +non-Gaussian two-mode photon-subtracted state given +by Equation (8). +For the sake of simplicity, we focus +here on the case φ = π/4. A generalization to any φ is +shown in Appendix A. + +ALICE +pA +pAB +pB +BOB5 +A. +Displacement +A displacement of θ along the axis xA = ±xB is pro- +duced by the following operator +ˆD(θ) = e−iθ(ˆpA±ˆpB)/2. +The Hamiltonian related to this displacement operator is +H± = (ˆpA ± ˆpB)/2. The optimal entanglement EQ ob- +tained estimating displacement along the axis xA = ±xB +for a pure photon-subtracted state given by Equation (8) +with φ = π/4 and squeezing parameters rA ̸= rB is +EQ = ±2erA+rB cos(ϵ), +(11) +with +cos(ϵ) = +2 sinh(rA) sinh(rB) +sinh2(rA) + sinh2(rB). +Displacement along either xA = xB or xA = −xB +detects entanglement respectively for in-phase squeezing +(rA, rB > 0) and in-quadrature –orthogonal– squeezing +(rA > 0, rB < 0). Figures 4a and 4b show contour plots +of optimal entanglement EQ in the two cases. States with +in-phase input squeezing show always a larger degree of +entanglement due to the argument rA + rB in Equa- +tion (11). +For in-phase squeezing the witness reaches +the maximum at rA = rB (dashed line along the diag- +onal in Figure 4a) and is given by EQ = 2e2rA. Like- +wise, for in-quadrature squeezing the maximum value +of EQ is not along the diagonal, but below it. +For +a given value of rA, the maximum EQ is obtained for +rB = log (1/(1 + 2 sinh (rA))1/2) (dashed line in Figure +4b) and is given by +EQ = +2 erA +1 + sinh (rA). +The shapes of Figures 4a and 4b can be explained in +terms of the symmetries of the two functions that com- +pose Equation (11): +± cos(ϵ) is a symmetric function +with respect to the diagonal sA = sB for every input +squeezing, whereas 2erA+rB is symmetric with respect to +the diagonal (antidiagonal, in this case along sA−sB = 6 +dB) for in-phase (in-quadrature) squeezing. +Importantly, we obtain the same result calculating the +entanglement through Equation (4), E = EQ, indicating +that the FI saturates the QFI. The result of Equation +(11) is particularly interesting because, following Equa- +tion (6), second order moments of the distribution reveal +entanglement with a non-Gaussian origin. +Recently, M. Tian et al. analyzed the multipartite en- +tanglement in a nondegenerate triple photon state using +a metrological criterion [35]. +They claimed there that +non-Gaussian entanglement cannot be sufficiently cap- +tured by linear quadratures, i.e. displacements. While +this is the case for triple photon states, we have shown +that it does not hold in general: displacements can detect +non-Gaussian entanglement of photon-subtracted states. +� +� +� +� +� +� +� +� +� +� +� +� +� +� +��(��) +��(��) +1 +2 +3 +4 +5 +6 +7 +� +� +� +� +� +� +� +� +-� +-� +-� +-� +-� +-� +��(��) +��(��) +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +FIG. 4. Optimal displacement-estimation entanglement EQ +given by Equation (11) versus squeezing of the input squeezed +states sA and sB. a) Displacement along xA = xB optimizes +EQ for states with in-phase input squeezing. b) Displacement +along xA = −xB optimizes EQ for states with in-quadrature +input squeezing. Maximum value of EQ in dashed. +B. +Phase shift +The phase-shift operator +ˆR(θ) = e−iθ( ˆ +NA± ˆ +NB) ∝ e−iθ(ˆx2 +A+ˆp2 +A±ˆx2 +B±ˆp2 +B)/4 +rotates the state by a phase θ in local phase sub- +spaces in either clockwise-clockwise (+) or clockwise- +counterclockwise (−) directions. +The related Hamilto- +nian is H± = ˆNA ± ˆNB. The optimal entanglement wit- +ness is in this case +EQ = ∓2 cosh(2rA) cosh(2rB) cos2(ϵ). +(12) +Entanglement +is +always +detected +for +clockwise- +counterclockwise (−) phase shifts, but not for clockwise- + +a)b)6 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +��(��) +��(��) +1 +2 +3 +4 +5 +6 +7 +8 +FIG. 5. Optimal phase-estimation entanglement EQ given by +Equation (12) versus squeezing of the input squeezed states +sA and sB. Maximum value of EQ in dashed. +clockwise (+) as it is just a global phase shift. Figure +5 shows contour plots of optimal entanglement EQ for +different values of squeezing. +Notably, the detection +of entanglement does not depend on the phase of the +input squeezed states as EQ is invariant under change +of sign of the squeezing parameters rA/B. The detected +entanglement is maximum for rA = rB (dashed line +along the diagonal in Figure 5) being EQ = 2 cosh2(2rA). +One can wonder if the Fisher information in Equa- +tion (4) reaches the QFI in this case. While measuring +the joint probability distribution in the (x1, x2)−plane +was enough to obtain the maximal value of the FI and +saturates the QFI for the displacement estimation, here +the situation is a bit more complicated. For simplicity, +we consider the case rA = rB in what follows. The FI +can be optimized by finding the set of angles (φA, φB) of +the measurement outcomes ξA = cos φAxA − sin φApA, +ξB = cos φBxB − sin φBpB for which the joint probabil- +ity distribution P(ξA, ξB|θ) leads to the best value of the +FI (see Appendix B). However, we find that such local +rotations are not enough to saturate the QFI, and that +only a mixing of modes A and B before the homodyne +detectors can lead to a saturation of the QFI. It is in- +deed possible to prove that a non-local rotation of −π/4 +between modes A and B and measuring the joint proba- +bility distribution P(x′ +A, p′ +B|θ), with x′ +A = (xA −xB)/ +√ +2 +and p′ +B = (pA + pB)/ +√ +2 is needed to saturate the QFI. +C. +Shearing +The shearing –also known as phase-gate– operator +ˆS(θ) = e−iθ(ˆx2 +A±ˆx2 +B)/4 +shears the state with respect to the axes xA and ±xB +by a gradient of θ. The related Hamiltonian is H± = +(ˆx2 +A ± ˆx2 +B)/4. The optimal entanglement is in this case +EQ = ∓e−2(rA+rB) +2 +cos2(ϵ). +(13) +Thus, shearing with respect to xA and xB does not de- +tect entanglement. However, shearing with respect to xA +and −xB captures it. Note that in this case the entangle- +ment is maximized for rA/B < 0, i.e. squeezing along the +quadratures pA/B, unlike displacement and phase estima- +tion where EQ is maximized for squeezing along xA/B. +Figures 6a and 6b show contour plots of optimal en- +tanglement EQ in the cases of input squeezing along +the same quadrature (a) or along different quadratures +(b). For input squeezing along the same quadratures the +detected entanglement is maximum again for rA = rB +(dashed line along the diagonal in Figure 6a) and given +by EQ = e−4rA/2. +However, the maximum entangle- +ment is below the diagonal for input squeezing along +different quadratures as for displacement. +For a given +value of rA, the maximum EQ is obtained for rB = +(−rA + log(1 + erA − e2rA))/2 (dashed line in Figure 6b) +and is given by +EQ = +e−2rA +2(−1 + sinh(rA))2 . +The shape of Figures 6a and 6b is explained in the same +way as for displacement. +Here again, we optimize the FI to see if it is possible +to reach the bound E = EQ. The same analysis as in the +case of the phase-shift operator (by performing local ro- +tations before the homodyne detection) is summarized in +Appendix B. The same conclusion follows: the FI never +reaches the QFI, and only a non-local rotation of -π/4 +between modes A and B leads to a saturation of the +QFI. +D. +Squeezing +The squeezing operator +ˆS(θ) = e−iθ(ˆxA ˆpA+ˆpAˆxA±ˆxB ˆpB±ˆpB ˆxB)/4 +squeezes the position quadratures of modes A and B by +a factor of eθ (+) or squeezes the position quadratures of +A by eθ and stretches those of B by e−θ (−). The related +Hamiltonian is H± = (ˆxAˆpA + ˆpAˆxA ± ˆxB ˆpB ± ˆpBˆxB)/4. +The optimal entanglement is here +EQ = 0. +Interestingly, the joint estimation of the squeezing pa- +rameter does not detect entanglement in any of the above +two cases. + +7 +� +-� +-� +-� +-� +-� +-� +� +-� +-� +-� +-� +-� +-� +��(��) +��(��) +�) +1 +2 +3 +4 +5 +6 +7 +� +-� +-� +-� +-� +-� +-� +� +� +� +� +� +� +� +��(��) +��(��) +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +FIG. 6. Optimal shearing-estimation entanglement EQ given +by Equation (13) versus squeezing of the input squeezed states +sA and sB. a) EQ for states with input squeezing along the +same quadrature. b) EQ for states with input squeezing along +different quadratures. Maximum value of EQ in dashed. +E. +Comparison and resource evaluation +In order to decide which parameter-estimation strat- +egy is best suited to detect entanglement we show in +Figure 7 the evolution of maximum entanglement EQ ver- +sus amount of squeezing in dB for the above four joint +parameter estimations in the case of in-phase and in- +quadrature input squeezing. For in-phase input squeez- +ing the maximum entanglement is obtained for rA = rB +(dashed line along the diagonal in Figures 4a, 5 and +6a). +For in-quadrature input squeezing the maximum +entanglement is obtained for rA = −rB for phase-shift +(dashed line along the diagonal in Figure 5), and for +rB = log (1/(1 + 2 sinh (rA))1/2) and rB = (−rA+log(1+ +� +� +� +� +� +� +� +� +� +� +� +� +|��|(��) +�� +FIG. 7. Maximum optimal entanglement EQ versus squeez- +ing in Alice’s mode (dashed curves of Figures 4,5, and +6): displacement estimation (blue), shearing estimation (or- +ange), phase shift estimation (green), and squeezing estima- +tion (gray). In-phase (in-quadrature) input squeezing in solid +(dashed). For phase shift estimation (green) the curve is the +same in both cases. EQ > 0 witnesses entanglement. +erA−e2rA))/2 for displacement and shearing, respectively +(dashed line below the diagonal in Figures 4b and 6b). +Remarkably, for values of input squeezing lower than ≈ 5 +dB, the best strategy is to jointly estimate the displace- +ment (solid, blue). For larger values of input squeezing, +phase shift and shearing estimation offer a greater sensi- +tivity to entanglement (green and orange, respectively). +On the contrary, as we saw above the joint estimation of +the squeezing parameter does not offer any information +on the entanglement of this state (solid, gray). +In terms of resources, displacement estimation is +also advantageous. +Both probability distributions and +quadrature variances corresponding to Alice and Bob can +be directly measured with homodyne detection. +Like- +wise, shearing estimation can be performed with homo- +dyne detection, but fourth-order moments of the distri- +butions (kurtosis) are necessary, which implies in general +larger data sets. In the case of phase estimation, photon- +number variances are necessary, which implies adding +complexity to the detection. +Another great advantage of displacement estimation is +that the displacement operation can be applied in post- +processing: once the probability distribution P(xA, xB|0) +is measured, the displaced probability distribution [un- +der the Hamiltonian ˆH± = (ˆpA ± ˆpB)/2] is directly given +by P(xA, xB|θ) = P(xA + θ, xB ± θ|0) [32], from which +one can compute the classical FI (see Sec. V C) – which +we know saturates the QFI in this case, leading the best +possible estimation. On the contrary, the shearing and +phase-shift operations can not be implemented in post- +processing using just the probability distribution as full +information about the state is needed for such operations. +Thus, shearing and phase-shift unitaries have to be im- +plemented at the level of the experimental setup or by + +b)8 +post-processing after measuring the full quantum state +by for instance double-homodyne detection [36]. In ad- +dition to this complication, contrary to the displacement +operation, as we pointed out in Sec. IV B and Sec. IV C, +the FI can be optimized with local rotations of the mea- +sured quadratures, but only saturates the QFI when one +mixes modes A and B. +V. +REALISTIC DETECTION OF +NON-GAUSSIAN ENTANGLEMENT +In this section we study the measurement of entan- +glement in a realistic scenario. As we found above, es- +timating displacement is the best strategy for ideal de- +tection at moderate values of squeezing. Moreover, it is +the simplest one, as the variances of the generators –field +quadratures– are directly measured with homodyne de- +tection. We thus focus on this option in the following. +A similar analysis could be carried out for shearing and +phase-shift estimation. Below we analyze the effect of un- +balancing the sensitivity in the displacement estimation, +the effect of losses on the detection of entanglement and +the discretization of the sampled data to build a joint +probability distribution and calculate the Fisher infor- +mation. +A. +Optimization of displacement axis for +entanglement witness +In the previous section, we analyzed the detection of +entanglement through displacement estimation when dis- +placing the input state along the axes xA = ±xB. How- +ever, we can optimize the entanglement detection displac- +ing the input state along an axis different to xA = ±xB +or, in other words, unbalancing the sensitivity related to +Alice and Bob in the joint parameter estimation. The +idea is the following: +instead of displacing the same +amount (1, ±1) in both amplitude quadratures, we dis- +place ( +√ +2 cos(δ + π/4), ± +√ +2 sin(δ + π/4)) along xA and +xB, respectively, where δ ∈ [0, π] is an angle that we can +optimize for each pair of values of rA and rB. This leads +to a new Hamiltonian ˆH± = (cos(δ + π/4)ˆpA ± sin(δ + +π/4)ˆpB)/ +√ +2. Calculating the optimal entanglement EQ +of Equation (6) we find now +Eδ +Q = EQ cos(2δ). +(14) +Therefore, displacing along xA = ±xB (δ = 0, π) is in- +deed the optimal strategy and displacement along any +other axis can only degrade the detection of entangle- +ment since | cos(2δ)| ≤ 1. +B. +Optical losses +The effect of optical losses can be entirely absorbed by +the covariance matrix when it is the same in both modes +[37]. The covariance matrix of the input squeezed state +V0 is modified in the following way Vη = (1 − η)V0 + η1, +where η represents the amount of losses. For instance, +the probability distribution related to the quantum state +given by Equation (8) with φ = π/4 and rA = rB ≡ r is +now +Pη(xA, xB) ∝ e− +x2 +A+x2 +B +2σ2 +(2ηe2rσ2 + (1 − η)(xA + xB)2), +with σ2 = (1 − η) e−2r + η. A similar but less straight- +forward result is obtained for general values of φ, rA and +rB. +Figure 8 shows the effect of losses on the detection of +entanglement for photon-subtracted states with φ = π/4 +and sA = sB (Figure 8a), sA = 1 dB and varying sB +(Figure 8b), and sA = 2 dB and varying sB (Figure +8c). +In Figure 9 we use the same state but now with +sA = −sB (Figure 9a), sA = 1 dB and varying sB +(Figure 9b), and sA = 2 dB and varying sB (Figure +9c). +In general, the effect of losses increases with the +amount of input squeezing, and the metrological entan- +glement is more resilient for input squeezing along differ- +ent quadratures. For sA > 0, sB > 0, the metrological +detection of entanglement is resilient up to ≈ 20% for in- +put values of squeezing between 1 and 2 dB, whereas for +sA > 0, sB < 0, the metrological detection of entangle- +ment is resilient up to ≈ 70% for highly asymmetric in- +put squeezing. Moreover, entanglement is more resilient +to losses in comparison with quantum steering, where the +losses threshold is about 7% for the same states [38]. +It must be emphasised that our entanglement witness +detects only entanglement related to the metrological +sensitivity of the state [25]. The losses produce quan- +tum decoherence and impair the metrological power of +the quantum state. We have checked that other entangle- +ment witness, such as the logarithmic negativity, detect +entanglement in regions where our metrological witness +cannot, but that entanglement is not a useful resource +for parameter estimation [39]. +C. +Discretization of sampled data +We need to obtain experimentally the FI and the vari- +ances associated to the local displacement generators +Var(ˆpA/B) in order to compute the entanglement witness +E of Equation (4). The variances are directly obtained +measuring the phase quadratures with homodyne detec- +tion. Estimating the FI experimentally from discrete out- +comes, in contrast with the theoretical computation that +assumes a continuum of outcomes, relies on the compu- +tation of a statistical distance –the Hellinger distance– +between a reference probability distribution and the +parameter-dependent one [22]. +The squared Hellinger +distance between a parameter-dependent probability dis- +tribution P(xA, xB|θ) and a reference P(xA, xB|0) is de- + +9 +1dB +2dB +3dB +4dB +5dB +6dB +���� +���� +���� +���� +���� +���� +���� +� +� +� +� +� +η +� +�) +��=��= +4dB +5dB +6dB +7dB +8dB +���� +���� +���� +���� +���� +���� +���� +��� +��� +��� +��� +η +� +�) +��=���� ��= +0.5dB +1dB +1.5dB +2dB +2.5dB +3dB +���� +���� +���� +���� +���� +���� +���� +� +� +� +� +η +� +�) +��=���� ��= +FIG. 8. Effect of losses η on displacement-estimation-based +entanglement for a photon-subtracted quantum state with +φ = π/4 and a) sA = sB (legend), b) sA = 1 dB, varying +sB (legend), and c) sA = 2 dB, varying sB (legend). E > 0 +witnesses entanglement. +fined as +d2 +H,P(θ) += 1 +2 +�� +R +( +� +P(xA, xB|θ) − +� +P(xA, xB|0))2dxAdxB. +The Taylor expansion of the squared Hellinger distance +to second order yields [22] +d2 +H,P(θ) = F +8 θ2 + O(θ3), +1dB +2dB +3dB +4dB +5dB +6dB +���� +���� +���� +���� +���� +���� +���� +��� +��� +��� +��� +��� +η +� +�) +��=-��= +-4dB +-5dB +-6dB +-7dB +-8dB +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +η +� +�) +��=���� ��= +-0.5dB +-1dB +-1.5dB +-2dB +-2.5dB +-3dB +���� +���� +���� +���� +���� +���� +���� +��� +��� +��� +��� +��� +η +� +�) +��=���� ��= +FIG. 9. Effect of losses η on displacement-estimation-based +entanglement for a photon-subtracted quantum state with +φ = π/4 and a) sA = −sB (legend), b) sA = 1 dB, varying +sB (legend), and c) sA = 2 dB, varying sB (legend). E > 0 +witnesses entanglement. +with F ≡ F(ˆρAB, ˆH) the FI. Thus, a quadratic fit is +enough to calculate the FI. +However, in an experimental implementation we do not +have exact probability distributions P(xA, xB|θ), but rel- +ative frequency distributions {F(xA, xB|θ)}, which ap- +proach the probability distributions for infinitely many +independent measurements. +In this case, due to sta- +tistical fluctuations δF, the squared Hellinger distance +varies when repeating the measurement. Taking the out- +come frequencies from a sample of M experimental re- + +10 +alizations, the sample average of the squared Hellinger +distance between two relative frequencies d2 +H,F(θ) is ap- +proximately [22] +⟨d2 +H,F(θ)⟩ = c0 + (F +8 + c2)θ2 + O(θ3, δF3), +(15) +with c0 = (n − 1)/4M, c2 ≈ F(1 + n)/32M and n the +number of pairs (xA, xB) for which F(xA, xB|θ) ̸= 0. +Note that ⟨d2 +H,F(θ)⟩ converges asymptotically to d2 +H,P(θ) +as M → ∞ and hence the estimation of F is asymptoti- +cally unbiased with the bias decreasing as M −1. +In the following we study the protocol by simulating +homodyne detection with rejection sampling of the theo- +retical probability distributions obtained from Equation +(8). We partition the real line corresponding to the out- +comes of the quadrature measured by Alice and Bob in +a series of bins with a given bin size ∆. We consider an +even number of bins as the mean value of the field is zero +for our non-Gaussian probe state. Figure 10 shows two +examples of sampled joint relative frequency distribu- +tions {F(xA, xB)} obtained through rejection sampling +of the probability distribution given by Equation (8) for +rA = rB = 0.2 (Figure 3a) and rA = −rB = 0.2 (Figure +3b). The number of samples is M = 5 × 105 and the bin +size ∆=0.2 (in the units of xA/B). +We list below the steps to follow in order to calculate +the FI: +1. we take the two sets of sampled data corresponding +to Alice ⃗xA and to Bob ⃗xB and split the sampled +data (⃗xA,⃗xB) of total size M in two equal sets. +2. we bin the data in areas of given size and compute +the relative frequencies {F(xA, xB|0)} of the first +set that is used as a reference. +3. we displace the data of the second set by an amount +θ –the displacement parameter–, bin the data and +compute the relative frequencies {F(xA, xB|θ)} +that are used as a probe. +4. we calculate the square root of each relative fre- +quency for the reference and the displaced data, +take the difference and square it. +5. we calculate the sample average of the squared +Hellinger distance ⟨d2 +H,F(θ)⟩ for a value of θ. +6. we repeat this process for different values of θ and +fit the results to a parabola, obtaining the FI with +its statistical error through Equation (15). +Using this value of FI and the sum of the variances +of the phase quadratures we calculate the entanglement +witness E through Equation (4). +We show in Figure 11 the effect of data discretiza- +tion and number of samples in the detection of entan- +glement for lossless and lossy cases. We use the quan- +tum state given by Equation (8) with rA = rB = 0.2 +(rA = −rB = 0.2) sampled in Figure 10a (10b). In each +FIG. 10. +Sampled joint relative frequency distributions +F(xA, xB) obtained through rejection sampling of the prob- +ability distribution of Figures 3 a and b. a) rA = rB = 0.2 +and b) rA = −rB = 0.2. 5 × 105 samples. Bin size ∆=0.2 (in +units of xA/B). +figure, the upper curves are for the lossless case, whereas +the lower curves are for η = 0.1. We displace our second +data set of size M/2 between θ ∈ {−0.05, 0.05} in steps +of 5×10−3, resulting in 20 data points that we fit with a +parabola using Equation (15). We partition the outcome +quadratures measured by Alice and Bob in a series of bins +of size ∆. We perform 30 simulations for each value of +bin size and total number of samples to obtain statistical +averages and errors. We show the value of entanglement +E obtained using a continuous probability distribution in +solid gray, and the values and errors obtained for differ- +ent bin size ∆ and total samples M in color. The colors +represent different number of samples: M = 106 (blue), +M = 2×106 (orange), M = 4×106 (green) and M = 107 +(red). We find that the distance between the computed +value from the simulated data and the theoretical value +decreases as the bin size shrinks. For large bin size, the +number of samples does not affect significantly the ac- +curacy of the measurement. +However, for smaller bin +size, the accuracy of the discretized estimation raises as + +F(XA,XB) +a) +0.15 +4 +0.10 +0.05 +2 +0.00 +-4 +0 +XB +-2 +-2 +0 +XA +2 +4F(XA,XB) +b) +0.10F +4 +0.05 +2 +0.00 +-4 +0 +XB +-2 +-2 +0 +XA +2 +411 +��� +��� +��� +��� +��� +� +� +� +� +� +��� ���� +� +�) +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� ���� +� +�) +FIG. 11. Effect of bin size ∆, number of total samples M +and losses η on the entanglement estimation E for a photon- +subtracted quantum state with φ = π/4 and a) rA = rB = +0.2, b) rA = −rB = 0.2 (|sA/B| = 1.74 dB). In each figure, +the upper curves are for the lossless case, whereas the lower +curves are for η = 0.1. Averages and errors are calculated +over 30 simulations. E > 0 witnesses entanglement. Solid, +gray: theoretical value. Blue: 106 samples. Orange: 2 × 106 +samples. Green: 4 × 106 samples. Red: 107 samples. +the number of samples increases. In general, the statis- +tical error obtained from the fit is lower as the bin size +increases. Note that a large discretization with an insuf- +ficient number of points can lead to an overestimation of +the entanglement E. We find that in the lossless case +the estimation is in good agreement with the theoretical +value for M ≥ 2 × 106 and ∆ < 0.1. In the lossy case, +more samples are necessary for the same value of bin size +∆ and overestimation is more significant. To not over- +estimate the entanglement we should use M ≥ 2 × 106 +and ∆ = 0.2. Notably, in both cases we detect entangle- +ment even using a coarse-grained bin size ∆ = 0.4 and a +relatively low number of samples M = 106. +VI. +DISCUSSION +Let us discuss possible practical implementations of +this protocol. There are few approaches depending on +the degree of freedom –or mode– selected to encode the +quantum information: path, polarization, frequency and +so on. The shared feature of the input modes is that they +are independent and excited in squeezed states. An event +measured by a single-photon detector fed by a small frac- +tion of power from Alice and Bob’s modes where which- +mode information is erased, heralds the subtraction of +a photon delocalized between the two modes [28]. Two +balanced homodyne detectors with a common local os- +cillator LO retrieve then the joint probability distribu- +tion. The sketch of Figure 1 is pretty accurate for path- +encoded modes where a common beam splitter erases the +which-path information. +In the case of spectral modes where the number of +modes is usually larger than two –for instance in a +multimode frequency-comb Gaussian resource [40, 41]– +mode-selective photon-subtraction is accomplished by +sum-frequency generation [33]. The detection of an up- +converted photon heralds the subtraction of a photon +from a multimode input state in one or various spectral +modes selected by a pump suitably tailored in frequency. +The joint probability distribution of photon-subtracted +spectral modes can be retrieved by spectrally-resolved +homodyne detection [42]. This approach allows to mea- +sure simultaneously the quadratures of the electric field +in a number of frequency-band modes. Then, applying a +change of basis between the photon-subtracted spectral +modes and these frequency-band modes one retrieves the +quadrature traces in the modes of interest and hence, the +joint probability distribution. +Moreover, we outline that in an experiment, in order +to prove that the entanglement results entirely from the +non-local photon subtraction, one would use the data +from the unconditioned state to test our entanglement +witness and demonstrate the independence of the two +input squeezed states. +Finally, comparing our simulations with the values +measured by Y.-S. Ra et al. [33], where the squeezing +of the first and second spectral modes is sA = −2.3 dB +and sB = −1.7 dB, respectively, with purities above 90% +and detection losses of the order of 12%, we conclude that +with a reasonable number of samples (≈ 106) it is pos- +sible to witness non-Gaussian entanglement using exclu- +sively homodyne detection with an experimentally feasi- +ble protocol. Moreover, we have found that entanglement +of modes with highly asymmetric input squeezing is re- +silient versus losses (Fig 9b). This can be advantageous +for entanglement detection in photon-subtracted spectral +multimode states for instance between the first and the +third spectral modes, where the input squeezing is highly +asymmetric. + +12 +VII. +CONCLUSIONS AND OUTLOOK +We proposed a protocol based on Fisher information +for witnessing entanglement in an important class of non- +Gaussian states: +single photon-subtracted CV states. +The protocol is based on the metrological entanglement +criterion proposed in [26], and its strength comes from its +simplicity, as it relies solely on homodyne detection. Our +approach witnesses entanglement not detected by Gaus- +sian criteria, like for instance Duan et al. criterion, using +the same resources, i.e. quadrature measurements. +We characterized the optimal metrologically-useful en- +tanglement of single photon-subtracted states analyzing +their metrological power in estimation of parameters gen- +erated by all single-mode Gaussian gates, namely: dis- +placement, phase shift, shearing and squeezing. We ana- +lyzed displacement estimation in details since it gives the +largest sensitivity for currently experimentally-relevant +values of squeezing (≤ 5 dB) and it can be applied in +postprocessing, thus minimizing the resources necessary +in non-Gaussian entanglement characterization and out- +performing other protocols where quantum-state tomog- +raphy is needed. +We demonstrated that our protocol is relevant and ex- +perimentally feasible using data from a simulated exper- +iment where the effect of losses, data discretization, and +number of samples were taken into account. Our results +show that non-Gaussian entanglement can be detected +with a feasible number of measurements and data bin- +ning. It is well known that losses impair the metrological +power of quantum states. However, we found that our +metrology-based entanglement detection is resilient up +to 70% losses in some cases. +The general setup of Figure 1 is versatile and can be +used to both implement Gaussian entanglement detection +protocols based on the covariance matrix and our metro- +logical protocol for non-Gaussian entanglement. For cer- +tain classes of states, we believe that this should be suf- +ficient to be able to detect entanglement in any mode +basis. However, to determine whether or not a state is +passively separable, as would be required for the sam- +pling protocols in [12], one would still need to certify the +presence of entanglement in every possible mode basis. +While our work certainly offers us a useful experimental +tool, we also hope that it will be a step towards finding +new techniques that allow us to certify entanglement in +every possible mode basis. After all, non-Gaussian en- +tangled states encompass a huge state space and we have +just started to dig it out. In order to gain insight about +general features of this exotic quantum feature, in future +work we will analyze entangled states based on multiple- +photon subtraction and connect our entanglement crite- +rion with others based on higher-order covariance matri- +ces [43]. +ACKNOWLEDGEMENTS +This received funding from the ANR JCJC project +NoRdiC (ANR-21-CE47-0005), the European Union’s +Horizon 2020 research and innovation programme un- +der Grant Agreement No. +899587, and the Quan- +tERA II project SPARQL that has received fund- +ing from the European Union’s Horizon 2020 research +and innovation programme under Grant Agreement No +101017733. +It was carried out during the tenure of +an ERCIM ‘Alain Bensoussan’ Fellowship Programme. +M.G. acknowledges funding by the Generalitat Valen- +ciana (CDEIGENT/2021/014). +Appendix A: General expression of EQ +1. +Displacement operator +In this appendix, we derive the general expression of +EQ for a given angle φ (which controls the probability of +subtraction in each mode) and for two different squeezing +parameters rA and rB when the Hamiltonian is given by +(ˆpA ± ˆpB)/2. +By performing a change of basis from (xA, pA, xB, pB) +to a new set of coordinates (x′ +A, p′ +A, x′ +B, p′ +B), it is possible +to map the general wavefunction (8) to the symmetric +case with equal squeezing parameters, i.e., +Ψ(x′ +A, x′ +B) ∝ e−e2r x′ +A +2+x′ +B +2 +4 +(x′ +A + x′ +B), +(A1) +where r ≡ (rA + rB)/2. This change of basis consists of +two operations +� +x′ +A +x′ +B +� += R(z) S(s) +� +xA +xB +� +, +(A2) +with +S(s) = +� +es +0 +0 e−s +� +, +s ≡ rA − rB +2 +, +(A3) +and +R(z) = +� +cos z +sin z +− sin z cos z +� +, +(A4) +where +cos z = +sinh rA cos φ + sinh rB sin φ +√ +2 +� +sinh2 rA cos2 φ + sinh2 rB sin2 φ +, +sin z = +− sinh rA cos φ + sinh rB sin φ +√ +2 +� +sinh2 rA cos2 φ + sinh2 rB sin2 φ +. +(A5) +Then, using expression (9) and the change of coordinates +(A2), one finds that a symplectic transformation con- +nects the second-order moments associated to the non- +symmetric and symmetric case: +σ = Λ σ′ Λ +T, +(A6) + +13 +with +σ = +� +⟨ˆp2 +A⟩ +⟨ˆpAˆpB⟩ +⟨ˆpAˆpB⟩ +⟨ˆp2 +B⟩ +� +, σ′ = +� +⟨ˆp′ +A +2⟩ +⟨ˆp′ +Aˆp′ +B⟩ +⟨ˆp′ +Aˆp′ +B⟩ +⟨ˆp′ +B +2⟩ +� +, +(A7) +and +Λ = S(s)R−1(z). +(A8) +In particular, one has +⟨ˆpAˆpB⟩ = cos(2z)⟨ˆp′ +Aˆp′ +B⟩ + sin(2z) +� +⟨ˆp′ +A +2⟩ − ⟨ˆp′ +B +2⟩ +� +. (A9) +Given that ⟨ˆp′ +A +2⟩ = ⟨ˆp′ +B +2⟩, we finally find the simple rela- +tion +EQ = cos(2z) E′ +Q = ±2 erA+rB cos(2z), +(A10) +with +cos(2z) = +sinh zA sinh zB sin(2φ) +sinh2 zA cos2 φ + sinh2 zB sin2 φ. +(A11) +In the symmetric case π/4 and different squeezing pa- +rameters, one finds exactly expression (11) of the main +text with ϵ = 2z. +Note that if we displace ( +√ +2 cos(δ+π/4), ± +√ +2 sin(δ+ +π/4)) along xA, xB, Eq. (A10) becomes +EQ(δ) ≡ ±2 erA+rB cos(2z) cos(2δ). +(A12) +One finds that in the general case EQ(δ) is maximized +when δ = φ + (n − 1/4)π, n ∈ Z, where Sgn is the sign +function. We recover for φ = π/4 the result discussed in +Section V A, i.e., that the optimal displacement is δ = +(0, π). +2. +Shearing and phase shift operators +The mapping between the non-symmetric and symmet- +ric case through transformations (A3) and (A4) can also +be used to compute the entanglement witness EQ in the +case of the shearing and phase shift operators. +Here, +one cannot use directly the symplectic transformation +(A6) since the expression of EQ involves higher order +moments. +However, one can still insert the change of +variables (A2) in Eq. (9). +Appendix B: Optimizing the choice of quadratures +for Alice and Bob +Alice and Bob can a priori measure the joint proba- +bility distribution in any basis (ξA, ξB) = (cos φAxA − +sin φApA, +cos φBxB − sin φBpB) [as defined in the main +text, see Section II]. The question that is answered in +this section is: what is the the optimal choice for the an- +gles (φA, φB) – for each of the three operators considered +in this paper – to maximize the Fisher information, and +thus the entanglement witness E (see Eq. 4)? +The result is actually straightforward for the displace- +ment estimation. +As stated in the main text, the FI +saturates the QFI when measuring the joint probability +distribution in the plane (xA, xB). Therefore, we only +treat below the more complicated cases of the shearing +and the phase shift operators. +1. +Shearing operator +Fig. 12 shows the FI as a function of both angles φA +and φB for r = rA = rB = −0.2 and φ = π/4. The red +dot pinpoints the maximal value (≃ 5.89) obtained in this +case for φA = 7 π/20 and φB = 13 π/20. The expected +QFI for this squeezing parameter should be 3 exp(4 r) ≃ +6.67. Therefore, contrary to the displacement estimation, +it is not possible to saturate the QFI with local rotations +of the quadratures. To generalize this result, we found +numerically the maximal value reached by the FI for a +large range of squeezing parameters (from s = sA = sB = +0 dB to s ≃ 6 dB – see Fig. 13). It is clear that, here +again, the FI does not saturate the QFI (dashed blue +curve) whatever the squeezing parameter is. +0 +1 +2 +3 +4 +5 +FIG. 12. FI computed in the case of the shearing operator for +rA = rB = −0.2, φ = π/4 and for different angles φA and φB. +These angles control the local rotations of the quadratures +(see text). The red dots indicate the maximal achievable value +for the FI. +2. +Phase shift operator +We reproduce here the same procedure as in the previ- +ous section for the phase shift operator. Fig.14 shows the +FI as a function of both angles φA and φB for r = rA = +rB = 0.2 and φ = π/4. The red dot pinpoints the maxi- +mal value (≃ 4.1) obtained in this case for φA = 13 π/100 + +3.0 +2.5 +2.0 +B +1.5 +1.0 +0.5 +0.0 +0.0 0.5 +1.0 +1.5 +2.0 +2.5 +3.014 +0 +1 +2 +3 +4 +5 +6 +0 +10 +20 +30 +40 +50 +s(dB) +FI, QFI +FIG. 13. Maximal value for the FI (red) in the case of the +shearing operator bounded by the QFI (dashed blue) for dif- +ferent squeezing parameters r = rA = rB < 0. +0 +1 +2 +3 +4 +FIG. 14. FI computed in the case of the phase shift operator +for rA = rB = 0.2, φ = π/4 and for different angles φA and +φB. These angles control the local rotations of the quadra- +tures (see text). The red dots indicate the maximal achievable +value for the FI. +and φB = 87 π/100. The expected QFI for this squeez- +ing parameter should be 2 cosh2(2r)+[−3+5 cosh(4r)] ≃ +6.02. Therefore, here again, it is not possible to saturate +the QFI with local rotations of the quadratures. As in the +previous section, Fig.15 shows the maximal value reached +by the FI for a large range of squeezing parameters (from +s = sA = sB = 0 dB to s ≃ 6 dB); the FI does not satu- +rate the QFI (dashed blue curve) whatever the squeezing +parameter is. +[1] A. Einstein, B. Podolsky, and N. Rosen, Can quantum- +mechanical description of physical reality be considered +complete?. Phys. Rev. 47, 777 (1935). +[2] A. Ac´ın, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, +J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, +S. Kuhr, M. Lewenstein, M. F. Riedel, P. O. Schmidt, R. +Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm. The +quantum technologies roadmap: a European community +view. New J. Phys. 20, 080201 (2018). +[3] R. Horodecki, P. Horodecki, M. Horodecki, and K. +Horodecki. Quantum entanglement. Rev. Mod. Phys. 81, +865 (2009). +[4] S. L. Braunstein and P. van Loock. Quantum informa- +tion with continuous variables, Rev. Mod. Phys. 77, 513 +(2005). +[5] M.V. Larsen, X. Guo, C.R. Breum, J.S. Neergaard- +Nielsen and U.L. Andersen. Deterministic generation of +a two-dimensional cluster state. Science 366, 369 - 372 +(2019). +[6] W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoen- +sombutamon, H. Emura, R.N. Alexander, S. Takeda, +J. Yoshikawa, N. C. Menicucci, H. Yonezawa and A. +Furusawa. Generation of time-domain-multiplexed two- +dimensional cluster state. Science 366, 373 - 376 (2019). + +3.0 +2.5 +2.0 +B +1.5 +1.0 +0.5 +0.0 +0.0 0.5 +1.0 +1.5 +2.0 +2.5 +3.015 +0 +1 +2 +3 +4 +5 +6 +0 +10 +20 +30 +40 +50 +s(dB) +FI, QFI +FIG. 15. Maximal value for the FI (red) in the case of the +phase shift operator bounded by the QFI (dashed blue) for +different squeezing parameters r = rA = rB > 0 – phase shift +operator. +[7] L.-M. Duan, G. Giedke, J.I. Cirac and P. Zoller. Insep- +arability criterion for continuous variable systems. Phys. +Rev. Lett. 84, 2722 (2000). +[8] R. Simon, Peres-Horodecki separability criterion for con- +tinuous variable systems. Phys. Rev. Lett. 84, 2726 +(2000). +[9] V. Giovannetti, S. Mancini, D. Vitali, and P. Tombesi. +Characterizing the entanglement of bipartite quantum +systems. Phys. Rev. A 67, 022320 (2003). +[10] P. van Loock and A. Furusawa. Detecting genuine mul- +tipartite continuous-variable entanglement.it Phys. Rev. +A 67, 052315 (2003). +[11] M. Walschaers, C. Fabre, V. Parigi, and N. Treps. En- +tanglement and Wigner Function Negativity of Multi- +mode Non-Gaussian States Phys. Rev. Lett. 119, 183601 +(2017). +[12] U. Chabaud and M. Walschaers. Resources for bosonic +quantum computational advantage. arXiv 2207.11781 +(2022). +[13] R. Namiki. Photonic families of non-Gaussian entangled +states and entanglement criteria for continuous-variable +systems. Phys. Rev. A 85, 062307 (2012). +[14] M. Walschaers. Non-Gaussian states and where to find +them. PRX Quantum 2, 030204 (2021). +[15] E. Shchukin and W. Vogel. Inseparability Criteria for +Continuous Bipartite Quantum States. Phys. Rev. Lett. +95, 230502 (2005). +[16] A. Miranowicz and M. Piani. Comment on “Inseparabil- +ity Criteria for Continuous Bipartite Quantum States”. +Phys. Rev. Lett. 97, 058901 (2006). +[17] E. Shchukin and P. van Loock. Higher-order Einstein- +Podolsky-Rosen correlations and inseparability condi- +tions for continuous variables. Phys. Rev. A 93, 032114 +(2016). +[18] G. S. Agarwal and A. Biswas. Inseparability inequalities +for higher order moments for bipartite systems. New J. +Phys. 7, 211 (2005). +[19] M. Hillery and M. S. Zubairy. Entanglement Conditions +for Two-Mode States. Phys. Rev. Lett. 96, 050503 (2006). +[20] S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and +R. L. de Matos Filho. Entropic Entanglement Criteria +for Continuous Variables. Phys. Rev. Lett. 103, 160505 +(2009). +[21] H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim. Efficient En- +tanglement Criteria beyond Gaussian Limits Using Gaus- +sian Measurements. Phys. Rev. Lett. 108, 030503 (2012). +[22] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, +D.B. Hume, L. Pezze, A. Smerzi and M.K. Oberthaler. +Fisher information and entanglement of non-Gaussian +spin states. Science 345, 424 (2014). +[23] L. Hu, +M. Al-amri, +Z. Liao, +and M. S. Zubairy. +Continuous-variable quantum key distribution with non- +Gaussian operations. Phys. Rev. A 102, 012608 (2020). +[24] T. Opatrny, G. Kurizki, and D.-G. Welsch. Improvement +on teleportation of continuous variables by photon sub- +traction via conditional measurement. Phys. Rev. A 61, +032302 (2000). +[25] L. Pezze and A. Smerzi. Entanglement, nonlinear dy- +namics, and the Heisenberg limit. Phys. Rev. Lett. 102, +100401 (2009). +[26] M. Gessner, L. Pezze, and A. Smerzi. Efficient entangle- +ment criteria for discrete, continuous, and hybrid vari- +ables. Phys. Rev. A 94, 020101(R) (2016). +[27] M. Gessner, L. Pezze, and A. Smerzi. Entanglement and +squeezing in continuous-variable systems. Quantum 1, 17 +(2017). +[28] A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri and +P. Grangier. Preparation of non-local superpositions +of quasi-classical light states. Nature Phys. 5, 189-192 +(2009). +[29] P. Hyllus and J. Eisert. Optimal entanglement wit- +nesses for continuous-variable systems New J. Phys. 8, +51 (2006). +[30] S.L. Braunstein and C.M. Caves, Statistical distance and +the geometry of quantum states. Phys. Rev. Lett. 72, +3439 (1994). +[31] M.M. Nieto. Displaced and squeezed number states. +Phys. Lett. A 229, 135-143 (1997). +[32] We can use the following expressions for operators acting +on a wavefunction Ψ(x) [31] +exp[θ∂x]Ψ(x) = Ψ(x − θ), +exp[θ(x∂x)]Ψ(x) = Ψ(xeθ), +exp[θ(∂x)2]Ψ(x) = +1 +√ +4πθ +� ∞ +−∞ +exp [−(y − x)2 +4θ +]Ψ(y)dy, +where ∂x is related to the phase quadrature ˆp through +the functional relation ˆp = −2i∂x. +[33] Y.S. Ra, A. Dufour, M. Walschaers, C. Jacquard, T. +Michel, C. Fabre and N. Treps. Non-Gaussian quantum +states of a multimode light field. Nature Phys. 16, 144- +147 (2020). +[34] G.S. Agarwal. Quantum Optics. Cambridge University +Press (2013). +[35] M. Tian, Y. Xiang, F.-X. Sun, M. Fadel, and Q. +He. Characterizing Multipartite non-Gaussian Entangle- +ment for a Three-Mode Spontaneous Parametric Down- +Conversion Process. Phys. Rev. Applied 18, +024065 +(2022). +[36] U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, +V. Parigi, D. Markham, and N. Treps. Certification of +Non-Gaussian States with Operational Measurements. +PRX Quantum. 2, 020333 (2021). + +16 +[37] M. Walschaers, Y.-S. Ra, and N. Treps. Mode-dependent- +loss model for multimode photon-subtracted states Phys. +Rev. A 100, 023828 (2019). +[38] C.E. Lopetegui, M. Gessner, M. Fadel, N. Treps and M. +Walschaers. Homodyne detection of non-Gaussian quan- +tum steering. PRX Quantum 3, 030347 (2022). +[39] Z. Qin, M. Gessner, Z. Ren, X. Deng, D. Han, W. Li, X. +Su, A. Smerzi and K. Peng . Characterizing the multi- +partite continuous-variable entanglement structure from +squeezing coefficients and the Fisher information. npj +Quantum. Inf. 5, 3 (2019). +[40] J. Roslund, R. Medeiros de Araujo, S. Jiang, C. Fabre +and N. Treps. Wavelength-multiplexed quantum net- +works with ultrafast frequency combs.Nature Phot. 8, 109 +- 112 (2014). +[41] Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. +Fabre and N. Treps. Multimode entanglement in reconfig- +urable graph states using optical frequency combs.Nature +Comm. 8, 15645 (2017). +[42] M. Ansquer, V. Thiel, S. De, B. Argence, G. Gredat, F. +Bretenaker and N. Treps. Unveiling the dynamics of opti- +cal frequency combs from phase-amplitude correlations. +Phys. Rev. Research 3, 033092 (2021). +[43] D. Zhang, D. Barral, Y. Cai, Y. Zhang, M. Xiao and K. +Bencheikh. Hierarchy of nonlinear entanglement dynam- +ics for continuous variables. Phys. Rev. Lett. 127, 150502 +(2021). + diff --git a/k9E2T4oBgHgl3EQfdwdn/content/tmp_files/load_file.txt b/k9E2T4oBgHgl3EQfdwdn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c63a9578dad8c1513314dc53602549c5c21f682 --- /dev/null +++ b/k9E2T4oBgHgl3EQfdwdn/content/tmp_files/load_file.txt @@ -0,0 +1,946 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf,len=945 +page_content='Metrological detection of purely-non-Gaussian entanglement David Barral,1, ∗ Mathieu Isoard,1 Giacomo Sorelli,1, 2 Manuel Gessner,3 Nicolas Treps,1 and Mattia Walschaers1, † 1Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS, ENS-PSL Research University, Coll`ege de France, 4 place Jussieu, F-75252 Paris, France 2Fraunhofer IOSB, Ettlingen, Fraunhofer Institute of Optronics, System Technologies and Image Exploitation, Gutleuthausstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 1, 76275 Ettlingen, Germany 3Departamento de F´ısica Te´orica, IFIC, Universidad de Valencia-CSIC, C/ Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Moliner 50, Burjassot, Valencia 46100, Spain (Dated: January 10, 2023) Entanglement and non-Gaussianity are physical resources essential for a large number of quantum- optics protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Non-Gaussian entanglement is indispensable for quantum-computing advantage and outperforms its Gaussian counterparts in a number of quantum-information protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The characterization of non-Gaussian entanglement is a critical matter as it is in general highly de- manding in terms of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We propose a simple protocol based on the Fisher information for witnessing entanglement in an important class of non-Gaussian entangled states: photon-subtracted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We demonstrate that our protocol is relevant for the detection of purely-non-Gaussian en- tanglement and that it is experimentally feasible through homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' INTRODUCTION Entanglement is considered one of the most striking breakthroughs of the 20th century science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The gedanken experiment proposed by Einstein, Podolsky and Rosen in 1935 [1] pointed out the notion of inseparability of a state composed by two quantum particles spatially distanced with maximally correlated momenta and maximally anti- correlated positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Nowadays, entanglement stands as a physical resource underpinning most of current develop- ment in quantum technologies [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The efficient detection and measurement of entanglement is a very active area of quantum physics [3], being far from simple especially for continuous variable (CV) systems which involve physical quantities with a continuous spectrum of values [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Multimode squeezed states of light are the cornerstone of CV quantum networks [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' They exhibit Gaus- sian statistics and their entanglement properties are com- pletely specified by their covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Criteria and witnesses for this Gaussian entanglement have been pro- posed and tested for decades [7, 8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, Gaus- sian entanglement can always be undone with passive lin- ear optics, a phenomenon generally refereed to as passive separabiliy [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' It was recently found that one requires states that are not passively separable as a resource for a quantum computational advantage [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Because all Gaussian states are passively separable, we can always find mode bases in which the covariance matrix of the state will not show any direct signature of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Yet, if the state is not passively separable, even the modes in those bases must be entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Because this entangle- ment is fully hidden in the non-Gaussian features of the state, we will here refer to it as non-Gaussian entangle- ment [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The goal of this work is to find a practical way to detect this type of non-Gaussian entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' ∗ david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='barral@lkb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='fr † mattia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='walschaers@lkb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='upmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='fr In order to characterize non-Gaussian entanglement a number of criteria based on high-order moments and on uncertainty relations of different classes of operators have been proposed [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Nevertheless, these crite- ria are far from being feasible with current experimen- tal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Other more experimentally-friendly crite- ria are based on the Shannon entropy and the fidelity of teleportation in quantum channels [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Here we tackle the problem from an operational point of view: non-Gaussian quantum correlations are also known to im- prove metrological sensitivity, the performance of quan- tum key distribution and quantum teleportation proto- cols [22–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The advantage of relying on the improve- ment of quantum protocols is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' On the one hand the detected entanglement is useful by design, and, on the other hand, the witness comes with a natural imple- mentation: executing the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In this Article, we will focus specifically on metrological protocols, where quantum estimation tools have been devised to witness entanglement [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' These witnesses are based on the fact that metrological sensitivity determines precision of measurements and this sensitivity is limited for separable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This can be used to detect entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Two powerful assets of these sensitivity-based witnesses are i) they do not make assumptions about the quantum state –Gaussianity, purity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=', and ii) they contain informa- tion about all high-order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We adapt the approach of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [26, 27] to the ex- perimental context and limitations of CV quantum op- tics and propose a general protocol based solely on ho- modyne detection, using both the variance of the mea- surement outcomes and the joint measurement statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Our protocol is efficient in terms of resources as the pa- rameter estimation is done in postprocessing using solely the data collected by homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We show its relevance analyzing an important class of non-Gaussian entangled states: photon-subtracted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We demon- strate that our protocol is pertinent for the detection of non-Gaussian entanglement and that it is experimentally arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='03909v1 [quant-ph] 10 Jan 2023 2 feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The article is organized as follows: We first present our protocol to detect entanglement through homodyne detection and postprocessing of the joint probability dis- tribution based on the metrological witness introduced in [26, 27] in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We then present in Section III the probe states we will use to test our non-Gaussian entanglement witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In Section IV we analyze which parameter is best suited to measure entanglement in our metrological protocol and calculate entanglement in an ideal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In Section IV we study a realistic case taking into account unbalanced input squeezing, losses and discretization of the measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We fi- nally discuss possible experimental implementations of our scheme, their limitations and feasibility in Section VI and we present our conclusions in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' ENTANGLEMENT DETECTION VIA LOCAL HOMODYNE DETECTION AND POSTPROCESSING We consider here the following problem: two experi- menters, Alice and Bob, who share an optical quantum state ˆρAB, want to elucidate if their shared state is entan- gled or not, while minimizing the amount of experimental resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' If the input state is Gaussian, they just need to measure the variances of linear combinations of optical- field quadratures and apply second-order moment-based criteria like for instance those of Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=', Simon or Giovanetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This can be easily imple- mented experimentally using homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' How- ever, the larger class of non-Gaussian states do not always present entanglement that can be revealed by second- order moment-based criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Particularly, the majority of entanglement criteria for quantum states with purely non-Gaussian correlations are based on either carrying out full quantum-state tomography [28] or measuring high-order moment correlations, protocols which are very demanding experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Here, we apply a metrological protocol to detect entan- glement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Alice and Bob share information in order to es- timate jointly a parameter θ generated by a Hamiltonian ˆH = ˆHA + ˆHB that acts locally on both subsystems such that ˆρθ AB = e−iθ ˆ H ˆρABeiθ ˆ H (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The metrolog- ical protocol consists of measuring the Fisher information (FI) defined as F(P(ξA, ξB|θ)) = � R2 P(ξA, ξB|θ) �∂L(ξA, ξB|θ) ∂θ �2 d2ξ, (1) where d2ξ = dξA dξB, L(ξA, ξB|θ) = log(P(ξA, ξB|θ)) represents the logarithmic likelihood related to the prob- ability density P(ξA, ξB|θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The latter quantity repre- sents the conditional probability to obtain a set of lo- cal measurement outcomes (ξA, ξB) given the parame- ter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The probability P(ξA, ξB|θ) can be rewritten as Tr[ˆρθ AB ˆΠξ], where ˆΠξ = |ξA, ξB⟩⟨ξA, ξB| is a positive- operator valued measure (POVM) such that � ˆΠξd2ξ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In our case, as illustrated in Figure 1, the observ- ables will correspond to local homodyne measurements ˆξA = cos φAˆxA+sin φAˆpA and ˆξB = cos φBˆxB+sin φB ˆpB, where φA, φB are two angles, and ˆxA, ˆxB, ˆpA, ˆpB are the amplitude and phase quadratures defined from the anni- hilation operators as ˆaI = ˆxI + iˆpI 2 , I ∈ {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (2) The quadrature operators thus satisfy the commutation rules [ˆxI, ˆpJ] = 2iδIJ, I, J ∈ {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Then, if ˆρAB is separable, the FI of Equation (1) for a state ˆρθ AB generated by ˆH is upper bounded by [26, 27] F(P(ξA, ξB|θ)) ≤ 4Var[ˆρA, ˆHA] + 4Var[ˆρB, ˆHB], (3) where ˆρA/B are the reduced density matrices for systems A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Because this is a necessary condi- tion for separability, its violation is a sufficient criterion for entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Therefore, we can introduce the following metrological witness of entanglement E = F(P(ξA, ξB|θ)) − 4(Var[ˆρA, ˆHA] + Var[ˆρB, ˆHB]) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (4) This inequality can reveal entanglement but not its origin –Gaussian or non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' From now on we will refer to non-Gaussian entanglement as the entanglement that is not detected by Gaussian entanglement witnesses based on second order moments –the covariance matrix– such as Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=', Simon and Giovanetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' criteria [7–9] or optimized witnesses such as presented in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The real interest of the witness (4) is that it also holds for any state, pure or mixed, and its major asset is the practicability of its computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Homodyne measure- ments in each mode with a common phase reference allow us to access experimentally i) the joint probability distri- bution P(ξA, ξB|θ), and thus the FI, and ii) the variances associated to the local generators, enabling to test the en- tanglement witness given by Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Moreover, in some cases (see Section IV) the parameter-dependence of the joint probability distribution P(ξA, ξB|θ) can be generated in postprocessing applying appropriate trans- formations directly to the joint probability distribution as P(ξA, ξB|θ) = P(Uθ(ξA), Uθ(ξB)), with Uθ(ξA/B) the transformation related to the Hamiltonian ˆHA/B in the quadrature space ξA/B [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This important feature avoids to apply impractical inline transformations to the state simplifying greatly the detection of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The entanglement witness (4) can be maximized by choosing an optimal measurement observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' It is well known in quantum metrology that the ultimate precision on the parameter θ is limited by the quantum Fisher In- formation FQ (QFI), which represents the sensitivity of the full quantum state ρAB to small perturbations gener- ated by ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' As a consequence, the FI is bounded by the 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Sketch of the proposed metrological protocol for entanglement detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Alice and Bob share a quantum state ˆρAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' They jointly estimate a parameter θ generated by two local Hamiltonians ˆHA/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Using two homodyne de- tectors with a common phase reference, Alice and Bob can re- trieve the parameter-dependent joint probability distribution P(xA, xB|θ), and thus the Fisher information related to this parameter estimation, and the local variances of the Hamil- tonians ˆHA/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' With this in hand, Alice and Bob can jointly compute the metrological witness of entanglement of Equa- tion (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' QFI as FQ[ˆρAB, ˆH] = max ˆΠ F(Tr[ˆρθ AB ˆΠ]), (5) which means that the entanglement witness (4) is max- imized when the FI saturates the QFI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=', when the measurement observable is optimized [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Note that we restrict ourselves to a POVM ˆΠξ corresponding to homo- dyne measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Thus, the FI related to this mea- surement observable does not saturate the QFI for every generator ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For pure states we can easily obtain the QFI from the variance of the generator ˆH of the parameter θ as FQ[ρAB, ˆH] = 4Var[ˆρAB, ˆH].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Applying this identity into Equation (4) we obtain the following simple condition for entanglement EQ ≡ max ˆΠ E = 8 Cov[ρAB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' ˆHA, ˆHB] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (6) This inequality for pure states should not come as an absolute surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' After all, any correlation that is seen in a bipartite pure state is a signature of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' APPLICATION TO PHOTON-SUBTRACTED STATES The protocol described in the previous section is valid for any CV system, regardless of the nature of the state under consideration, as long as one has access to the probability distributions of each subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In this section we introduce the states that we will use as a probe of our non-Gaussian entanglement criterion, namely, photon-subtracted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In particular we will analyze bipartite states without Gaussian correlations in order to focus on their non-Gaussian features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We consider two-mode photon subtracted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This class of states has been demonstrated in optical systems using different degrees of freedom, such as polarization or frequency modes [28, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In Section VI we will explain in detail different experimental methods for their pro- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Let us consider two independent single-mode squeezed states respectively related to Alice and Bob |Ψ0⟩ = ˆSA(rA, θA) ˆSB(rB, θB)|00⟩, (7) where ˆSI(rI, θI) = exp{(−rI/2)(ˆa2 Ie−2iθI − ˆa†2 I e2iθI)} is the single-mode squeezing operator, and rI ∈ R+ and θI ∈ R are respectively the squeezing parameter and squeezing phase for each mode I = A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The amount of squeezing in decibels is given by sI = 10 log10(e−2rI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In what follows we analyze two cases: in-phase squeez- ing (θB = θA) and in-quadrature squeezing (θB = θA + π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Without loss of generality we set θA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We include the information about the squeezing phase by extending the domain of the squeezing parameter to rI ∈ R such that ˆSI(rI, θI) ≡ ˆSI(rI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Thus, Equa- tion (7) corresponds to a Gaussian state and all its information is encoded in the covariance matrix V0 = diag(e−2rA, e2rA, e−2rB, e2rB), written with respect to the vector of amplitude and phase quadratures in each mode ⃗ξ = (xA, pA, xB, pB)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Note that V0 does not present off-diagonal terms, thus the input Gaussian state is fully separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Next, we perform a delocalized subtraction of one pho- ton on this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This operation produces a superposi- tion of two-mode squeezed vacuum and squeezed single- photon states that one can show that up to normaliza- tions is [34] |Ψ⟩ ∝ (cos(φ)ˆaA + sin(φ)ˆaB)|Ψ0⟩ = ˆSA(rA) ˆSB(rB)(cos(φ) sinh(rA)|10⟩ + sinh(rB) sin(φ)|01⟩), where the parameter φ controls the probability of sub- traction in each mode and we have considered in-phase subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A sketch of this operation is shown in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The wavefunction of this state in the amplitude quadratures of the optical field is given by Ψ(xA,xB) ≡ ⟨xA, xB|Ψ⟩ ∝ e− e2rA x2 A+e2rB x2 B 4 × ((e2rA − 1) cos (φ)xA + (e2rB − 1) sin (φ)xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (8) Examples of joint probability distributions P(xA, xB) = |Ψ(xA, xB)|2 for a photon subtracted state given by Equation (8) with φ = π/4 and rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2, rA = −rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2, are respectively shown in Figure 3 a) and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' ALICE i0H P(CA,CBO) pAB LO Var[βA, HA] Var[pB, HB] BOB4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Sketch of an optical setup for delocalized single- photon subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Alice and Bob prepare two squeezed states in given optical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A small fraction of each mode power is diverted to a common beam splitter with a trans- mittivity controlled by a parameter φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' An event measured by a single-photon detector heralds the subtraction of a photon delocalized between the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The local Hamiltonians ˆHA/B are in general polyno- mials of amplitude x and phase p local quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The separability bound related to their variances can be cal- culated using wavefunctions via ⟨ˆxn i ˆpm j ⟩ = (9) (−2i)m �� R xn i Ψ(xA, xB)∗ ∂mΨ(xA, xB) ∂xm j dxAdxB, where the functional relation ˆpj → −2i∂/∂xj is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The entanglement present in these states is not grasped by Gaussian entanglement witnesses: this can be gener- ally understood from the covariance matrix of a photon- subtracted state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [11] shows that this covariance matrix can generally be written as V = V0 + 2(V0 − 1)P(V0 − 1) Tr[(V0 − 1)P] , (10) where V0 is the initial Gaussian state’s covariance matrix and P is a matrix that projects on the phase space axes associated with the mode of photon subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In our present case, we find that P = � � � � cos2(φ) 0 1 2 sin(2φ) 0 0 cos2(φ) 0 1 2 sin(2φ) 1 2 sin(2φ) 0 sin2(φ) 0 0 1 2 sin(2φ) 0 sin2(φ) � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Thus, we see in Equation (10) that on the level of the co- variance matrix the photon subtraction only adds Gaus- sian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This implies that no additional entangle- ment can be witnessed by purely looking at the covari- ance matrix [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' As a consequence, since we set V0 = diag(e−2rA, e2rA, e−2rB, e2rB), we find that V should not display any entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' � -� -� � � � � � � � � � � � �� �� �) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='14 � -� -� � � � � � � � � � � � �� �� �) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Contour plots of the joint probability distribution for a photon subtracted state given by Equation (8) with φ = π/4 and a) rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2, b) rA = −rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 (|sA/B| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='74 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' IDEAL DETECTION OF NON-GAUSSIAN ENTANGLEMENT In order to decide which Hamiltonian ˆH is best suited to witness entanglement we can calculate theoretically EQ through Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This can guide us decid- ing which parameter is best suited to detect entangle- ment of a given quantum state in a realistic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Below, we use the estimation of parameters related to the four single-mode Gaussian gates in CV quantum op- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Namely: displacement, phase-shift, shearing and squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We analyze in which cases the joint estima- tion of these parameters reveals the entanglement of the non-Gaussian two-mode photon-subtracted state given by Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For the sake of simplicity, we focus here on the case φ = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A generalization to any φ is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' ALICE pA pAB pB BOB5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Displacement A displacement of θ along the axis xA = ±xB is pro- duced by the following operator ˆD(θ) = e−iθ(ˆpA±ˆpB)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The Hamiltonian related to this displacement operator is H± = (ˆpA ± ˆpB)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The optimal entanglement EQ ob- tained estimating displacement along the axis xA = ±xB for a pure photon-subtracted state given by Equation (8) with φ = π/4 and squeezing parameters rA ̸= rB is EQ = ±2erA+rB cos(ϵ), (11) with cos(ϵ) = 2 sinh(rA) sinh(rB) sinh2(rA) + sinh2(rB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Displacement along either xA = xB or xA = −xB detects entanglement respectively for in-phase squeezing (rA, rB > 0) and in-quadrature –orthogonal– squeezing (rA > 0, rB < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Figures 4a and 4b show contour plots of optimal entanglement EQ in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' States with in-phase input squeezing show always a larger degree of entanglement due to the argument rA + rB in Equa- tion (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For in-phase squeezing the witness reaches the maximum at rA = rB (dashed line along the diag- onal in Figure 4a) and is given by EQ = 2e2rA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Like- wise, for in-quadrature squeezing the maximum value of EQ is not along the diagonal, but below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For a given value of rA, the maximum EQ is obtained for rB = log (1/(1 + 2 sinh (rA))1/2) (dashed line in Figure 4b) and is given by EQ = 2 erA 1 + sinh (rA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The shapes of Figures 4a and 4b can be explained in terms of the symmetries of the two functions that com- pose Equation (11): ± cos(ϵ) is a symmetric function with respect to the diagonal sA = sB for every input squeezing, whereas 2erA+rB is symmetric with respect to the diagonal (antidiagonal, in this case along sA−sB = 6 dB) for in-phase (in-quadrature) squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Importantly, we obtain the same result calculating the entanglement through Equation (4), E = EQ, indicating that the FI saturates the QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The result of Equation (11) is particularly interesting because, following Equa- tion (6), second order moments of the distribution reveal entanglement with a non-Gaussian origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Recently, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' analyzed the multipartite en- tanglement in a nondegenerate triple photon state using a metrological criterion [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' They claimed there that non-Gaussian entanglement cannot be sufficiently cap- tured by linear quadratures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' While this is the case for triple photon states, we have shown that it does not hold in general: displacements can detect non-Gaussian entanglement of photon-subtracted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' � � � � � � � � � � � � � � ��(��) ��(��) 1 2 3 4 5 6 7 � � � � � � � � � � � � � � ��(��) ��(��) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='25 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Optimal displacement-estimation entanglement EQ given by Equation (11) versus squeezing of the input squeezed states sA and sB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' a) Displacement along xA = xB optimizes EQ for states with in-phase input squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' b) Displacement along xA = −xB optimizes EQ for states with in-quadrature input squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Maximum value of EQ in dashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phase shift The phase-shift operator ˆR(θ) = e−iθ( ˆ NA± ˆ NB) ∝ e−iθ(ˆx2 A+ˆp2 A±ˆx2 B±ˆp2 B)/4 rotates the state by a phase θ in local phase sub- spaces in either clockwise-clockwise (+) or clockwise- counterclockwise (−) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The related Hamilto- nian is H± = ˆNA ± ˆNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The optimal entanglement wit- ness is in this case EQ = ∓2 cosh(2rA) cosh(2rB) cos2(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (12) Entanglement is always detected for clockwise- counterclockwise (−) phase shifts, but not for clockwise- a)b)6 � � � � � � � � � � � � � � ��(��) ��(��) 1 2 3 4 5 6 7 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Optimal phase-estimation entanglement EQ given by Equation (12) versus squeezing of the input squeezed states sA and sB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Maximum value of EQ in dashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' clockwise (+) as it is just a global phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Figure 5 shows contour plots of optimal entanglement EQ for different values of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Notably, the detection of entanglement does not depend on the phase of the input squeezed states as EQ is invariant under change of sign of the squeezing parameters rA/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The detected entanglement is maximum for rA = rB (dashed line along the diagonal in Figure 5) being EQ = 2 cosh2(2rA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' One can wonder if the Fisher information in Equa- tion (4) reaches the QFI in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' While measuring the joint probability distribution in the (x1, x2)−plane was enough to obtain the maximal value of the FI and saturates the QFI for the displacement estimation, here the situation is a bit more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For simplicity, we consider the case rA = rB in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The FI can be optimized by finding the set of angles (φA, φB) of the measurement outcomes ξA = cos φAxA − sin φApA, ξB = cos φBxB − sin φBpB for which the joint probabil- ity distribution P(ξA, ξB|θ) leads to the best value of the FI (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, we find that such local rotations are not enough to saturate the QFI, and that only a mixing of modes A and B before the homodyne detectors can lead to a saturation of the QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' It is in- deed possible to prove that a non-local rotation of −π/4 between modes A and B and measuring the joint proba- bility distribution P(x′ A, p′ B|θ), with x′ A = (xA −xB)/ √ 2 and p′ B = (pA + pB)/ √ 2 is needed to saturate the QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Shearing The shearing –also known as phase-gate– operator ˆS(θ) = e−iθ(ˆx2 A±ˆx2 B)/4 shears the state with respect to the axes xA and ±xB by a gradient of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The related Hamiltonian is H± = (ˆx2 A ± ˆx2 B)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The optimal entanglement is in this case EQ = ∓e−2(rA+rB) 2 cos2(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (13) Thus, shearing with respect to xA and xB does not de- tect entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, shearing with respect to xA and −xB captures it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Note that in this case the entangle- ment is maximized for rA/B < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' squeezing along the quadratures pA/B, unlike displacement and phase estima- tion where EQ is maximized for squeezing along xA/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Figures 6a and 6b show contour plots of optimal en- tanglement EQ in the cases of input squeezing along the same quadrature (a) or along different quadratures (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For input squeezing along the same quadratures the detected entanglement is maximum again for rA = rB (dashed line along the diagonal in Figure 6a) and given by EQ = e−4rA/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, the maximum entangle- ment is below the diagonal for input squeezing along different quadratures as for displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For a given value of rA, the maximum EQ is obtained for rB = (−rA + log(1 + erA − e2rA))/2 (dashed line in Figure 6b) and is given by EQ = e−2rA 2(−1 + sinh(rA))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The shape of Figures 6a and 6b is explained in the same way as for displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Here again, we optimize the FI to see if it is possible to reach the bound E = EQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The same analysis as in the case of the phase-shift operator (by performing local ro- tations before the homodyne detection) is summarized in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The same conclusion follows: the FI never reaches the QFI, and only a non-local rotation of -π/4 between modes A and B leads to a saturation of the QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Squeezing The squeezing operator ˆS(θ) = e−iθ(ˆxA ˆpA+ˆpAˆxA±ˆxB ˆpB±ˆpB ˆxB)/4 squeezes the position quadratures of modes A and B by a factor of eθ (+) or squeezes the position quadratures of A by eθ and stretches those of B by e−θ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The related Hamiltonian is H± = (ˆxAˆpA + ˆpAˆxA ± ˆxB ˆpB ± ˆpBˆxB)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The optimal entanglement is here EQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Interestingly, the joint estimation of the squeezing pa- rameter does not detect entanglement in any of the above two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 7 � � � � � � � � � � � � � � ��(��) ��(��) �) 1 2 3 4 5 6 7 � � � � � � � � � � � � � � ��(��) ��(��) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Optimal shearing-estimation entanglement EQ given by Equation (13) versus squeezing of the input squeezed states sA and sB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' a) EQ for states with input squeezing along the same quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' b) EQ for states with input squeezing along different quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Maximum value of EQ in dashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Comparison and resource evaluation In order to decide which parameter-estimation strat- egy is best suited to detect entanglement we show in Figure 7 the evolution of maximum entanglement EQ ver- sus amount of squeezing in dB for the above four joint parameter estimations in the case of in-phase and in- quadrature input squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For in-phase input squeez- ing the maximum entanglement is obtained for rA = rB (dashed line along the diagonal in Figures 4a, 5 and 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For in-quadrature input squeezing the maximum entanglement is obtained for rA = −rB for phase-shift (dashed line along the diagonal in Figure 5), and for rB = log (1/(1 + 2 sinh (rA))1/2) and rB = (−rA+log(1+ � � � � � � � � � � � � |��|(��) �� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Maximum optimal entanglement EQ versus squeez- ing in Alice’s mode (dashed curves of Figures 4,5, and 6): displacement estimation (blue), shearing estimation (or- ange), phase shift estimation (green), and squeezing estima- tion (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In-phase (in-quadrature) input squeezing in solid (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For phase shift estimation (green) the curve is the same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' EQ > 0 witnesses entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' erA−e2rA))/2 for displacement and shearing, respectively (dashed line below the diagonal in Figures 4b and 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Remarkably, for values of input squeezing lower than ≈ 5 dB, the best strategy is to jointly estimate the displace- ment (solid, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For larger values of input squeezing, phase shift and shearing estimation offer a greater sensi- tivity to entanglement (green and orange, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' On the contrary, as we saw above the joint estimation of the squeezing parameter does not offer any information on the entanglement of this state (solid, gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In terms of resources, displacement estimation is also advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Both probability distributions and quadrature variances corresponding to Alice and Bob can be directly measured with homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Like- wise, shearing estimation can be performed with homo- dyne detection, but fourth-order moments of the distri- butions (kurtosis) are necessary, which implies in general larger data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In the case of phase estimation, photon- number variances are necessary, which implies adding complexity to the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Another great advantage of displacement estimation is that the displacement operation can be applied in post- processing: once the probability distribution P(xA, xB|0) is measured, the displaced probability distribution [un- der the Hamiltonian ˆH± = (ˆpA ± ˆpB)/2] is directly given by P(xA, xB|θ) = P(xA + θ, xB ± θ|0) [32], from which one can compute the classical FI (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' V C) – which we know saturates the QFI in this case, leading the best possible estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' On the contrary, the shearing and phase-shift operations can not be implemented in post- processing using just the probability distribution as full information about the state is needed for such operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Thus, shearing and phase-shift unitaries have to be im- plemented at the level of the experimental setup or by b)8 post-processing after measuring the full quantum state by for instance double-homodyne detection [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In ad- dition to this complication, contrary to the displacement operation, as we pointed out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' IV B and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' IV C, the FI can be optimized with local rotations of the mea- sured quadratures, but only saturates the QFI when one mixes modes A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' REALISTIC DETECTION OF NON-GAUSSIAN ENTANGLEMENT In this section we study the measurement of entan- glement in a realistic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' As we found above, es- timating displacement is the best strategy for ideal de- tection at moderate values of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Moreover, it is the simplest one, as the variances of the generators –field quadratures– are directly measured with homodyne de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We thus focus on this option in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A similar analysis could be carried out for shearing and phase-shift estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Below we analyze the effect of un- balancing the sensitivity in the displacement estimation, the effect of losses on the detection of entanglement and the discretization of the sampled data to build a joint probability distribution and calculate the Fisher infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Optimization of displacement axis for entanglement witness In the previous section, we analyzed the detection of entanglement through displacement estimation when dis- placing the input state along the axes xA = ±xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' How- ever, we can optimize the entanglement detection displac- ing the input state along an axis different to xA = ±xB or, in other words, unbalancing the sensitivity related to Alice and Bob in the joint parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The idea is the following: instead of displacing the same amount (1, ±1) in both amplitude quadratures, we dis- place ( √ 2 cos(δ + π/4), ± √ 2 sin(δ + π/4)) along xA and xB, respectively, where δ ∈ [0, π] is an angle that we can optimize for each pair of values of rA and rB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This leads to a new Hamiltonian ˆH± = (cos(δ + π/4)ˆpA ± sin(δ + π/4)ˆpB)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Calculating the optimal entanglement EQ of Equation (6) we find now Eδ Q = EQ cos(2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (14) Therefore, displacing along xA = ±xB (δ = 0, π) is in- deed the optimal strategy and displacement along any other axis can only degrade the detection of entangle- ment since | cos(2δ)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Optical losses The effect of optical losses can be entirely absorbed by the covariance matrix when it is the same in both modes [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The covariance matrix of the input squeezed state V0 is modified in the following way Vη = (1 − η)V0 + η1, where η represents the amount of losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For instance, the probability distribution related to the quantum state given by Equation (8) with φ = π/4 and rA = rB ≡ r is now Pη(xA, xB) ∝ e− x2 A+x2 B 2σ2 (2ηe2rσ2 + (1 − η)(xA + xB)2), with σ2 = (1 − η) e−2r + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A similar but less straight- forward result is obtained for general values of φ, rA and rB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Figure 8 shows the effect of losses on the detection of entanglement for photon-subtracted states with φ = π/4 and sA = sB (Figure 8a), sA = 1 dB and varying sB (Figure 8b), and sA = 2 dB and varying sB (Figure 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In Figure 9 we use the same state but now with sA = −sB (Figure 9a), sA = 1 dB and varying sB (Figure 9b), and sA = 2 dB and varying sB (Figure 9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In general, the effect of losses increases with the amount of input squeezing, and the metrological entan- glement is more resilient for input squeezing along differ- ent quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For sA > 0, sB > 0, the metrological detection of entanglement is resilient up to ≈ 20% for in- put values of squeezing between 1 and 2 dB, whereas for sA > 0, sB < 0, the metrological detection of entangle- ment is resilient up to ≈ 70% for highly asymmetric in- put squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Moreover, entanglement is more resilient to losses in comparison with quantum steering, where the losses threshold is about 7% for the same states [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' It must be emphasised that our entanglement witness detects only entanglement related to the metrological sensitivity of the state [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The losses produce quan- tum decoherence and impair the metrological power of the quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We have checked that other entangle- ment witness, such as the logarithmic negativity, detect entanglement in regions where our metrological witness cannot, but that entanglement is not a useful resource for parameter estimation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Discretization of sampled data We need to obtain experimentally the FI and the vari- ances associated to the local displacement generators Var(ˆpA/B) in order to compute the entanglement witness E of Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The variances are directly obtained measuring the phase quadratures with homodyne detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Estimating the FI experimentally from discrete out- comes, in contrast with the theoretical computation that assumes a continuum of outcomes, relies on the compu- tation of a statistical distance –the Hellinger distance– between a reference probability distribution and the parameter-dependent one [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The squared Hellinger distance between a parameter-dependent probability dis- tribution P(xA, xB|θ) and a reference P(xA, xB|0) is de- 9 1dB 2dB 3dB 4dB 5dB 6dB ���� ���� ���� ���� ���� ���� ���� � � � � � η � �) ��=��= 4dB 5dB 6dB 7dB 8dB ���� ���� ���� ���� ���� ���� ���� ��� ��� ��� ��� η � �) ��=���� ��= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5dB 1dB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5dB 2dB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5dB 3dB ���� ���� ���� ���� ���� ���� ���� � � � � η � �) ��=���� ��= FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Effect of losses η on displacement-estimation-based entanglement for a photon-subtracted quantum state with φ = π/4 and a) sA = sB (legend), b) sA = 1 dB, varying sB (legend), and c) sA = 2 dB, varying sB (legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' E > 0 witnesses entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' fined as d2 H,P(θ) = 1 2 �� R ( � P(xA, xB|θ) − � P(xA, xB|0))2dxAdxB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The Taylor expansion of the squared Hellinger distance to second order yields [22] d2 H,P(θ) = F 8 θ2 + O(θ3), 1dB 2dB 3dB 4dB 5dB 6dB ���� ���� ���� ���� ���� ���� ���� ��� ��� ��� ��� ��� η � �) ��=-��= 4dB 5dB 6dB 7dB 8dB ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� η � �) ��=���� ��= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5dB 1dB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5dB 2dB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5dB 3dB ���� ���� ���� ���� ���� ���� ���� ��� ��� ��� ��� ��� η � �) ��=���� ��= FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Effect of losses η on displacement-estimation-based entanglement for a photon-subtracted quantum state with φ = π/4 and a) sA = −sB (legend), b) sA = 1 dB, varying sB (legend), and c) sA = 2 dB, varying sB (legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' E > 0 witnesses entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' with F ≡ F(ˆρAB, ˆH) the FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Thus, a quadratic fit is enough to calculate the FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, in an experimental implementation we do not have exact probability distributions P(xA, xB|θ), but rel- ative frequency distributions {F(xA, xB|θ)}, which ap- proach the probability distributions for infinitely many independent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In this case, due to sta- tistical fluctuations δF, the squared Hellinger distance varies when repeating the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Taking the out- come frequencies from a sample of M experimental re- 10 alizations, the sample average of the squared Hellinger distance between two relative frequencies d2 H,F(θ) is ap- proximately [22] ⟨d2 H,F(θ)⟩ = c0 + (F 8 + c2)θ2 + O(θ3, δF3), (15) with c0 = (n − 1)/4M, c2 ≈ F(1 + n)/32M and n the number of pairs (xA, xB) for which F(xA, xB|θ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Note that ⟨d2 H,F(θ)⟩ converges asymptotically to d2 H,P(θ) as M → ∞ and hence the estimation of F is asymptoti- cally unbiased with the bias decreasing as M −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In the following we study the protocol by simulating homodyne detection with rejection sampling of the theo- retical probability distributions obtained from Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We partition the real line corresponding to the out- comes of the quadrature measured by Alice and Bob in a series of bins with a given bin size ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We consider an even number of bins as the mean value of the field is zero for our non-Gaussian probe state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Figure 10 shows two examples of sampled joint relative frequency distribu- tions {F(xA, xB)} obtained through rejection sampling of the probability distribution given by Equation (8) for rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 (Figure 3a) and rA = −rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The number of samples is M = 5 × 105 and the bin size ∆=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 (in the units of xA/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We list below the steps to follow in order to calculate the FI: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' we take the two sets of sampled data corresponding to Alice ⃗xA and to Bob ⃗xB and split the sampled data (⃗xA,⃗xB) of total size M in two equal sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' we bin the data in areas of given size and compute the relative frequencies {F(xA, xB|0)} of the first set that is used as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' we displace the data of the second set by an amount θ –the displacement parameter–, bin the data and compute the relative frequencies {F(xA, xB|θ)} that are used as a probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' we calculate the square root of each relative fre- quency for the reference and the displaced data, take the difference and square it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' we calculate the sample average of the squared Hellinger distance ⟨d2 H,F(θ)⟩ for a value of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' we repeat this process for different values of θ and fit the results to a parabola, obtaining the FI with its statistical error through Equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Using this value of FI and the sum of the variances of the phase quadratures we calculate the entanglement witness E through Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We show in Figure 11 the effect of data discretiza- tion and number of samples in the detection of entan- glement for lossless and lossy cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We use the quan- tum state given by Equation (8) with rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 (rA = −rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2) sampled in Figure 10a (10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In each FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Sampled joint relative frequency distributions F(xA, xB) obtained through rejection sampling of the prob- ability distribution of Figures 3 a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' a) rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 and b) rA = −rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 5 × 105 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Bin size ∆=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 (in units of xA/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' figure, the upper curves are for the lossless case, whereas the lower curves are for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We displace our second data set of size M/2 between θ ∈ {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='05} in steps of 5×10−3, resulting in 20 data points that we fit with a parabola using Equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We partition the outcome quadratures measured by Alice and Bob in a series of bins of size ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We perform 30 simulations for each value of bin size and total number of samples to obtain statistical averages and errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We show the value of entanglement E obtained using a continuous probability distribution in solid gray, and the values and errors obtained for differ- ent bin size ∆ and total samples M in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The colors represent different number of samples: M = 106 (blue), M = 2×106 (orange), M = 4×106 (green) and M = 107 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We find that the distance between the computed value from the simulated data and the theoretical value decreases as the bin size shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For large bin size, the number of samples does not affect significantly the ac- curacy of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, for smaller bin size, the accuracy of the discretized estimation raises as F(XA,XB) a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='15 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='05 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='00 4 0 XB 2 2 0 XA 2 4F(XA,XB) b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='10F 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='05 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='00 4 0 XB 2 2 0 XA 2 411 ��� ��� ��� ��� ��� � � � � � ��� ���� � �) ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ���� � �) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Effect of bin size ∆, number of total samples M and losses η on the entanglement estimation E for a photon- subtracted quantum state with φ = π/4 and a) rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2, b) rA = −rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 (|sA/B| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='74 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In each figure, the upper curves are for the lossless case, whereas the lower curves are for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Averages and errors are calculated over 30 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' E > 0 witnesses entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Solid, gray: theoretical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Blue: 106 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Orange: 2 × 106 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Green: 4 × 106 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Red: 107 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' the number of samples increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In general, the statis- tical error obtained from the fit is lower as the bin size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Note that a large discretization with an insuf- ficient number of points can lead to an overestimation of the entanglement E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We find that in the lossless case the estimation is in good agreement with the theoretical value for M ≥ 2 × 106 and ∆ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In the lossy case, more samples are necessary for the same value of bin size ∆ and overestimation is more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' To not over- estimate the entanglement we should use M ≥ 2 × 106 and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Notably, in both cases we detect entangle- ment even using a coarse-grained bin size ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='4 and a relatively low number of samples M = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' DISCUSSION Let us discuss possible practical implementations of this protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' There are few approaches depending on the degree of freedom –or mode– selected to encode the quantum information: path, polarization, frequency and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The shared feature of the input modes is that they are independent and excited in squeezed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' An event measured by a single-photon detector fed by a small frac- tion of power from Alice and Bob’s modes where which- mode information is erased, heralds the subtraction of a photon delocalized between the two modes [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Two balanced homodyne detectors with a common local os- cillator LO retrieve then the joint probability distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The sketch of Figure 1 is pretty accurate for path- encoded modes where a common beam splitter erases the which-path information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In the case of spectral modes where the number of modes is usually larger than two –for instance in a multimode frequency-comb Gaussian resource [40, 41]– mode-selective photon-subtraction is accomplished by sum-frequency generation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The detection of an up- converted photon heralds the subtraction of a photon from a multimode input state in one or various spectral modes selected by a pump suitably tailored in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The joint probability distribution of photon-subtracted spectral modes can be retrieved by spectrally-resolved homodyne detection [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This approach allows to mea- sure simultaneously the quadratures of the electric field in a number of frequency-band modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Then, applying a change of basis between the photon-subtracted spectral modes and these frequency-band modes one retrieves the quadrature traces in the modes of interest and hence, the joint probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Moreover, we outline that in an experiment, in order to prove that the entanglement results entirely from the non-local photon subtraction, one would use the data from the unconditioned state to test our entanglement witness and demonstrate the independence of the two input squeezed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Finally, comparing our simulations with the values measured by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [33], where the squeezing of the first and second spectral modes is sA = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='3 dB and sB = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='7 dB, respectively, with purities above 90% and detection losses of the order of 12%, we conclude that with a reasonable number of samples (≈ 106) it is pos- sible to witness non-Gaussian entanglement using exclu- sively homodyne detection with an experimentally feasi- ble protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Moreover, we have found that entanglement of modes with highly asymmetric input squeezing is re- silient versus losses (Fig 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This can be advantageous for entanglement detection in photon-subtracted spectral multimode states for instance between the first and the third spectral modes, where the input squeezing is highly asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 12 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' CONCLUSIONS AND OUTLOOK We proposed a protocol based on Fisher information for witnessing entanglement in an important class of non- Gaussian states: single photon-subtracted CV states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The protocol is based on the metrological entanglement criterion proposed in [26], and its strength comes from its simplicity, as it relies solely on homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Our approach witnesses entanglement not detected by Gaus- sian criteria, like for instance Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' criterion, using the same resources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' quadrature measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We characterized the optimal metrologically-useful en- tanglement of single photon-subtracted states analyzing their metrological power in estimation of parameters gen- erated by all single-mode Gaussian gates, namely: dis- placement, phase shift, shearing and squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We ana- lyzed displacement estimation in details since it gives the largest sensitivity for currently experimentally-relevant values of squeezing (≤ 5 dB) and it can be applied in postprocessing, thus minimizing the resources necessary in non-Gaussian entanglement characterization and out- performing other protocols where quantum-state tomog- raphy is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We demonstrated that our protocol is relevant and ex- perimentally feasible using data from a simulated exper- iment where the effect of losses, data discretization, and number of samples were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Our results show that non-Gaussian entanglement can be detected with a feasible number of measurements and data bin- ning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' It is well known that losses impair the metrological power of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, we found that our metrology-based entanglement detection is resilient up to 70% losses in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The general setup of Figure 1 is versatile and can be used to both implement Gaussian entanglement detection protocols based on the covariance matrix and our metro- logical protocol for non-Gaussian entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' For cer- tain classes of states, we believe that this should be suf- ficient to be able to detect entanglement in any mode basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, to determine whether or not a state is passively separable, as would be required for the sam- pling protocols in [12], one would still need to certify the presence of entanglement in every possible mode basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' While our work certainly offers us a useful experimental tool, we also hope that it will be a step towards finding new techniques that allow us to certify entanglement in every possible mode basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' After all, non-Gaussian en- tangled states encompass a huge state space and we have just started to dig it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' In order to gain insight about general features of this exotic quantum feature, in future work we will analyze entangled states based on multiple- photon subtraction and connect our entanglement crite- rion with others based on higher-order covariance matri- ces [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This received funding from the ANR JCJC project NoRdiC (ANR-21-CE47-0005), the European Union’s Horizon 2020 research and innovation programme un- der Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 899587, and the Quan- tERA II project SPARQL that has received fund- ing from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No 101017733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' It was carried out during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' acknowledges funding by the Generalitat Valen- ciana (CDEIGENT/2021/014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Appendix A: General expression of EQ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Displacement operator In this appendix, we derive the general expression of EQ for a given angle φ (which controls the probability of subtraction in each mode) and for two different squeezing parameters rA and rB when the Hamiltonian is given by (ˆpA ± ˆpB)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' By performing a change of basis from (xA, pA, xB, pB) to a new set of coordinates (x′ A, p′ A, x′ B, p′ B), it is possible to map the general wavefunction (8) to the symmetric case with equal squeezing parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=', Ψ(x′ A, x′ B) ∝ e−e2r x′ A 2+x′ B 2 4 (x′ A + x′ B), (A1) where r ≡ (rA + rB)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' This change of basis consists of two operations � x′ A x′ B � = R(z) S(s) � xA xB � , (A2) with S(s) = � es 0 0 e−s � , s ≡ rA − rB 2 , (A3) and R(z) = � cos z sin z − sin z cos z � , (A4) where cos z = sinh rA cos φ + sinh rB sin φ √ 2 � sinh2 rA cos2 φ + sinh2 rB sin2 φ , sin z = − sinh rA cos φ + sinh rB sin φ √ 2 � sinh2 rA cos2 φ + sinh2 rB sin2 φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (A5) Then, using expression (9) and the change of coordinates (A2), one finds that a symplectic transformation con- nects the second-order moments associated to the non- symmetric and symmetric case: σ = Λ σ′ Λ T, (A6) 13 with σ = � ⟨ˆp2 A⟩ ⟨ˆpAˆpB⟩ ⟨ˆpAˆpB⟩ ⟨ˆp2 B⟩ � , σ′ = � ⟨ˆp′ A 2⟩ ⟨ˆp′ Aˆp′ B⟩ ⟨ˆp′ Aˆp′ B⟩ ⟨ˆp′ B 2⟩ � , (A7) and Λ = S(s)R−1(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (A8) In particular, one has ⟨ˆpAˆpB⟩ = cos(2z)⟨ˆp′ Aˆp′ B⟩ + sin(2z) � ⟨ˆp′ A 2⟩ − ⟨ˆp′ B 2⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (A9) Given that ⟨ˆp′ A 2⟩ = ⟨ˆp′ B 2⟩, we finally find the simple rela- tion EQ = cos(2z) E′ Q = ±2 erA+rB cos(2z), (A10) with cos(2z) = sinh zA sinh zB sin(2φ) sinh2 zA cos2 φ + sinh2 zB sin2 φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (A11) In the symmetric case π/4 and different squeezing pa- rameters, one finds exactly expression (11) of the main text with ϵ = 2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Note that if we displace ( √ 2 cos(δ+π/4), ± √ 2 sin(δ+ π/4)) along xA, xB, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (A10) becomes EQ(δ) ≡ ±2 erA+rB cos(2z) cos(2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (A12) One finds that in the general case EQ(δ) is maximized when δ = φ + (n − 1/4)π, n ∈ Z, where Sgn is the sign function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' We recover for φ = π/4 the result discussed in Section V A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=', that the optimal displacement is δ = (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Shearing and phase shift operators The mapping between the non-symmetric and symmet- ric case through transformations (A3) and (A4) can also be used to compute the entanglement witness EQ in the case of the shearing and phase shift operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Here, one cannot use directly the symplectic transformation (A6) since the expression of EQ involves higher order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' However, one can still insert the change of variables (A2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Appendix B: Optimizing the choice of quadratures for Alice and Bob Alice and Bob can a priori measure the joint proba- bility distribution in any basis (ξA, ξB) = (cos φAxA − sin φApA, cos φBxB − sin φBpB) [as defined in the main text, see Section II].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The question that is answered in this section is: what is the the optimal choice for the an- gles (φA, φB) – for each of the three operators considered in this paper – to maximize the Fisher information, and thus the entanglement witness E (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 4)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The result is actually straightforward for the displace- ment estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' As stated in the main text, the FI saturates the QFI when measuring the joint probability distribution in the plane (xA, xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Therefore, we only treat below the more complicated cases of the shearing and the phase shift operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Shearing operator Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 12 shows the FI as a function of both angles φA and φB for r = rA = rB = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 and φ = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The red dot pinpoints the maximal value (≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='89) obtained in this case for φA = 7 π/20 and φB = 13 π/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The expected QFI for this squeezing parameter should be 3 exp(4 r) ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Therefore, contrary to the displacement estimation, it is not possible to saturate the QFI with local rotations of the quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' To generalize this result, we found numerically the maximal value reached by the FI for a large range of squeezing parameters (from s = sA = sB = 0 dB to s ≃ 6 dB – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' It is clear that, here again, the FI does not saturate the QFI (dashed blue curve) whatever the squeezing parameter is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 0 1 2 3 4 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' FI computed in the case of the shearing operator for rA = rB = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2, φ = π/4 and for different angles φA and φB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' These angles control the local rotations of the quadratures (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The red dots indicate the maximal achievable value for the FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phase shift operator We reproduce here the same procedure as in the previ- ous section for the phase shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='14 shows the FI as a function of both angles φA and φB for r = rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2 and φ = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The red dot pinpoints the maxi- mal value (≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='1) obtained in this case for φA = 13 π/100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='014 0 1 2 3 4 5 6 0 10 20 30 40 50 s(dB) FI, QFI FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Maximal value for the FI (red) in the case of the shearing operator bounded by the QFI (dashed blue) for dif- ferent squeezing parameters r = rA = rB < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 0 1 2 3 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' FI computed in the case of the phase shift operator for rA = rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='2, φ = π/4 and for different angles φA and φB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' These angles control the local rotations of the quadra- tures (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The red dots indicate the maximal achievable value for the FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' and φB = 87 π/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The expected QFI for this squeez- ing parameter should be 2 cosh2(2r)+[−3+5 cosh(4r)] ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Therefore, here again, it is not possible to saturate the QFI with local rotations of the quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' As in the previous section, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='15 shows the maximal value reached by the FI for a large range of squeezing parameters (from s = sA = sB = 0 dB to s ≃ 6 dB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' the FI does not satu- rate the QFI (dashed blue curve) whatever the squeezing parameter is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Einstein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Podolsky, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rosen, Can quantum- mechanical description of physical reality be considered complete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='. Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 47, 777 (1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ac´ın, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Bloch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Buhrman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Calarco, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Eichler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Eisert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Esteve, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Gisin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Glaser, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Jelezko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Kuhr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lewenstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Riedel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Schmidt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Thew, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Wallraff, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walmsley, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Wilhelm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' The quantum technologies roadmap: a European community view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 20, 080201 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Horodecki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Horodecki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Horodecki, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Horodecki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 81, 865 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Braunstein and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' van Loock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Quantum informa- tion with continuous variables, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 77, 513 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Larsen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Breum, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Neergaard- Nielsen and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Andersen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Deterministic generation of a two-dimensional cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Science 366, 369 - 372 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [6] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Asavanant, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Shiozawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Yokoyama, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Charoen- sombutamon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Emura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Alexander, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Takeda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Yoshikawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Menicucci, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Yonezawa and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Furusawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Generation of time-domain-multiplexed two- dimensional cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Science 366, 373 - 376 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 B 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='015 0 1 2 3 4 5 6 0 10 20 30 40 50 s(dB) FI, QFI FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Maximal value for the FI (red) in the case of the phase shift operator bounded by the QFI (dashed blue) for different squeezing parameters r = rA = rB > 0 – phase shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Duan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Giedke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Cirac and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Zoller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Insep- arability criterion for continuous variable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 84, 2722 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Simon, Peres-Horodecki separability criterion for con- tinuous variable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 84, 2726 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Giovannetti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Mancini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Vitali, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Tombesi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Characterizing the entanglement of bipartite quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 67, 022320 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' van Loock and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Furusawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Detecting genuine mul- tipartite continuous-variable entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='it Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 67, 052315 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walschaers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fabre, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Parigi, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' En- tanglement and Wigner Function Negativity of Multi- mode Non-Gaussian States Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 119, 183601 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [12] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Chabaud and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walschaers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Resources for bosonic quantum computational advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' arXiv 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='11781 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Namiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Photonic families of non-Gaussian entangled states and entanglement criteria for continuous-variable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 85, 062307 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walschaers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Non-Gaussian states and where to find them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' PRX Quantum 2, 030204 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Shchukin and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Vogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Inseparability Criteria for Continuous Bipartite Quantum States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 95, 230502 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Miranowicz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Piani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Comment on “Inseparabil- ity Criteria for Continuous Bipartite Quantum States”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 97, 058901 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Shchukin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' van Loock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Higher-order Einstein- Podolsky-Rosen correlations and inseparability condi- tions for continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 93, 032114 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Agarwal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Biswas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Inseparability inequalities for higher order moments for bipartite systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 7, 211 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Hillery and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Zubairy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Entanglement Conditions for Two-Mode States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 96, 050503 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walborn, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Taketani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Salles, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Toscano, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' de Matos Filho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Entropic Entanglement Criteria for Continuous Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 103, 160505 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Nha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ji, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Efficient En- tanglement Criteria beyond Gaussian Limits Using Gaus- sian Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 108, 030503 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Strobel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Muessel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Linnemann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Zibold, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Hume, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Pezze, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Smerzi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Oberthaler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fisher information and entanglement of non-Gaussian spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Science 345, 424 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Hu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Al-amri, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Liao, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Zubairy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Continuous-variable quantum key distribution with non- Gaussian operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 102, 012608 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Opatrny, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Kurizki, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Welsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Improvement on teleportation of continuous variables by photon sub- traction via conditional measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 61, 032302 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Pezze and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Smerzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Entanglement, nonlinear dy- namics, and the Heisenberg limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 102, 100401 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Gessner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Pezze, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Smerzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Efficient entangle- ment criteria for discrete, continuous, and hybrid vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 94, 020101(R) (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Gessner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Pezze, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Smerzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Entanglement and squeezing in continuous-variable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Quantum 1, 17 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ourjoumtsev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ferreyrol, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Tualle-Brouri and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Grangier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Preparation of non-local superpositions of quasi-classical light states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Nature Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 5, 189-192 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [29] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Hyllus and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Eisert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Optimal entanglement wit- nesses for continuous-variable systems New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 8, 51 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Braunstein and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Caves, Statistical distance and the geometry of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 72, 3439 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Nieto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Displaced and squeezed number states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 229, 135-143 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [32] We can use the following expressions for operators acting on a wavefunction Ψ(x) [31] exp[θ∂x]Ψ(x) = Ψ(x − θ), exp[θ(x∂x)]Ψ(x) = Ψ(xeθ), exp[θ(∂x)2]Ψ(x) = 1 √ 4πθ � ∞ −∞ exp [−(y − x)2 4θ ]Ψ(y)dy, where ∂x is related to the phase quadrature ˆp through the functional relation ˆp = −2i∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [33] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Dufour, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walschaers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Jacquard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Michel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fabre and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Non-Gaussian quantum states of a multimode light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Nature Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 16, 144- 147 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [34] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Agarwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Quantum Optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Cambridge University Press (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Tian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Xiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fadel, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Characterizing Multipartite non-Gaussian Entangle- ment for a Three-Mode Spontaneous Parametric Down- Conversion Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Applied 18, 024065 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [36] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Chabaud, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Roeland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walschaers, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Grosshans, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Parigi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Markham, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Certification of Non-Gaussian States with Operational Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' PRX Quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 2, 020333 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 16 [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walschaers, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ra, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Mode-dependent- loss model for multimode photon-subtracted states Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' A 100, 023828 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lopetegui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Gessner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fadel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Walschaers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Homodyne detection of non-Gaussian quan- tum steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' PRX Quantum 3, 030347 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [39] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Qin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Gessner, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Deng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Han, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Su, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Smerzi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Peng .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Characterizing the multi- partite continuous-variable entanglement structure from squeezing coefficients and the Fisher information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' npj Quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 5, 3 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Roslund, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Medeiros de Araujo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fabre and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Wavelength-multiplexed quantum net- works with ultrafast frequency combs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='Nature Phot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 8, 109 112 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [41] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Roslund, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ferrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Arzani, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Fabre and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Multimode entanglement in reconfig- urable graph states using optical frequency combs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content='Nature Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 8, 15645 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Ansquer, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Thiel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' De, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Argence, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Gredat, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Bretenaker and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Unveiling the dynamics of opti- cal frequency combs from phase-amplitude correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Research 3, 033092 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' [43] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Barral, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Xiao and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Bencheikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Hierarchy of nonlinear entanglement dynam- ics for continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} +page_content=' 127, 150502 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfdwdn/content/2301.03909v1.pdf'} diff --git a/kNE1T4oBgHgl3EQf0QWt/content/tmp_files/2301.03454v1.pdf.txt b/kNE1T4oBgHgl3EQf0QWt/content/tmp_files/2301.03454v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..48e36ed69fd610f78371576297329df9d256550c --- /dev/null +++ b/kNE1T4oBgHgl3EQf0QWt/content/tmp_files/2301.03454v1.pdf.txt @@ -0,0 +1,1955 @@ +Godunov–like numerical fluxes for conservation laws on networks∗ +Luk´aˇs Vacek† +Charles University, Faculty of Mathematics and Physics +Sokolovsk´a 83, Praha 8, 186 75, Czech Republic +and +V´aclav Kuˇcera‡ +Charles University, Faculty of Mathematics and Physics +Sokolovsk´a 83, Praha 8, 186 75, Czech Republic +January 10, 2023 +Abstract +This paper deals with the construction of a discontinuous Galerkin scheme for the solution of Lighthill- +Whitham-Richards traffic flows on networks. +The focus of the paper is the construction of two new +numerical fluxes at junctions, which are based on the Godunov numerical flux. We analyze the basic +properties of the two Godunov-based fluxes and the resulting scheme, namely conservativity and the +traffic distribution property. We prove that if the junction is not congested, the traffic flows according +to predetermined preferences of the drivers. Otherwise a small traffic distribution error is present, which +we interpret as either the existence of dedicated turning lanes, or factoring of human behavior into the +model. We compare our approach to that of ˇCani´c et al. (J. Sci. Comput., 2015). Numerical experiments +are provided. +1 +Introduction +In this paper, we are concerned with the simulation of the movement of traffic on networks of roads. We +take the macroscopic approach, where the traffic is modeled as a uniform continuum which moves through +the roads. This is opposed to the microscopic approach, where each individual vehicle is modeled separately. +Since the total number of vehicles is conserved, the basic mathematical model for us will be that of partial +differential equations (PDEs) describing conservation laws, namely nonlinear first order hyperbolic PDEs. +Specifically, we are concerned with the so-called Lighthill-Whitham-Richards (LWR) traffic flow model, where +the traffic moves according to an equilibrium flow of homogeneous traffic, which is described by a so-called +fundamental diagram, cf. [5], [6], [7] or [11] for an overview. Such approaches to modelling traffic flows on a +single road are more or less standard. What is considerably newer and less studied is the generalization of the +LWR models to networks of roads, which can be described by an oriented graph, where on each road we have +the equations for the LWR model and we need to supply some kind of boundary condition at intersections +which correspond to vertices of the graph cf. [4]. +In our case, we use the discontinuous Galerkin (DG) method to discretize the LWR model on each +individual road and the behavior of traffic at junctions is determined by prescribing a numerical flux at the +junction. In such a case, one must take into account not only the necessity for the resulting scheme to be +∗The work of L. Vacek is supported by the Charles University, project GA UK No. 1114119. The work of V. Kuˇcera is +supported by the Czech Science Foundation, project No. 20-01074S. +†Corresponding author. Email: lvacek@karlin.mff.cuni.cz +‡Email: kucera@karlin.mff.cuni.cz +1 +arXiv:2301.03454v1 [math.NA] 9 Jan 2023 + +conservative (vehicles are not lost or formed at intersections), but also other properties, such as taking into +account the preferences of individual drivers as to which outgoing road they wish to take from the junction, +etc. Such numerical fluxes were constructed and applied in [4] and [1], where it is assumed that the drivers +behave in such a way as to maximize the total flux through the intersection. This leads to a complicated linear +programming problem, which can be explicitly solved (giving an explicit construction of the numerical flux) +only in the simplest cases, cf. [4] and [1]. We take a slightly different approach, where the resulting numerical +fluxes are explicitly constructed on an arbitrary junction based on the traffic distribution requirements. The +construction seems more natural for human drivers, who are mainly concerned with the traffic density at their +specific pair of incoming and outgoing roads and are (somewhat selfishly) not concerned with maximizing +the total flux of all the traffic through the junction. The latter case is much more realistic e.g. for a swarm +of centrally coordinated or communicating autonomous vehicles. +We have already described the basic construction of our numerical flux in the paper [10] which was +however based on a generic classical numerical flux, as used in DG methods on single roads. In this paper, +we refine the construction and base it on the Godunov numerical flux, which is based on the exact solution +of a local Riemann problem, and is therefore a natural numerical flux also from the point of view of the +PDE theory, since it corresponds to the so called Bardos-LeRoux-N´ed´elec boundary conditions, which is the +correct way how to prescribed Dirichlet data on a boundary. In this paper we take the classical Godunov +numerical flux and use it to construct a Godunov-like numerical flux at a general junction, which is based +on the drivers’ preferences described by traffic distribution coefficients. Actually, we derived two similar +numerical fluxes that differ in the way the traffic distribution coefficients are treated. We then proceed to +analyze the resulting DG scheme, namely we prove a discrete analogue of the Rankine-Hugoniot condition +at the junction, from which we derive global conservativity of the scheme across the whole network. As it +turns out, our numerical flux(es) do not exactly satisfy in all situations the apriori preferences of the drivers +in the form of relations given by the traffic distribution coefficients. We analyze this effect in detail and +derive relations for the traffic distribution error, which we then interpret. Namely, we can show that if (in +some sense) a junction is not congested, the drivers follow their predetermined preferences. However this +is not true when there is a congestion in the junction, and the original traffic distribution coefficients are +not satisfied exactly. We interpret this in two ways – (a) as typical human behavior, where some drivers +decide to change their route if they see that the one they originally chose is blocked, or (b) that there are +dedicated turning lanes in the incoming roads before the junctions. Such turning lanes allow some drivers, +whose route of choice is not blocked to pass the ones whose preferred outgoing road is congested. If there +are no dedicated turning lanes than a traffic jam on one of the outgoing roads blocks all of the traffic in +the entire junction, even though other outgoing roads may be completely free. This is what happens in the +numerical flux considered in [4], [1], but not in our case. Finally, we demonstrate the mentioned phenomena +on simple numerical test cases. +The structure of the paper is as follows. In Section 2 we introduce the basic concepts and notation needed +to describe traffic flows on networks along with the traffic distribution coefficients describing the drivers’ +preferences. In Section 3, we introduce the discontinuous Galerkin method on single roads and introduce +and reformulate the classical Godunov numerical flux. In Section 4 we review the approach of [4], [1] and +construct our Godunov-like numerical fluxes at junctions along with the corresponding DG formulation on +networks. In Section 5, we prove the conservativity of the DG scheme on networks with the considered +Godunov fluxes and analyze the traffic distribution error. Finally, we present numerical results in Section 6. +2 +Macroscopic traffic flow models on networks +We begin with the mathematical description of macroscopic vehicular traffic, cf. +[6], [7] and [11] for a +more detailed treatment of the subject. First, we consider a single road described mathematically as a one- +dimensional interval. In basic macroscopic models, traffic flow is described by three fundamental quantities +– traffic flow Q, traffic density ρ and mean traffic flow velocity V , all of these being functions of both the +spatial position x and time t. +The basic governing equation of traffic flow is derived using the assumption that the number of cars in +2 + +an arbitrary segment [x1, x2] of the road changes only due to the flux through the endpoints, i.e. +d +dt +� x2 +x1 +ρ(x, t) dx = Q(x1, t) − Q(x2, t). +(1) +Writing the right-hand side as an integral and eliminating the integral over the arbitrarily chosen [x1, x2] +gives the conservation law for ρ in the form +∂ +∂tρ(x, t) + ∂ +∂xQ(x, t) = 0. +(2) +Equation (2) must be supplemented by an initial condition and appropriate boundary conditions which we +will treat in detail in the case of networks of roads. +Equation (2) is underdetermined, as there is a single equation for two unknowns. Thus we need to supply +another equation or relation between the variables. Greenshields described a relation between traffic density +and traffic flow in the paper [5]. He made the assumption derived from observations that in homogeneous +traffic (traffic with no changes in time and space), the traffic flow Q is a function which depends only on the +density ρ. Let us denote the equilibrium flow of homogeneous traffic as Qe, derived from Q. The relationship +between the ρ and Qe is described by the so-called fundamental diagram. The approach where we use the +equilibrium traffic flow Qe in equation (2) is called the Lighthill-Whitham-Richards (LWR) traffic flow model +and results in the equation +ρt + +� +Qe(ρ) +� +x = 0, +x ∈ R, t > 0, +ρ(x, 0) = ρ0(x), +x ∈ R. +(3) +Equation (3) belongs to the class of nonlinear first order hyperbolic equations and, for practical purposes +will be considered on finite intervals with appropriate boundary conditions. +There are many different proposals for the equilibrium traffic flow Qe derived from real traffic data, cf. +[7]. Here we present only Greenshields model, which defines the equilibrium traffic flow as +Qe(ρ) = vmax ρ +� +1 − +ρ +ρmax +� +, +where vmax is the maximal velocity and ρmax is the maximal density. We can see the fundamental diagram +in Figure 1, where vmax = ρmax = 1. +Now we consider a road network represented by a directed graph, following [4]. The graph is a finite +collection of directed edges (roads), connected together at vertices (intersections). Each vertex has a finite +set of incoming edges and outgoing edges. +0.2 +0.4 +0.6 +0.8 +1.0 +ρ +0.2 +0.4 +0.6 +0.8 +1.0 +Ve +(a) Velocity–density diagram. +0.2 +0.4 +0.6 +0.8 +1.0 +ρ +0.05 +0.10 +0.15 +0.20 +0.25 +Qe +(b) Flow–density diagram. +Figure 1: Fundamental diagrams of the Greenshields model. +3 + +Figure 2: Example of a network. +Definition 1 (Network). We define a network as a couple (I, J ), where I = {In}N +n=1 is a finite set of +edges and J = {Jm}M +m=1 is a finite set of vertices. Each edge In is represented by an interval [an, bn] ⊆ +[−∞, ∞], n = 1, . . . , N. Each vertex Jm is a union of two non–empty subsets Inc(Jm) and Out(Jm) of +{1, . . . , N} representing incoming and outgoing edges, respectively. We assume the following: +(i) For all Ji, Jj ∈ J , i ̸= j : Inc(Ji) ∩ Inc(Jj) = ∅ and Out(Ji) ∩ Out(Jj) = ∅. +(ii) If i /∈ ∪J∈J Inc(J), i ∈ {1, . . . , N}, then bi = ∞ and if i /∈ ∪J∈J Out(J), i ∈ {1, . . . , N}, then ai = −∞. +Moreover, for all i ∈ {1, . . . , N} : i ∈ ∪J∈J Inc(J) or i ∈ ∪J∈J Out(J). +Condition (i) states that each edge can be incoming for at most one vertex and outgoing for at most one +vertex. Condition (ii) states that edges that are connected to only one vertex extend to ±∞. Of course in +practice artificial inflow/outflow boundaries are introduced on such edges in the numerical solution. We can +see an example in Figure 2. +As we are dealing with traffic flows described by LWR models, we assume that the traffic on each edge +number i ∈ {1, . . . , N} is described by a conservation law of the general form +(ui)t + (f(ui))x = 0, +x ∈ (ai, bi), t > 0, +ui(x, 0) = u0,i(x), +x ∈ (ai, bi), +(4) +where ui : (ai, bi) × [0, ∞) → R is the traffic density on the i-th edge (road). We note that although our +primary interest are traffic flow models, the system of equations (4) can represent general conservation laws, +which is the reason why we have abandoned the notation based on traffic density ρ and traffic flow Q and use +the generic notation u and f in the governing equations. We note that we will assume that the convective +flux f has a global maximum at some critical value (critical density) u∗ and f is non–decreasing on the +interval (−∞, u∗] and non–increasing on [u∗, ∞), cf. Section 3.1. This assumption is typical for traffic flow +models. +What remains is to describe the behavior of traffic at junctions. +For this purpose it is sufficient to +first consider a single vertex (junction) and its incoming and outgoing roads for simplicity. The resulting +considerations can then be applied separately to each vertex of the general network. +We consider a network (I, J ) and fix a vertex J ∈ J for which we assume that Inc(J) = {1, . . . , n} and +Out(J) = {n + 1, . . . , n + m}. We define the spatial limits of traffic densities on individual roads at the +common vertex J as +u− +i (bi, t) := lim +x→bi− ui(x, t) +and +u+ +j (aj, t) := +lim +x→aj+ uj(x, t) +for all i = 1, . . . , n and j = n + 1, . . . , n + m. Now we can define the solution at a junction. +Definition 2 (Traffic solution at a junction). Let J be a junction with incoming roads I1, . . . , In and outgoing +road In+1, . . . , In+m. Then we define a weak solution at J as a collection of functions ul : Il × [0, ∞) → R, +4 + +l = 1, . . . , n + m such that +n+m +� +l=1 +� bl +al +� ∞ +0 +� +ul +∂ϕl +∂t + f(ul)∂ϕl +∂x +� +dt dx = 0 +holds for every ϕl ∈ C1 +0([al, bl] × [0, ∞)), l = 1, . . . , n + m, that are also smooth across the junction, i.e. +ϕ− +i (bi, ·) = ϕ+ +j (aj, ·), +�∂ϕi +∂x +�− +(bi, ·) = +�∂ϕj +∂x +�+ +(aj, ·), +for all i ∈ {1, . . . , n} and j ∈ {n + 1, . . . , n + m}. +The basic property of the weak solution from Definition 2 is that it satisfies the Rankine-Hugoniot +condition which is essentially the conservation of vehicles at the junction. +Lemma 3. Let u = (u1, . . . , un+m)T be a weak solution at the junction J such that each x → ui(x, t) has +bounded variation. Then u satisfies the Rankine-Hugoniot condition +n +� +i=1 +f(u− +i (bi, t)) = +n+m +� +j=n+1 +f(u+ +j (aj, t)) +(5) +for almost every t > 0 at the junction J. +Proof. The proof is a simple application of integration by parts and can by found in [4, Lemma 5.1.9]. +Definition 2 simply enforces the conservation of vehicles at J. +However it is also necessary to take +into account the preferences of drivers how the traffic from incoming roads is distributed to outgoing roads +according to some predetermined coefficients. +Definition 4 (Traffic distribution matrix). Let J be a fixed vertex with n incoming edges and m outgoing +edges. We define a traffic distribution matrix A as +A = +� +�� +αn+1,1 +· · · +αn+1,n +... +... +... +αn+m,1 +· · · +αn+m,n +� +�� , +where 0 ≤ αj,i ≤ 1 for all i ∈ {1, . . . , n}, j ∈ {n + 1, . . . , n + m} and +n+m +� +j=n+1 +αj,i = 1 +(6) +holds for all i ∈ {1, . . . , n}. +The ith column of A describes how the traffic from the incoming road Ii distributes to the outgoing roads +at the junction J. In other words, if X is the amount of traffic coming from road Ii then αj,iX is the desired +amount of traffic going form Ii towards road Ij. +As stated, for simplicity, we assume a simple network with only one junction in the rest of this paper. +All our definitions and theorems can then be extended straightforwardly to an arbitrary network (I, J ). +3 +Discontinuous Galerkin method +We discretize the governing equation (3) using the discontinuous Galerkin (DG) method. +This method +introduced by Reed and Hill in [8] represents a robust, reliable and accurate numerical method for the solution +of first order hyperbolic problems. The DG method uses discontinuous piecewise polynomial approximations +5 + +of the exact solution along with a suitable weak form of the governing equations and can thus be viewed as +a combination of the the finite element and finite volume methods, cf. [3], [9]. One advantage of the DG +method over standard finite elements is its robustness with respect to the Gibbs phenomenon. This occurs +when a continuous approximation is used to approximate a discontinuous function – these typically arise as +solutions to nonlinear first order hyperbolic problems, such as those considered here. +In general, the DG method is described on a polygonal (polyhedral) domain Ω ⊂ Rd, d ∈ N. Let Th be +a partition of Ω into a finite number of closed elements K with mutually disjoint interiors, such that +Ω = +� +K∈Th +K. +Since the traffic model is defined on a line, we consider Ω ⊂ R, Ω = (a, b). In the 1D case, an element K is +an interval [aK, bK], where aK and bK are boundary points of K. We set hK = |bK − aK|, h = maxK∈T hK. +We denote the set of all boundary faces (points in 1D) of all elements by Fh. Further, we define the set of +all inner points by +FI +h = {x ∈ Fh; x ∈ Ω} +and the set of boundary points FB +h = {a, b}. Obviously Fh = FI +h ∪FB +h . We use a suitable weak formulation +of (3) on the broken Sobolev space Hk(Ω, Th) = {v; v|K ∈ Hk(K), ∀K ∈ Th}, where Hk(I), k ∈ N, is the +Sobolev space over an interval I. Functions from this space will be approximated by discontinuous piecewise +polynomial functions from the space +Sh = {v; v|K ∈ P p(K), ∀K ∈ Th}, +where P p(K) denotes the space of all polynomials on K of degree at most p. +For each point x ∈ FI +h there exist two neighbours K− +x , K+ +x ∈ Th such that x = K− +x ∩K+ +x . Every function +v ∈ Hk(Ω, Th) is generally discontinuous at x ∈ FI +h. Thus, for all x ∈ FI +h, we introduce the following notation +for traces and jumps: +v−(x) = lim +y→x− v(y), +v+(x) = lim +y→x+ v(y), +[v]x = v−(x) − v+(x). +In order to have consistent notation, in the point x ∈ FB +h we define +v+(a) = lim +y→a+ v(y), +v(a) = − [v]a = v−(a) := v+(a), +v−(b) = lim +y→b− v(y), +v(b) = [v]b = v+(b) := v−(b). +The definition of the jump [v]a := −v+(a) or [v]b := v−(b) may seem inconsistent with the definition on +interior points. This notation is used due to the integration by parts in the following sections. For simplicity, +if [·]x appear in a sum of the form � +x∈Fh . . ., we omit the index x and write [·]. +We formulate the DG method for first order hyperbolic problems of the form +ut + f(u)x = 0, +x ∈ Ω, t ∈ (0, T), +(7) +u = uD, +x ∈ FD +h , t ∈ (0, T), +(8) +u(x, 0) = u0(x), +x ∈ Ω, +(9) +where the Dirichlet boundary condition uD : FD +h × (0, T) → R and the initial condition u0 : Ω → R are +given functions. The Dirichlet boundary condition is prescribed only on the inlet FD +h ⊆ FB +h , respecting the +direction of information propagation. The function f ∈ C1(R) is called the convective flux. Our aim is to +seek a function u : Ω × (0, T) → R such that (7)–(9) is satisfied. As we have seen, problem (7) is the main +part of macroscopic equations for traffic. +6 + +In order to derive the DG formulation of (7), we multiply by a test function ϕ ∈ H1(Ω, Th) and integrate +over an arbitrary element K ∈ Th. Then we apply integration by parts and obtain +� +K +utϕ dx − +� +K +f(u)ϕ′ dx + f(u(bK, t))ϕ−(bK) − f(u(aK, t))ϕ+(aK) = 0. +(10) +Finally, we sum over all K ∈ Th and obtain +� +Ω +utϕ dx − +� +K∈Th +� +K +f(u)ϕ′ dx + +� +x∈Fh +f(u) [ϕ] = 0. +We wish to approximate u by a function uh ∈ H1(Ω, Th) which is in general discontinuous on Fh. Thus, +we need to give proper meaning to the function f(uh) in points x ∈ Fh. We proceed similarly as in the finite +volume method and use the approximation +f(uh) ≈ H(u− +h , u+ +h ), +(11) +where H(· , · ) is a numerical flux, cf. [3]. Finally, we define the DG solution of problem (7). +Definition 5 (DG solution). The function uh : Ω × (0, T) → R is called a DG finite element solution of +hyperbolic problem (7)–(9) if the following properties hold: +(i) uh ∈ C1 ([0, T] ; Sh). +(ii) uh(0) = uh0, where uh0 denotes an Sh approximation of the initial condition u0. +(iii) uh = uD for all x ∈ FD +h , t ∈ (0, T). +(iv) For all ϕ ∈ Sh and for all t ∈ (0, T), uh satisfies +� +Ω +(uh)tϕ dx − +� +K∈Th +� +K +f(uh)ϕ′ dx + +� +x∈Fh +H(u− +h , u+ +h ) [ϕ] = 0. +(12) +3.1 +Godunov numerical flux +In our implementation, we use the Godunov numerical flux, cf. [9]. This is defined as the flux f evaluated +at the exact solution of the Riemann problem with the piecewise defined initial condition u− and u+. This +can be shown to be equivalent to the more practical form, cf. [9], +HGod +orig +� +u−, u+� += +� +minu−≤u≤u+ f(u), +if u− < u+, +maxu+≤u≤u− f(u), +if u− ≥ u+. +(13) +We call this form the original form and for our purposes, we use an alternative form which is inspired by +the maximum possible traffic flow approach (see Section 4.1) in the case of one incoming and one outgoing +road. +Definition 6 (Alternative form of the Godunov numerical flux). Let the convective flux f have a global +maximum at u∗ and f is non–decreasing on the interval (−∞, u∗] and non–increasing on [u∗, ∞). Then the +Godunov numerical flux is defined as +HGod � +u−, u+� += min +� +fin(u−), fout(u+) +� +, +(14) +where +fin(u−) = +� +f(u−), +if u− < u∗, +f(u∗), +if u− ≥ u∗, +fout(u+) = +� +f(u∗), +if u+ ≤ u∗, +f(u+), +if u+ > u∗. +7 + +ρ +Qe +fin +fout +Figure 3: fin and fout for Greenshields traffic flow. +This can be interpreted as the maximal possible flow through the common boundary, where fin is the +maximal possible inflow from the left element and fout is the maximal possible outflow to the right element. +We can see fin and fout for Greenshields traffic flow in Figure 3. +Formulas (13) and (14) are equivalent, which is shown in the following lemma. +Lemma 7. If the convective flux f attains its global maximum at u∗ and is non–decreasing on (−∞, u∗] and +non–increasing on [u∗, ∞), then HGod +orig (u−, u+) = HGod (u−, u+) for all u−, u+ ∈ R. +Proof. We divide the proof into six different cases. There are two main cases based on the ordering of u− +and u+. Then for each case we present three sub-cases, which place u∗ in different positions. +1. +u− < u+ +a) +If u− < u+ ≤ u∗ then HGod +orig = f(u−) and HGod = min {f(u−), f(u∗)} = f(u−). +b) +If u− ≤ u∗ < u+ then HGod +orig = min {f(u−), f(u+)} and HGod = min {f(u−), f(u+)}. +c) +If u∗ < u− < u+ then HGod +orig = f(u+) and HGod = min {f(u∗), f(u+)} = f(u+). +2. +u− ≥ u+ +a) +If u+ ≤ u− < u∗ then HGod +orig = f(u−) and HGod = min {f(u−), f(u∗)} = f(u−). +b) +If u+ ≤ u∗ ≤ u− then HGod +orig = f(u∗) and HGod = min {f(u∗), f(u∗)} = f(u∗). +c) +If u∗ < u+ ≤ u− then HGod +orig = f(u+) and HGod = min {f(u∗), f(u+)} = f(u+). +We showed that HGod +orig (u−, u+) = HGod (u−, u+) in every possible situation. +In the rest of the paper, the numerical flux H(· , · ) will always be the Godunov numerical flux written in +the alternative form (14). +4 +Numerical fluxes at junctions +In order to formulate the DG scheme on a simple network, we first need to construct the numerical fluxes +at the junction. Such a numerical flux is considered in [1] and [4] which is based on the assumption that the +drivers wish to maximize the total flux through the junction while respecting the traffic distribution exactly. +We discuss this approach in Section 4.1. +In this paper we take a different approach which has the advantage that it is simple and explicitly +constructed for all junction types unlike that of [1] and [4], which leads to the solution of a linear programming +8 + +problem. We will present two different constructions of the numerical fluxes at the junction based on the +Godunov numerical flux in Sections 4.2 and 4.3. We shall prove the basic properties of our construction and +discuss the differences with the approach of [1] and [4]. +4.1 +Maximum possible traffic flow +Based on the traffic distribution matrix, the authors of [4] define the following admissible traffic solution at +a junction, also used for numerical simulations in [1]. +Definition 8 (Admissible traffic solution at a junction, following [4]). Let u = (u1, . . . , un+m)T be such that +ui(·, t) is of bounded variation for every t ≥ 0. Then u is called an admissible weak solution of (4) related +to the matrix A at the junction J if the following properties hold: +(i) u is a weak solution at the junction J. +(ii) f(u+ +j (aj, ·)) = �n +i=1 αj,if(u− +i (bi, ·)), for all j = n + 1, . . . , n + m. +(iii) �n +i=1 f(u− +i (bi, ·)) is maximal subject to (i) and (ii). +Remark 1. Condition (ii) simply states that traffic from incoming roads is distributed to outgoing roads +according to the traffic distribution matrix. Condition (iii) is a mathematical formulation of the assumption +made in [4], that respecting (ii), “drivers choose so as to maximize the total flux” through the junction. +One problem with the approach of [4] and [1] is that explicitly constructing the fluxes from Definition +8 requires the solution of a linear programming problem on the incoming fluxes. This is done in [4] for the +purposes of constructing a Riemann solver at the junction and in [1] for the purposes of obtaining numerical +fluxes at the junction in order to formulate the DG scheme. Closed-form solutions are provided in [1] in the +special cases n = 1, m = 2 and n = 2, m = 1 and n = 2, m = 2. In Section 4, we present an alternative +construction of fluxes at the junction which has the advantage of a simple formulation for general n, m. We +will give an interpretation of our construction, which shows that it is more suited for certain situations, +giving more realistic behavior of the drivers, than the approach from Definition 8. We compare the two +approaches in Section 6. +4.1.1 +One incoming and two outgoing roads +This example is important to us, because it inspires us in the construction of the α-inside Godunov flux (see +Section 4.3). We use the method described in [1, Section 2.2] with our notation. +In this case, we have distribution coefficient α2,1 = α and α3,1 = 1 − α. Then according to [4], we +calculate the maximum possible inflow to the junction from the incoming road as +H1(t) = min +� +fin(u− +1 (b1, t)), fout(u+ +2 (a2, t)) +α +, fout(u+ +3 (a3, t)) +1 − α +� +. +(15) +The outflow from the junction to an outgoing road is calculated as H1 multiplied by the corresponding +distribution coefficient, i.e. H2(t) = αH1(t) and H3(t) = (1 − α)H1(t). +Remark 2. We note, that traffic congestion on one of the outgoing roads influences the traffic flow to the +second outgoing road. For example, when fout(u+ +2 ) = 0, then H1 = H2 = H3 = 0. Therefore, the intersection +is completely blocked, even in the case when the other outgoing road I3 is completely empty (u3 ≡ 0). This +is caused by the assumption in Definition 8 that drivers strictly adhere to their original preferences for the +choice of the outgoing road, even on a congested intersection. Thus the flows to the outgoing roads I2, I3 +must always be in the ratio α to 1 − α, irrespective of the traffic situation. +9 + +4.2 +α-outside Godunov flux +In our previous paper [10], we based the construction of the numerical flux at the junction on the Lax- +Friedrichs numerical flux. In this paper we start from the Godunov numerical flux, which will have advantages +that we will see in Section 5.2. In either case, we construct the junction fluxes as follows. +At the junction, we consider an incoming road Ii and an outgoing road Ij. If these roads were the only +roads at the junction, i.e. if they were directly connected to each other, the (numerical) flux of traffic from +Ii to Ij would simply be H +� +u− +hi(bi, t), u+ +hj(aj, t) +� +, where uhi and uhj are the DG solutions on Ii and Ij, +respectively. From the traffic distribution matrix, we know the ratios of the traffic flow distribution to the +outgoing roads. Thus, we take the numerical flux Hj(t) at the left point of the outgoing road Ij, i.e. at the +junction, at time t as +Hj(t) := +n +� +i=1 +αj,iH +� +u− +hi(bi, t), u+ +hj(aj, t) +� +, +(16) +for j = n+1, . . . , n+m. The numerical flux Hj(t) can be viewed as the DG analogue of taking the combined +traffic outflow �n +i=1 αj,if +� +u− +i (bi, t) +� +from all incoming roads and prescribing it as the inflow of traffic to the +road Ij. +Similarly, we take the numerical flux Hi(t) at the right point of the incoming road Ii, i.e. at the junction, +at time t as +Hi(t) := +n+m +� +j=n+1 +αj,iH +� +u− +hi(bi, t), u+ +hj(aj, t) +� +, +(17) +for i = 1, . . . , n. Again, this can be viewed as an approximation of the traffic flow �n+m +j=n+1 αj,if +� +u+ +j (aj, t) +� +being prescribed as the outflow of traffic from Ii. +Now we can formulate the DG method for the simple network with one junction using the numerical +fluxes defined in (16) and (17). Then the case of general networks is a straightforward generalization, where +the aforementioned construction of numerical fluxes at junctions is applied on each junction separately. +We consider the DG formulation (12) on every incoming and outgoing road represented by the intervals +(ai, bi), i = 1, . . . , n and (aj, bj), j = n + 1, . . . , n + m, respectively. Since the DG method is applied on finite +intervals, we replace the endpoints at ±∞ from Definition 1 by artificial inflow/outflow boundaries at finite +points along with inflow Dirichlet data. For every interval (ak, bk), k = 1, . . . , n + m, we consider a partition +Thk along with the corresponding discrete space Shk. We write the DG formulation directly for the case of +LWR models (3) with unknown density u and flux f(u). +Definition 9 (DG formulation on a simple network). We seek functions uhk ∈ C1 ([0, T] ; Shk), k = 1, . . . , n+ +m satisfying the following. +(i) Incoming roads: For all i = 1 . . . , n and all ϕi ∈ Shi +� bi +ai +(uhi)tϕi dx − +� +K∈Thi +� +K +f(uhi)ϕ′ +i dx + +� +x∈FI +hi +H +� +u− +hi, u+ +hi +� +[ϕi] ++Hiϕ− +i (bi) − H +� +uDi, u+ +hi(ai) +� +ϕ+ +i (ai) = 0, +(18) +where Hi = Hi(t) is the numerical flux defined in (17) and uDi is the Dirichlet datum corresponding +to the left artificial inflow boundary point ai of (ai, bi). +(ii) Outgoing roads: For all j = n + 1, . . . , n + m and all ϕj ∈ Shj +� bj +aj +(uhj)tϕj dx − +� +K∈Thj +� +K +f(uhj)ϕ′ +j dx + +� +x∈FI +hj +H +� +u− +hj, u+ +hj +� +[ϕj] ++H +� +u− +hj(bj), uDj +� +ϕ− +j (bj) − Hjϕ+ +j (aj) = 0, +(19) +where Hj = Hj(t) is the numerical flux defined in (16). +10 + +Remark 3. We note that the choice of the arguments in the numerical flux at the artificial boundary point +bj in (19) corresponds to an outflow boundary condition. Typically, uDj = u− +hj(bj). This term could be +rewritten using the original physical flux f(u− +hj(bj)) due to consistency of the numerical flux H. +Definitions (16) and (17) are independent of the specific choice of the numerical flux H. In the paper [10], +we used the Lax-Friedrichs numerical flux, while in this paper we use Godunov’s flux. Since the distribution +coefficients αj,i are outside of the numerical flux H in (16) and (17), we call this construction the α-outside +Godunov flux, as opposed to the following section. +4.3 +α-inside Godunov flux +When comparing the maximum possible traffic flow (15) with the α-outside Godunov flux (17), we find +the main difference in the position of the distribution coefficient. +While in (17) the traffic distribution +coefficient is outside of the minimization (14) defining the Godunov flux, in (15) this coefficient is inside the +minimization defining the Godunov-like flux. This leads to the idea of moving the distribution coefficients +in (16) and (17) inside the Godunov numerical fluxes. To this end we define an auxiliary Godunov-like flux +with a third variable for the traffic distribution coefficient. +Definition 10 (Godunov numerical flux with three variables). The Godunov numerical flux with three +variables is defined as +HGod � +u−, u+, α +� += min +� +αfin(u−), fout(u+) +� +, +(20) +where fin(u−) and fout(u+) are defined as in Definition 6. +The reason why we put the distribution coefficient in front of fin, is the representation of the real flow +from the incoming road. Only αj,ifin(u− +i (bi, t)) cars per time unit want to go from incoming road i to +outgoing road j. So it makes sense to use αj,ifin in the definition of the Godunov flux instead of fin as +in (14). For simplicity, we drop the superscript “God” and define the numerical flux with three variables +H(· , · , · ) as the flux (20) in the rest of this paper. +Now we can take the numerical flux Hj(t) with α-inside at the left point of the outgoing road Ij, i.e. at +the junction, at time t as +Hj(t) := +n +� +i=1 +H +� +u− +hi(bi, t), u+ +hj(aj, t), αj,i +� +, +(21) +for j = n + 1, . . . , n + m. Similarly, we take the numerical flux Hi(t) with α-inside at the right point of the +incoming road Ii, i.e. at the junction, as +Hi(t) := +n+m +� +j=n+1 +H +� +u− +hi(bi, t), u+ +hj(aj, t), αj,i +� +, +(22) +for i = 1, . . . , n. We can use the same DG formulation as in Definition 9 with Hi defined as (22) and Hj +defined as (21). +4.4 +One incoming and two outgoing roads +We use the same example as in Section 4.1.1 with one incoming and two outgoing road and compare all three +approaches. Our aim is to identify and discuss the differences between them and describe their behavior. +For simplicity, we use the notation f (1) +in := fin(u− +1 (b1, t)), f (2) +out := fout(u+ +2 (a2, t)) and f (3) +out := fout(u+ +3 (a3, t)). +In the case of the α-outside Godunov flux, we calculate the inflow to the junction from the incoming road +as +H1(t) = α min +� +f (1) +in , f (2) +out +� ++ (1 − α) min +� +f (1) +in , f (3) +out +� +. +(23) +We compare this flux value to that obtained by the maximum possible flow (15). At first glance equations +(15) and (23) seem completely different, cf. Table 1. On the other hand, if fin is less than both f (2) +out and +11 + +Approach +H1 +H2 +H3 +Maximum +min{f (1) +in , f (2) +out +α , f (3) +out +1−α} +α min{f (1) +in , f (2) +out +α , f (3) +out +1−α} +(1 − α) min{f (1) +in , f (2) +out +α , f (3) +out +1−α} +possible +α-outside +α min{f(1) +in , f(2) +out} + (1 − α) min{f(1) +in , f(3) +out} +α min{f (1) +in , f (2) +out} +(1 − α) min{f (1) +in , f (3) +out} +Godunov +α-inside +min{αf(1) +in , f(2) +out} + min{(1 − α)f(1) +in , f(3) +out} +min{αf (1) +in , f (2) +out} +min{(1 − α)f (1) +in , f (3) +out} +Godunov +Table 1: Comparison of the three approaches on the example with one incoming and two outgoing roads. +f (3) +out then equations (15) and (23) give the same value fin. The outflows from the junction are the individual +terms in the right hand side of (23). Hence, H2(t) = α min{f (1) +in , f (2) +out} and H3(t) = (1 − α) min{f (1) +in , f (3) +out}. +If we compare H2 and H3 from the maximum possible flow and α-outside Godunov flux, we can see the +distribution coefficient α and 1 − α in front of the minimization in both cases. Again, if fin is less than +both f (2) +out and f (3) +out, both of the approaches give the same values of H2 and H3. But if the minimizer is +f (2) +out and f (2) +out divided by the corresponding distribution coefficient, respectively, then the maximum possible +flow gives us H2 = f (2) +out while α-outside Godunov gives us H2 = αf (2) +out, which is a lower flux value than the +maximum possible traffic flow. We can sum up these considerations informally in the statement that if the +outgoing roads are emptier than the incoming road then the maximum possible traffic flow and the α-outside +Godunov flux coincide. Once one of the outgoing roads is fuller than the incoming one, the two approaches +differ, the latter one giving a smaller traffic flux. +In the case of the α-inside Godunov flux, we calculate the inflow to the junction from the incoming road +as +H1(t) = min +� +αf (1) +in , f (2) +out +� ++ min +� +((1 − α) f (1) +in , f (3) +out +� +. +(24) +This approach is somewhere between the α-outside Godunov flux and maximum possible flow, see Table 1. +Again, if fin is less than both f (2) +out and f (3) +out, equations (15), (23) and (24) give us same value. But if one +of the fout is the minimizer, we typically get three different values. Also in this case the outflows from the +junction are the individual terms in (24), hence H2(t) = min{αf (1) +in , f (2) +out} and H3(t) = min{(1−α)f (1) +in , f (3) +out}. +Here is the main difference between the α-outside and α-inside Godunov fluxes. Of course, when fin is less +than both f (2) +out and f (3) +out, we get the same values. But if f (2) +out is the minimizer, α-inside Godunov gives us +the same H2 as the maximum possible flow (H1 and H3 are typically different). This is why we say that +α-inside Godunov lies between the other two approaches and takes positives from both. +5 +Properties +In this section, we look at the basic properties of the numerical fluxes at junctions that we considered in +Section 4. Namely, the Rankine-Hugoniot conditions and the satisfaction of the traffic distribution according +to the coefficients in the traffic distribution matrix. +5.1 +Rankine-Hugoniot condition +First, we show that our Godunov-based fluxes satisfy the discrete analogues of the Rankine-Hugoniot con- +dition (5), which leads to conservativity of the resulting DG scheme. +Lemma 11 (Discrete Rankine–Hugoniot condition for α-outside Godunov flux). The numerical fluxes (16) +and (17) with α outside satisfy the discrete version of the Rankine–Hugoniot condition (5): +n +� +i=1 +Hi(t) = +n+m +� +j=n+1 +Hj(t). +12 + +Proof. From the definition of Hi and Hj, we immediately obtain +n +� +i=1 +Hi(t) = +n +� +i=1 +n+m +� +j=n+1 +αj,iH +� +u− +hi(bi, t), u+ +hj(aj, t) +� += +n+m +� +j=n+1 +n +� +i=1 +αj,iH +� +u− +hi(bi, t), u+ +hj(aj, t) +� += +n+m +� +j=n+1 +Hj(t). +Lemma 12 (Discrete Rankine–Hugoniot condition for α-inside Godunov flux). The numerical fluxes (21) +and (22) with α inside satisfy the discrete version of the Rankine–Hugoniot condition (5): +n +� +i=1 +Hi(t) = +n+m +� +j=n+1 +Hj(t). +Proof. From the definition of Hi and Hj, we immediately obtain +n +� +i=1 +Hi(t) = +n +� +i=1 +n+m +� +j=n+1 +H +� +u− +hi(bi, t), u+ +hj(aj, t), αj,i +� += +n+m +� +j=n+1 +n +� +i=1 +H +� +u− +hi(bi, t), u+ +hj(aj, t), αj,i +� += +n+m +� +j=n+1 +Hj(t). +The previous two lemmas allow us to prove that for the DG scheme, the total number of vehicles in the +network is conserved (modulo inlet and outlet boundary conditions), which is the basic property we expect +from the conservation laws. Since the DG scheme is naturally conservative on each individual road, the +question boils down to conservativity of the scheme at the junction. +Corollary 13 (Conservation property of the DG scheme). Let uDj = u− +hj(bj). +The DG scheme from +Definition 9 conserves the total number of vehicles in the network in the sense that +d +dt +n+m +� +k=1 +� bk +ak +uhk dx = +n +� +i=1 +H +� +uDi, u+ +hi(ai) +� +− +n+m +� +j=n+1 +H +� +u− +hj(bj), uDj +� +for both α–outside and α–inside Godunov fluxes. +Proof. We set all test functions as ϕk ≡ 1 for all k = 1, . . . , n + m and sum together all of the equations (18) +and (19) for all i and j. We get +d +dt +n+m +� +k=1 +� bk +ak +uhk dx + +n +� +i=1 +Hi − +n+m +� +j=n+1 +Hj + +n+m +� +j=n+1 +H +� +u− +hj(bj), uDj +� +− +n +� +i=1 +H +� +uDi, u+ +hi(ai) +� += 0. +The second and third terms cancel one another since � +i Hi − � +j Hj = 0 due to Lemma 11 or 12. This +completes the proof. +13 + +5.2 +Traffic distribution error +The purpose of the traffic distribution coefficients αj,i from Definition 4 is to describe the preferences of the +drivers from each incoming road for which outgoing road they want to take. In other words, how the traffic +from each incoming road is distributed among the outgoing roads. In the end this means we want to satisfy +condition (ii) from Definition 8 in terms of the numerical fluxes at the junction: +Hj(t) = +n +� +i=1 +αj,iHi(t), +j = n + 1, . . . , n + m +(25) +The maximum possible flux approach from [4] is based on maximizing the total flux through the junction +while satisfying (25) exactly under any circumstances. +In this section we show that in the case of our +Godunov-based fluxes, this relation is satisfied if, loosely speaking, the junction is not congested. In other +words, whenever the outgoing roads can accept the incoming traffic, the drivers drive according to their +original preferences. If any of the outgoing roads cannot accept the incoming traffic, relation (25) is no +longer satisfied exactly, but with a small error (traffic distribution error). We interpret this as a natural +behavior of human drivers – when one of the preferred outgoing roads is full, some drivers will change their +original preferences and take a different route. In this context, strictly adhering to (25) irrespective of the +traffic situation would correspond to non-human drivers, e.g. autonomous vehicles with a prescribed course +which cannot be changed. +Another interpretation is the presence of dedicated turning lanes in front of the junction. These allow +other cars to pass the standing vehicles which want to go to a congested outgoing road. In a single-lane road, +even one standing vehicle can block the whole road if it cannot proceed to its desired outgoing road. Such a +vehicle blocks the way for other drivers, even those who could otherwise proceed since their desired outgoing +roads are free. This was discussed in Remark 2 for the maximum possible flux, where a single congested +road blocks the entire junction. +In this section, we analyze when the traffic distribution condition (25) is satisfied exactly for our Godunov- +based fluxes. In the following lemmas, we express the traffic distribution error for these fluxes. +Lemma 14 (Traffic distribution error for α–outside Godunov). The numerical fluxes (16) and (17) with α +outside satisfy +Hj(t) = +n +� +i=1 +αj,iHi(t) + Ej(t) +for all j = n + 1, . . . , n + m, where the error term is +Ej(t) = +n +� +i=1 +n+m +� +l=n+1 +l̸=j +αj,iαl,i +� +Hi,j(t) − Hi,l(t) +� +, +(26) +where Hi,j(t) := H +� +u− +hi(bi, t), u+ +hj(aj, t) +� +. +Proof. By definition (16), +Hj(t) = +n +� +i=1 +αj,iHi,j(t) = +n +� +i=1 +αj,iHi(t) + +n +� +i=1 +αj,i +� +Hi,j(t) − Hi(t) +� +� +�� +� +Ej(t) +, +where Ej(t) is the error term which we will show has the form (26): By Definition (17), we have +Ej(t) = +n +� +i=1 +αj,i +� +Hi,j(t) − +n+m +� +l=n+1 +αl,iHi,l(t) +� += +n +� +i=1 +αj,i +n+m +� +l=n+1 +αl,i +� +Hi,j(t) − Hi,l(t) +� += +n +� +i=1 +n+m +� +l=n+1 +l̸=j +αj,iαl,i +� +Hi,j(t) − Hi,l(t) +� +, +14 + +since �n+m +l=n+1 αl,i = 1 due to (6). This completes the proof. +Lemma 15 (Traffic distribution error for α–inside Godunov). The numerical fluxes (21) and (22) with α +inside satisfy +Hj(t) = +n +� +i=1 +αj,iHi(t) + Ej(t) +for all j = n + 1, . . . , n + m, where the error term is +Ej(t) = +n +� +i=1 +n+m +� +l=n+1 +l̸=j +(αl,iHi,j(t) − αj,iHi,l(t)) , +(27) +where Hi,j(t) := H +� +u− +hi(bi, t), u+ +hj(aj, t), αj,i +� +. +Proof. By definition (21), +Hj(t) = +n +� +i=1 +Hi,j(t) = +n +� +i=1 +αj,iHi(t) + +n +� +i=1 +� +Hi,j(t) − αj,iHi(t) +� +� +�� +� +Ej(t) +, +where Ej(t) is the error term which we will show has the form (27): By Definition (22), we have +Ej(t) = +n +� +i=1 +� +Hi,j(t) − αj,i +n+m +� +l=n+1 +Hi,l(t) +� += +n +� +i=1 +n+m +� +l=n+1 +l̸=j +� +αl,iHi,j(t) − αj,iHi,l(t) +� +, +since �n+m +l=n+1 αl,i = 1 due to (6). This completes the proof. +Now we discuss general situations when the traffic distribution errors are zero. We note that we have also +analyzed the traffic distribution errors in our previous paper [10], where the numerical fluxes at the junction +were based on the Lax-Friedrichs flux instead of Godunov. In that case, the errors were in general always +nonzero but small. We therefore view the Godunov-like approach of this paper as more natural, since the +traffic distribution relation (25) is satisfied exactly in many situation. Namely, in Theorem 1 we show that +if the density on all the outgoing roads is smaller than u∗, i.e. the density where the traffic flow is maximal, +then the traffic is distributed according to (25). +In the following, we shall consider a fixed time t and omit the argument t from the functions uh, Hi,j, Ej +and similar, in order to simplify the notation. +Theorem 1 (Zero traffic distribution error). Let u+ +hj(aj) ≤ u∗ for all j ∈ {n + 1, . . . , n + m}. Then Ej = 0 +for all j ∈ {n + 1, . . . , n + m} for both the α–inside and α–outside Godunov fluxes. +Proof. For the α-outside flux, due to the form (26) of the error term, if Hi,j = Hi,l for all i, j, l, then +Ej = 0. This happens, in general, when the inflow term is the minimizer in (14) in both numerical fluxes, i.e. +Hi,j = Hi,l = fin(u− +hi(bi)). The simplest general case when this happens is if fout(u+ +hj(aj)) = fout(u+ +hl(al)) = +f(u∗) ≥ fin(u− +hi(bi)), or in other words if u+ +hj(aj) ≤ u∗ and u+ +hl(al) ≤ u∗ for all l, j ∈ {n + 1, . . . , n + m}. +For the α-inside flux, due to the form (26) of the error term, if αl,iHi,j = αj,iHi,l for all i, j, l, then +Ej = 0. +Again,if the inflow term is the minimizer in (20) in both numerical fluxes, we get αl,iHi,j = +αl,iαj,ifin(u− +hi(bi)) = αj,iHi,l. This occurs if fout(u+ +hj(aj)) = f(u∗) ≥ αj,ifin(u− +hi(bi)), or in other words if +u+ +hj(aj) ≤ u∗ for all j ∈ {n + 1, . . . , n + m}. +Theorem 1 states that the traffic is distributed exactly according to the original preferences if the outgoing +roads are sufficiently free. The result is valid for both of the presented variants of the Godunov flux. If we +take into account the specific form of the flux, where f(0) = f(umax = 0, we can perform a finer analysis, +which allows heavier traffic through the junction, while still satisfying (25) exactly. +15 + +Theorem 2 (Zero traffic distribution error for α–outside). Assume that for each i ∈ {1, . . . , n} one of the +following conditions is satisfied: +1. u− +hi(bi) ≥ u∗ and u+ +hj(aj) ≤ u∗ for all j ∈ {n + 1, . . . , n + m}. +2. u− +hi(bi) < u∗ and u+ +hj(aj) ≤ ˜u for all j ∈ {n+1, . . . , n+m}, where ˜u > u∗ is such that f(˜u) = f(u− +hi(bi)). +Then Ej = 0 for all j ∈ {n + 1, . . . , n + m}. +Proof. As in the proof of Theorem 1, we wish to have Hi,j = Hi,l for all i, j, l. And again we achieve this by +assuming the inflow term is the minimizer in both numerical fluxes in (14), i.e. Hi,j = Hi,l = fin(u− +hi(bi)). +Consult Figure 4(a) in the following. +Case 1. If u− +hi(bi) ≥ u∗ then fin(u− +hi(bi)) = f(u∗) and for Hi,j = Hi,l = fin(u∗) to hold, we must +necessarily have also fout(u+ +hj(aj)) = f(u∗), i.e. u+ +hj(aj) ≤ u∗ for all j ∈ {n + 1, . . . , n + m}. +Case 2. Since u− +hi(bi) < u∗ and since f is non-increasing on [u∗, ∞), there exists ˜u > u∗ such that +f(˜u) = f(u− +hi(bi)), cf. Figure 4(a). Then if u+ +hj(aj) ≤ ˜u for all j ∈ {n+1, . . . , n+m}, we have fout(u+ +hj(aj)) ≥ +f(˜u) = f(u− +hi(bi)) = fin(u− +hi(bi)). +Therefore, Hi,j = fin(u− +hi(bi)) for all j ∈ {n + 1, . . . , n + m}, hence +Hi,j = Hi,l for all j, l. +Theorem 3 (Zero traffic distribution error for α–inside). Assume that for each i ∈ {1, . . . , n} one of the +following conditions is satisfied: +1. u− +hi(bi) ≥ u∗ and u+ +hj(aj) ≤ ˜uj for all j ∈ {n + 1, . . . , n + m}, where ˜uj ≥ u∗ is such that f(˜uj) = +αj,if(u∗). +2. u− +hi(bi) < u∗ and u+ +hj(aj) ≤ ˜uj for all j ∈ {n + 1, . . . , n + m}, where ˜uj > u∗ is such that f(˜uj) = +αj,if(u− +hi(bi)). +Then Ej = 0 for all j ∈ {n + 1, . . . , n + m}. +Proof. Due to Lemma 15, we wish to have αl,iHi,j = αj,iHi,l for all i, j, l. And again we achieve this by +assuming the inflow term is the minimizer in both numerical fluxes in the Godunov flux (20), since then we +will have αl,iHi,j = αj,iHi,l = αj,iαl,ifin(u− +hi(bi)). Consult Figure 4(b) in the following. +ρ +f +ui +-(bi) +u* +u +fin +fout +(a) α–outside. +ρ +f +ui +-(bi) +u* +u +α fin +fout +(b) α–inside. +Figure 4: Illustration of Theorems 2 and 3. +16 + +Case 1. +If u− +hi(bi) ≥ u∗ then fin(u− +hi(bi)) = f(u∗) and fout(u+ +hj(aj)) ≥ f(˜uj) = αj,if(u∗). +Hence +Hi,j = αj,if(u∗) and similarly Hi,l = αl,if(u∗). Therefore αl,iHi,j = αj,iHi,l. +Case 2. If u− +hi(bi) < u∗ then fin(u− +hi(bi)) = f(u− +hi(bi)) and fout(u+ +hj(aj)) ≥ f(˜uj) = αj,if(u− +hi(bi)). Hence +Hi,j = αj,if(u− +hi(bi)) and similarly Hi,l = αl,if(u− +hi(bi)). Therefore αl,iHi,j = αj,iHi,l. +We note that Theorems 2 and 3 are more general in that they allow larger densities on the outgoing roads +than Theorem 1, while still satisfying the traffic distribution condition (25) exactly. Theorem 1 requires +u+ +hj(aj) ≤ u∗ on all outgoing roads. However, Theorems 2 and 3 allow a weaker condition u+ +hj(aj) ≤ ˜uj +for some ˜uj > u∗ provided that u− +hi(bi) < u∗, i.e. if the incoming roads are not too full. Moreover, the +value of this allowed density ˜uj is higher for the α–inside Godunov than for α–outside, as can be seen when +comparing the values of ˜u in Figures 4(a) and 4(b). Thus the α–inside numerical flux allows for a larger +traffic flow through the junction, while still satisfying the traffic distribution condition (25). +6 +Numerical results +Here we present numerical experiments comparing the three Godunov-like numerical fluxes considered in this +paper. As for the implementation, integrals over individual elements in (12) are evaluated using Gaussian +quadrature rules. Basis functions of the space Sh are taken as Legendre polynomials on individual elements, +where the support of each basis function is a single element. By writing uh in terms of basis functions in +space and setting the test function ϕ to elements of the basis, equation (12) reduces to a system of ordinary +differential equations which is solved by Adams–Bashforth methods. +The DG method is much less susceptible to the Gibbs phenomenon than the finite element method, +however spurious oscillations can still occur locally in the vicinity of discontinuities or steep gradients in +the solution. There are several approaches how to treat these local oscillations, e.g. adding local artificial +diffusion. In our case, we apply limiters to the DG solution. In our implementation, we use the modified +minmod limiter from [2], cf. also [9]. +Often the solution of (7) is a physical quantity which satisfies some admissibility conditions, e.g. the +physical density must be positive. If we obtain a solution which is not in the admissible interval, e.g. due +to overshoots or undershoots, the problem can become ill-posed or even undefined. This is our case, since +the traffic density u must naturally satisfy u ∈ [0, umax]. The DG method by itself does not guaranty such +bounds are satisfied for the discrete solution. Limiters usually prevent this from happening, however in traffic +flows, it is natural that entire regions of the computational domain have u = 0 or u = umax and it is easy +for the algorithm to produce e.g. negative density due to round-off errors. To prevent this from happening, +we use the following procedure. If the average density on an element K is in the admissible interval, we +decrease the slope of our solution so that the modified density lies in [umin, umax] similarly as in the limiting +procedure. The important property is that the element average does not change after the application of +the limiter. As further insurance, if the average density on an element K is not in the admissible interval +[0, umax], then we change the solution such that u ≡ 0 or u ≡ umax on the whole element K. The latter case, +when the average density on an element is not in the admissible interval is extremely rare and, for us, serves +as an indicator that the time step is too large or the mesh is too coarse. Since in this case the described +procedure does not conserve the total number of vehicles, we rather decrease the time step or increase the +number of elements. For polynomials of higher degree, we can use the method described in the paper [12] +by Zhang, Xia and Shu, which reduces to the procedure described above in the simplest case of piecewise +linear approximations in 1D. +We consider a simple network with one incoming road (Road 1, colored red in Figures 5 and 6) and two +outgoing roads, Road 2 (green) and Road 3 (blue). The network will be closed at their endpoints (a1, b2 and +b3) meaning that the inflow density at a1 is equal to zero and the outflow densities at b2 and b3 are maximal +(i.e. equal to 1). Thus, we can check the total number of cars, because we have neither inflow nor outflow into +the network. We choose α2,1 = 0.75 and α3,1 = 0.25. The length of all roads is 1. We use the combination +of the explicit Euler method (step size τ = 10−4) and DG method (number of elements N = 150 on each +17 + +road). We calculate the piecewise linear approximations of solutions and we use two Gaussian quadrature +points in each element. We use Greenshields model with vmax = 1 and ρmax = 1. +In the following section, we set the initial condition to see the differences between approaches. We choose +congested examples to demonstrate the distribution error from Lemma 15. In non–congested cases, the +traffic distribution error is zero. +6.1 +α-outside vs. α-inside Godunov flux +First, we compare the α-outside and α-inside Godunov fluxes. This example shows that the traffic flow from +the incoming road in the α-outside case isn’t as high as in the α-inside case. We also expect a distribution +error in α-outside, which corresponds to Lemma 14. We use the initial conditions +u0,1(x) = +� +0.5, +0.5, +u0,2(x) = +� +0.75, +0, +u0,3(x) = +� +0.25, +x ∈ [1, 1.5], +0, +x ∈ (1.5, 2], +cf. Fig. 5a. The total amount of vehicles is 0.5 on Road 1. These cars are distributed into Road 2 (it has +0.375 cars already) and Road 3 (it has 0.125 cars already) by the distribution coefficients. At the end, we +can expect 0.75 cars on Road 2 and 0.25 cars on Road 3. +We can see the results in Fig. 5. We can observe, that the α-outside Godunov numerical flux creates +a traffic congestion at the end of Road 1, see Figure 5b and 5c. This effect is very subtle, but important +– in Figures 5b and 5c, for the α-outside flux there is a slight increase in density at the end of Road 1 +immediately before the bifurcation to Roads 2 and 3. The key point is that the density there locally rises +above u∗, which is, in a certain sense, the definition of a congestion. This effect does not occur for α-inside +(the density remains beneath u∗) and is artificial in the α-outside case, as this congestion should not occur +– Roads 2 and 3 are able to take in even the larger inflow which the α-inside Godunov flux prescribes there. +However, in this case the α-outside gives a smaller inflow to Roads 2 and 3, leading to the congestion at the +end of Road 1. This is due to the position of the distribution coefficient and evaluation of the minimum in +the calculation of the numerical flux between Road 1 and Road 2. Because f (2) +out < f (1) +in and α2,1f (1) +in < f (2) +out, +the α-inside case takes α2,1f (1) +in as the minimizer in the flux definition, whereas the α-outside case takes f (2) +out +and multiplies it by α2,1. Thus, α-outside has lower inflow from Road 1 and makes a congestion. Once the +traffic density on Road 2 decreases, the inflow is same in both cases, cf. Figure 5d and 5e. +The final results are in Fig. 5f. The numerical flux with α–inside has 0.75 cars on Road 2 and 0.25 +on Road 3, i.e. there is no distribution error and traffic is distributed exactly according to the drivers’ +preferences. The numerical flux with α–outside gives approx. 0.7498 cars on Road 2 and 0.2502 cars on +Road 3. In this case, we have a small distribution error which is caused by a traffic congestion between Road +1 and Road 2, and therefore some drivers prefer Road 3 instead of their original preference of Road 2. Note +that this congestion is caused by the choice of the α–outside Godunov flux, not by the traffic situation in +itself. +6.2 +α-inside Godunov flux vs. Maximum possible traffic flow +Second, we compare the α-inside Godunov flux and Maximum possible traffic flow flux. +We use initial +conditions +u0,1(x) = +� +0, +1, +u0,2(x) = +� +1, +0, +u0,3(x) = +� +0, +x ∈ [0, 0.5], +0, +x ∈ (0.5, 1], +cf. Fig. 6a. The total amount of vehicles is 0.5 cars on Road 1. These cars are distributed into Road 2 (it +has 0.5 cars already) and Road 3 according to the distribution coefficients. At the end, we can expect 0.875 +cars on Road 2 and 0.125 cars on Road 3. +We can see the results in Fig. 6. If we compare the flow through the junction in Fig. 6b and 6c, we +can see that our numerical flux allows flow between Road 1 and Road 3 while the Maximum possible flow +18 + +(a) t = 0. +(b) t = 0.6. +(c) t = 1.2. +(d) t = 1.8. +(e) t = 2.4. +(f) t = 3. +Figure 5: Comparison of α-inside (left) and α-outside (right) Godunov flux on network with Road 1, Road +2 and Road 3. +19 + +p +p +1.0 F +1.0 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0α-inside +α-outside +d +p +1.0 E +1.0 E +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +20p +d +1.0 E +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 ++ +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0(a) t = 0. +(b) t = 0.25. +(c) t = 0.5. +(d) t = 1.25. +(e) t = 2.5. +(f) t = 4. +Figure 6: Comparison of α-inside Godunov flux (left) and Maximum possible traffic flow (right) on network +with Road 1, Road 2 and Road 3. +20 + +α-inside +Maximum possible flow +p +p +1.0 E +1.0 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X ++ +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 E +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0p +p +1.0 F +1.0 F +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +X +X +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0doesn’t. From the time t = 0.5, the congestion on Road 2 decreases and the traffic flows from Road 1 to +Road 2 and 3. The inflow from Road 1 in the case with maximum possible flow is still lower than in case with +our numerical flux. In Figure 6d, the inflow is the same in both cases. At t = 2.5, Road 1 is almost empty +in the case of our Godunov flux (approximately 0.0003 cars), whereas in the case of Maximum possible flow +there are still approx. 0.0414 cars, cf. Figure 6e. It is obvious, that the movement of all cars finished earlier +in the case of the Godunov flux. +The final results are in Fig. 6f. The maximum possible traffic flow has 0.875 cars on Road 2 and 0.125 +on Road 3, i.e. there is no distribution error. The Godunov numerical flux with α–inside has approx. 0.8438 +cars on Road 2 and approx. 0.1562 cars on Road 3. In this case, we have a non–zero distribution error which +is caused by a traffic jam on Road 2, and therefore some drivers prefer rather Road 3. +We note that for both of the compared fluxes, the solution on Road 2 (green) is identical in Figures 6a–6d. +The difference is in the distribution of cars on Road 1 and Road 3 (blue), where the maximum possible flux +gives a zero flow due to the traffic jam, unlike the α-inside flux which allows a small number of cars to enter +Road 3 due to the traffic distribution error. The results on Road 2 start to differ in Figure 6e, since then +all cars are evacuated from Road 1 (here the nonzero flux to Road 3 helped ‘drain’ Road 1 more quickly) in +the α-inside Godunov case, unlike the maximum possible flux where there are still some cars left on Road 1 +supplying an inflow to Roads 2 and 3. +As can be seen from these two examples, the effect of the traffic distribution errors is rather subtle, but +in our view leads to more realistic results, where human drivers tend to adapt to current traffic situations +and change their original preferences in the presence of traffic jams, e.g. by taking alternative routes. In the +approach used in the maximum possible traffic flow, the drivers strictly adhere to their original preferences +under all circumstances, even in extreme situations when one of the outgoing roads is completely jammed +and the other is empty. We propose two interpretations of these phenomena. The first one is that the +more flexible Godunov flux describes human drivers which adapt to current situations, while the Maximum +flow describes e.g. a fleet of communicating autonomous vehicles, which optimize (maximize) the total flow +through the junction, while strictly adhering to the predetermined routes. On the other hand, a typical +human driver does not care about maximizing the total flux through the junction, he simply wants to get +through the junction to his desired outgoing road and does not really care what happens on the other roads. +Another possible interpretation is the presence of dedicated turning lanes in front of the junction (Go- +dunov) and their absence (maximum flow). If dedicated turning lanes are not present, a traffic jam on one of +the outgoing roads causes a congestion in the whole junction, as cars which want to go to another possibly +empty road cannot do so, since they cannot overtake the standing vehicles. It is the presence of turning lanes +that allows these cars to pass the other standing cars, resulting in a nonzero flow through the junction and a +small violation of the predetermined traffic distribution coefficients. Once again we remind that if the roads +are sufficiently free (in the sense of Section 5.2), the drivers strictly adhere to their original preferences even +in the case of the Godunov flux. It is only in the presence of congestions that the flexibility of the Godunov +approach manifests itself (in the form of the traffic distribution error). +7 +Conclusion +In this paper, we dealt with the construction and analysis of two new numerical fluxes for traffic flows on +networks. +We use the discontinuous Galerkin method to discretize the governing equations in the form +of first order nonlinear hyperbolic conservation laws describing the traffic flow on individual roads. The +main contribution is the construction of two new numerical fluxes at the network junctions that are a +generalization of the Godunov numerical flux. The construction is an extension of our previous work [10] +which was based on a generic numerical flux, rather than Godunov. +We prove basic properties of the +two newly proposed Godunov-like fluxes, namely the conservativity of the resulting numerical method (via +discrete Rankine-Hugoniot conditions) and analyze situations when the predetermined drivers’ preferences +are satisfied or possibly violated. +Specifically, the drivers’ preferences at junctions are given by traffic +distribution coefficients. We show that if the junction is not congested, the traffic flows according to these +preferences. Once the junction becomes congested, there can be a small traffic distribution error which we +21 + +interpret either as factoring of human behavior into the model or the existence of dedicated turning lanes +in front of the junction, as opposed to single-lane roads. We demonstrate these phenomena numerically and +compare with the approach to the construction of numerical fluxes taken in [1]. One of the advantages of our +Godunov-like numerical fluxes is the simplicity of their explicit construction for all types of junctions, unlike +the approach of [1] and [4], which requires the solution of a linear programming problem. In subsequent +papers we will prove an entropy inequality for the scheme along with L2 stability and error estimates and +analyze the behavior of limiters at junctions. +References +[1] ˇCani´c, S., Piccoli, B., Qiu, J., Ren, T.: Runge-Kutta Discontinuous Galerkin Method for Traffic Flow +Model on Networks. Journal of Scientific Computing 63, 233–255 (2015) +[2] Cockburn, B., Shu, C.W.: TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element +Method for Conservation Laws II: General Framework. Mathematics of Computation 52(186), 411–435 +(1989) +[3] Dolejˇs´ı, V., Feistauer, M.: Discontinuous Galerkin Method – Analysis and Applications to Compressible +Flow. Springer, Heidelberg (2015) +[4] Garavello, M., Piccoli, B.: Traffic Flow on Networks, vol. 1. American Institute of Mathematical Sciences +(AIMS), Springfield, MO (2006) +[5] Greenshields, B.D.: A Study of Traffic Capacity. Highway Research Board 14, 448–477 (1935) +[6] J¨ungel, A.: Modeling and Numerical Approximation of Traffic Flow Problems. Universit¨at Mainz (2002). +Available online: https://www.asc.tuwien.ac.at/~juengel/scripts/trafficflow.pdf, accessed: +2020-08-14 +[7] Kachroo, P., Sastry, S.: Traffic Flow Theory: Mathematical Framework. +University of California +Berkeley (2012) +[8] Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neutron Transport Equation. Tech. rep., Los +Alamos Scientific Lab., N. Mex.(USA) (1973) +[9] Shu, C.W.: Discontinuous Galerkin Methods: General Approach and Stability. Numerical Solutions of +Partial Differential Equations pp. 149–201 (2009) +[10] Vacek, L., Kuˇcera, V.: Discontinuous Galerkin method for macroscopic traffic flow models on networks. +Communications on Applied Mathematics and Computation 4(3), 986–1010 (2022) +[11] van Wageningen-Kessels, F., van Lint, H., Vuik, K., Hoogendoorn, S.: Genealogy of Traffic Flow Models. +EURO Journal on Transportation and Logistics 4(4), 445–473 (2015) +[12] Zhang, X., Xia, Y., Shu, C.W.: Maximum–Principle–Satisfying and Positivity–Preserving High Order +Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes. Journal of Scientific +Computing 50, 29–62 (2012) +22 + diff --git a/kNE1T4oBgHgl3EQf0QWt/content/tmp_files/load_file.txt b/kNE1T4oBgHgl3EQf0QWt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ee5f200b727d2efd82f82cc93324d119c70d10e --- /dev/null +++ b/kNE1T4oBgHgl3EQf0QWt/content/tmp_files/load_file.txt @@ -0,0 +1,1164 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf,len=1163 +page_content='Godunov–like numerical fluxes for conservation laws on networks∗ Luk´aˇs Vacek† Charles University, Faculty of Mathematics and Physics Sokolovsk´a 83, Praha 8, 186 75, Czech Republic and V´aclav Kuˇcera‡ Charles University, Faculty of Mathematics and Physics Sokolovsk´a 83, Praha 8, 186 75, Czech Republic January 10, 2023 Abstract This paper deals with the construction of a discontinuous Galerkin scheme for the solution of Lighthill- Whitham-Richards traffic flows on networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The focus of the paper is the construction of two new numerical fluxes at junctions, which are based on the Godunov numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We analyze the basic properties of the two Godunov-based fluxes and the resulting scheme, namely conservativity and the traffic distribution property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We prove that if the junction is not congested, the traffic flows according to predetermined preferences of the drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Otherwise a small traffic distribution error is present, which we interpret as either the existence of dedicated turning lanes, or factoring of human behavior into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We compare our approach to that of ˇCani´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Numerical experiments are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 1 Introduction In this paper, we are concerned with the simulation of the movement of traffic on networks of roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We take the macroscopic approach, where the traffic is modeled as a uniform continuum which moves through the roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is opposed to the microscopic approach, where each individual vehicle is modeled separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Since the total number of vehicles is conserved, the basic mathematical model for us will be that of partial differential equations (PDEs) describing conservation laws, namely nonlinear first order hyperbolic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Specifically, we are concerned with the so-called Lighthill-Whitham-Richards (LWR) traffic flow model, where the traffic moves according to an equilibrium flow of homogeneous traffic, which is described by a so-called fundamental diagram, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [5], [6], [7] or [11] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Such approaches to modelling traffic flows on a single road are more or less standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' What is considerably newer and less studied is the generalization of the LWR models to networks of roads, which can be described by an oriented graph, where on each road we have the equations for the LWR model and we need to supply some kind of boundary condition at intersections which correspond to vertices of the graph cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In our case, we use the discontinuous Galerkin (DG) method to discretize the LWR model on each individual road and the behavior of traffic at junctions is determined by prescribing a numerical flux at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In such a case, one must take into account not only the necessity for the resulting scheme to be ∗The work of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Vacek is supported by the Charles University, project GA UK No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 1114119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The work of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Kuˇcera is supported by the Czech Science Foundation, project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 20-01074S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Email: lvacek@karlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='cz ‡Email: kucera@karlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='cz 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='03454v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='NA] 9 Jan 2023 conservative (vehicles are not lost or formed at intersections), but also other properties, such as taking into account the preferences of individual drivers as to which outgoing road they wish to take from the junction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Such numerical fluxes were constructed and applied in [4] and [1], where it is assumed that the drivers behave in such a way as to maximize the total flux through the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This leads to a complicated linear programming problem, which can be explicitly solved (giving an explicit construction of the numerical flux) only in the simplest cases, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [4] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We take a slightly different approach, where the resulting numerical fluxes are explicitly constructed on an arbitrary junction based on the traffic distribution requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The construction seems more natural for human drivers, who are mainly concerned with the traffic density at their specific pair of incoming and outgoing roads and are (somewhat selfishly) not concerned with maximizing the total flux of all the traffic through the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The latter case is much more realistic e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' for a swarm of centrally coordinated or communicating autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We have already described the basic construction of our numerical flux in the paper [10] which was however based on a generic classical numerical flux, as used in DG methods on single roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this paper, we refine the construction and base it on the Godunov numerical flux, which is based on the exact solution of a local Riemann problem, and is therefore a natural numerical flux also from the point of view of the PDE theory, since it corresponds to the so called Bardos-LeRoux-N´ed´elec boundary conditions, which is the correct way how to prescribed Dirichlet data on a boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this paper we take the classical Godunov numerical flux and use it to construct a Godunov-like numerical flux at a general junction, which is based on the drivers’ preferences described by traffic distribution coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Actually, we derived two similar numerical fluxes that differ in the way the traffic distribution coefficients are treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We then proceed to analyze the resulting DG scheme, namely we prove a discrete analogue of the Rankine-Hugoniot condition at the junction, from which we derive global conservativity of the scheme across the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As it turns out, our numerical flux(es) do not exactly satisfy in all situations the apriori preferences of the drivers in the form of relations given by the traffic distribution coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We analyze this effect in detail and derive relations for the traffic distribution error, which we then interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Namely, we can show that if (in some sense) a junction is not congested, the drivers follow their predetermined preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' However this is not true when there is a congestion in the junction, and the original traffic distribution coefficients are not satisfied exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We interpret this in two ways – (a) as typical human behavior, where some drivers decide to change their route if they see that the one they originally chose is blocked, or (b) that there are dedicated turning lanes in the incoming roads before the junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Such turning lanes allow some drivers, whose route of choice is not blocked to pass the ones whose preferred outgoing road is congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If there are no dedicated turning lanes than a traffic jam on one of the outgoing roads blocks all of the traffic in the entire junction, even though other outgoing roads may be completely free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is what happens in the numerical flux considered in [4], [1], but not in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Finally, we demonstrate the mentioned phenomena on simple numerical test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In Section 2 we introduce the basic concepts and notation needed to describe traffic flows on networks along with the traffic distribution coefficients describing the drivers’ preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In Section 3, we introduce the discontinuous Galerkin method on single roads and introduce and reformulate the classical Godunov numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In Section 4 we review the approach of [4], [1] and construct our Godunov-like numerical fluxes at junctions along with the corresponding DG formulation on networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In Section 5, we prove the conservativity of the DG scheme on networks with the considered Godunov fluxes and analyze the traffic distribution error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Finally, we present numerical results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 2 Macroscopic traffic flow models on networks We begin with the mathematical description of macroscopic vehicular traffic, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [6], [7] and [11] for a more detailed treatment of the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' First, we consider a single road described mathematically as a one- dimensional interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In basic macroscopic models, traffic flow is described by three fundamental quantities – traffic flow Q, traffic density ρ and mean traffic flow velocity V , all of these being functions of both the spatial position x and time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The basic governing equation of traffic flow is derived using the assumption that the number of cars in 2 an arbitrary segment [x1, x2] of the road changes only due to the flux through the endpoints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' d dt � x2 x1 ρ(x, t) dx = Q(x1, t) − Q(x2, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (1) Writing the right-hand side as an integral and eliminating the integral over the arbitrarily chosen [x1, x2] gives the conservation law for ρ in the form ∂ ∂tρ(x, t) + ∂ ∂xQ(x, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (2) Equation (2) must be supplemented by an initial condition and appropriate boundary conditions which we will treat in detail in the case of networks of roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Equation (2) is underdetermined, as there is a single equation for two unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus we need to supply another equation or relation between the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Greenshields described a relation between traffic density and traffic flow in the paper [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' He made the assumption derived from observations that in homogeneous traffic (traffic with no changes in time and space), the traffic flow Q is a function which depends only on the density ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let us denote the equilibrium flow of homogeneous traffic as Qe, derived from Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The relationship between the ρ and Qe is described by the so-called fundamental diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The approach where we use the equilibrium traffic flow Qe in equation (2) is called the Lighthill-Whitham-Richards (LWR) traffic flow model and results in the equation ρt + � Qe(ρ) � x = 0, x ∈ R, t > 0, ρ(x, 0) = ρ0(x), x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (3) Equation (3) belongs to the class of nonlinear first order hyperbolic equations and, for practical purposes will be considered on finite intervals with appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' There are many different proposals for the equilibrium traffic flow Qe derived from real traffic data, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Here we present only Greenshields model, which defines the equilibrium traffic flow as Qe(ρ) = vmax ρ � 1 − ρ ρmax � , where vmax is the maximal velocity and ρmax is the maximal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can see the fundamental diagram in Figure 1, where vmax = ρmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Now we consider a road network represented by a directed graph, following [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The graph is a finite collection of directed edges (roads), connected together at vertices (intersections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Each vertex has a finite set of incoming edges and outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='0 ρ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='0 Ve (a) Velocity–density diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='0 ρ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='25 Qe (b) Flow–density diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Figure 1: Fundamental diagrams of the Greenshields model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 3 Figure 2: Example of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 1 (Network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We define a network as a couple (I, J ), where I = {In}N n=1 is a finite set of edges and J = {Jm}M m=1 is a finite set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Each edge In is represented by an interval [an, bn] ⊆ [−∞, ∞], n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Each vertex Jm is a union of two non–empty subsets Inc(Jm) and Out(Jm) of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , N} representing incoming and outgoing edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We assume the following: (i) For all Ji, Jj ∈ J , i ̸= j : Inc(Ji) ∩ Inc(Jj) = ∅ and Out(Ji) ∩ Out(Jj) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (ii) If i /∈ ∪J∈J Inc(J), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , N}, then bi = ∞ and if i /∈ ∪J∈J Out(J), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , N}, then ai = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Moreover, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , N} : i ∈ ∪J∈J Inc(J) or i ∈ ∪J∈J Out(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Condition (i) states that each edge can be incoming for at most one vertex and outgoing for at most one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Condition (ii) states that edges that are connected to only one vertex extend to ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Of course in practice artificial inflow/outflow boundaries are introduced on such edges in the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can see an example in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As we are dealing with traffic flows described by LWR models, we assume that the traffic on each edge number i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , N} is described by a conservation law of the general form (ui)t + (f(ui))x = 0, x ∈ (ai, bi), t > 0, ui(x, 0) = u0,i(x), x ∈ (ai, bi), (4) where ui : (ai, bi) × [0, ∞) → R is the traffic density on the i-th edge (road).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We note that although our primary interest are traffic flow models, the system of equations (4) can represent general conservation laws, which is the reason why we have abandoned the notation based on traffic density ρ and traffic flow Q and use the generic notation u and f in the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We note that we will assume that the convective flux f has a global maximum at some critical value (critical density) u∗ and f is non–decreasing on the interval (−∞, u∗] and non–increasing on [u∗, ∞), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This assumption is typical for traffic flow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' What remains is to describe the behavior of traffic at junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For this purpose it is sufficient to first consider a single vertex (junction) and its incoming and outgoing roads for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The resulting considerations can then be applied separately to each vertex of the general network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We consider a network (I, J ) and fix a vertex J ∈ J for which we assume that Inc(J) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n} and Out(J) = {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We define the spatial limits of traffic densities on individual roads at the common vertex J as u− i (bi, t) := lim x→bi− ui(x, t) and u+ j (aj, t) := lim x→aj+ uj(x, t) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n and j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Now we can define the solution at a junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 2 (Traffic solution at a junction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let J be a junction with incoming roads I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , In and outgoing road In+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , In+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then we define a weak solution at J as a collection of functions ul : Il × [0, ∞) → R, 4 l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m such that n+m � l=1 � bl al � ∞ 0 � ul ∂ϕl ∂t + f(ul)∂ϕl ∂x � dt dx = 0 holds for every ϕl ∈ C1 0([al, bl] × [0, ∞)), l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m, that are also smooth across the junction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' ϕ− i (bi, ·) = ϕ+ j (aj, ·), �∂ϕi ∂x �− (bi, ·) = �∂ϕj ∂x �+ (aj, ·), for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n} and j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The basic property of the weak solution from Definition 2 is that it satisfies the Rankine-Hugoniot condition which is essentially the conservation of vehicles at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , un+m)T be a weak solution at the junction J such that each x → ui(x, t) has bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then u satisfies the Rankine-Hugoniot condition n � i=1 f(u− i (bi, t)) = n+m � j=n+1 f(u+ j (aj, t)) (5) for almost every t > 0 at the junction J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The proof is a simple application of integration by parts and can by found in [4, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 2 simply enforces the conservation of vehicles at J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' However it is also necessary to take into account the preferences of drivers how the traffic from incoming roads is distributed to outgoing roads according to some predetermined coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 4 (Traffic distribution matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let J be a fixed vertex with n incoming edges and m outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We define a traffic distribution matrix A as A = � �� αn+1,1 · · αn+1,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' αn+m,1 · · αn+m,n � �� , where 0 ≤ αj,i ≤ 1 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n}, j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m} and n+m � j=n+1 αj,i = 1 (6) holds for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The ith column of A describes how the traffic from the incoming road Ii distributes to the outgoing roads at the junction J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In other words, if X is the amount of traffic coming from road Ii then αj,iX is the desired amount of traffic going form Ii towards road Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As stated, for simplicity, we assume a simple network with only one junction in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' All our definitions and theorems can then be extended straightforwardly to an arbitrary network (I, J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 3 Discontinuous Galerkin method We discretize the governing equation (3) using the discontinuous Galerkin (DG) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This method introduced by Reed and Hill in [8] represents a robust, reliable and accurate numerical method for the solution of first order hyperbolic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The DG method uses discontinuous piecewise polynomial approximations 5 of the exact solution along with a suitable weak form of the governing equations and can thus be viewed as a combination of the the finite element and finite volume methods, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [3], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' One advantage of the DG method over standard finite elements is its robustness with respect to the Gibbs phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This occurs when a continuous approximation is used to approximate a discontinuous function – these typically arise as solutions to nonlinear first order hyperbolic problems, such as those considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In general, the DG method is described on a polygonal (polyhedral) domain Ω ⊂ Rd, d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let Th be a partition of Ω into a finite number of closed elements K with mutually disjoint interiors, such that Ω = � K∈Th K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Since the traffic model is defined on a line, we consider Ω ⊂ R, Ω = (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the 1D case, an element K is an interval [aK, bK], where aK and bK are boundary points of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We set hK = |bK − aK|, h = maxK∈T hK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We denote the set of all boundary faces (points in 1D) of all elements by Fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Further, we define the set of all inner points by FI h = {x ∈ Fh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' x ∈ Ω} and the set of boundary points FB h = {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Obviously Fh = FI h ∪FB h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We use a suitable weak formulation of (3) on the broken Sobolev space Hk(Ω, Th) = {v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' v|K ∈ Hk(K), ∀K ∈ Th}, where Hk(I), k ∈ N, is the Sobolev space over an interval I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Functions from this space will be approximated by discontinuous piecewise polynomial functions from the space Sh = {v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' v|K ∈ P p(K), ∀K ∈ Th}, where P p(K) denotes the space of all polynomials on K of degree at most p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For each point x ∈ FI h there exist two neighbours K− x , K+ x ∈ Th such that x = K− x ∩K+ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Every function v ∈ Hk(Ω, Th) is generally discontinuous at x ∈ FI h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus, for all x ∈ FI h, we introduce the following notation for traces and jumps: v−(x) = lim y→x− v(y), v+(x) = lim y→x+ v(y), [v]x = v−(x) − v+(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In order to have consistent notation, in the point x ∈ FB h we define v+(a) = lim y→a+ v(y), v(a) = − [v]a = v−(a) := v+(a), v−(b) = lim y→b− v(y), v(b) = [v]b = v+(b) := v−(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The definition of the jump [v]a := −v+(a) or [v]b := v−(b) may seem inconsistent with the definition on interior points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This notation is used due to the integration by parts in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For simplicity, if [·]x appear in a sum of the form � x∈Fh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', we omit the index x and write [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We formulate the DG method for first order hyperbolic problems of the form ut + f(u)x = 0, x ∈ Ω, t ∈ (0, T), (7) u = uD, x ∈ FD h , t ∈ (0, T), (8) u(x, 0) = u0(x), x ∈ Ω, (9) where the Dirichlet boundary condition uD : FD h × (0, T) → R and the initial condition u0 : Ω → R are given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The Dirichlet boundary condition is prescribed only on the inlet FD h ⊆ FB h , respecting the direction of information propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The function f ∈ C1(R) is called the convective flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Our aim is to seek a function u : Ω × (0, T) → R such that (7)–(9) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As we have seen, problem (7) is the main part of macroscopic equations for traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6 In order to derive the DG formulation of (7), we multiply by a test function ϕ ∈ H1(Ω, Th) and integrate over an arbitrary element K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then we apply integration by parts and obtain � K utϕ dx − � K f(u)ϕ′ dx + f(u(bK, t))ϕ−(bK) − f(u(aK, t))ϕ+(aK) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (10) Finally, we sum over all K ∈ Th and obtain � Ω utϕ dx − � K∈Th � K f(u)ϕ′ dx + � x∈Fh f(u) [ϕ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We wish to approximate u by a function uh ∈ H1(Ω, Th) which is in general discontinuous on Fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus, we need to give proper meaning to the function f(uh) in points x ∈ Fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We proceed similarly as in the finite volume method and use the approximation f(uh) ≈ H(u− h , u+ h ), (11) where H(· , · ) is a numerical flux, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Finally, we define the DG solution of problem (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 5 (DG solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The function uh : Ω × (0, T) → R is called a DG finite element solution of hyperbolic problem (7)–(9) if the following properties hold: (i) uh ∈ C1 ([0, T] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Sh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (ii) uh(0) = uh0, where uh0 denotes an Sh approximation of the initial condition u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (iii) uh = uD for all x ∈ FD h , t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (iv) For all ϕ ∈ Sh and for all t ∈ (0, T), uh satisfies � Ω (uh)tϕ dx − � K∈Th � K f(uh)ϕ′ dx + � x∈Fh H(u− h , u+ h ) [ϕ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1 Godunov numerical flux In our implementation, we use the Godunov numerical flux, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is defined as the flux f evaluated at the exact solution of the Riemann problem with the piecewise defined initial condition u− and u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This can be shown to be equivalent to the more practical form, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' [9], HGod orig � u−, u+� = � minu−≤u≤u+ f(u), if u− < u+, maxu+≤u≤u− f(u), if u− ≥ u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (13) We call this form the original form and for our purposes, we use an alternative form which is inspired by the maximum possible traffic flow approach (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1) in the case of one incoming and one outgoing road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 6 (Alternative form of the Godunov numerical flux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let the convective flux f have a global maximum at u∗ and f is non–decreasing on the interval (−∞, u∗] and non–increasing on [u∗, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then the Godunov numerical flux is defined as HGod � u−, u+� = min � fin(u−), fout(u+) � , (14) where fin(u−) = � f(u−), if u− < u∗, f(u∗), if u− ≥ u∗, fout(u+) = � f(u∗), if u+ ≤ u∗, f(u+), if u+ > u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 7 ρ Qe fin fout Figure 3: fin and fout for Greenshields traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This can be interpreted as the maximal possible flow through the common boundary, where fin is the maximal possible inflow from the left element and fout is the maximal possible outflow to the right element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can see fin and fout for Greenshields traffic flow in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Formulas (13) and (14) are equivalent, which is shown in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If the convective flux f attains its global maximum at u∗ and is non–decreasing on (−∞, u∗] and non–increasing on [u∗, ∞), then HGod orig (u−, u+) = HGod (u−, u+) for all u−, u+ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We divide the proof into six different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' There are two main cases based on the ordering of u− and u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then for each case we present three sub-cases, which place u∗ in different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' u− < u+ a) If u− < u+ ≤ u∗ then HGod orig = f(u−) and HGod = min {f(u−), f(u∗)} = f(u−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' b) If u− ≤ u∗ < u+ then HGod orig = min {f(u−), f(u+)} and HGod = min {f(u−), f(u+)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' c) If u∗ < u− < u+ then HGod orig = f(u+) and HGod = min {f(u∗), f(u+)} = f(u+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' u− ≥ u+ a) If u+ ≤ u− < u∗ then HGod orig = f(u−) and HGod = min {f(u−), f(u∗)} = f(u−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' b) If u+ ≤ u∗ ≤ u− then HGod orig = f(u∗) and HGod = min {f(u∗), f(u∗)} = f(u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' c) If u∗ < u+ ≤ u− then HGod orig = f(u+) and HGod = min {f(u∗), f(u+)} = f(u+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We showed that HGod orig (u−, u+) = HGod (u−, u+) in every possible situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the rest of the paper, the numerical flux H(· , · ) will always be the Godunov numerical flux written in the alternative form (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 4 Numerical fluxes at junctions In order to formulate the DG scheme on a simple network, we first need to construct the numerical fluxes at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Such a numerical flux is considered in [1] and [4] which is based on the assumption that the drivers wish to maximize the total flux through the junction while respecting the traffic distribution exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We discuss this approach in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this paper we take a different approach which has the advantage that it is simple and explicitly constructed for all junction types unlike that of [1] and [4], which leads to the solution of a linear programming 8 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We will present two different constructions of the numerical fluxes at the junction based on the Godunov numerical flux in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We shall prove the basic properties of our construction and discuss the differences with the approach of [1] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1 Maximum possible traffic flow Based on the traffic distribution matrix, the authors of [4] define the following admissible traffic solution at a junction, also used for numerical simulations in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 8 (Admissible traffic solution at a junction, following [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , un+m)T be such that ui(·, t) is of bounded variation for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then u is called an admissible weak solution of (4) related to the matrix A at the junction J if the following properties hold: (i) u is a weak solution at the junction J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (ii) f(u+ j (aj, ·)) = �n i=1 αj,if(u− i (bi, ·)), for all j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (iii) �n i=1 f(u− i (bi, ·)) is maximal subject to (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Condition (ii) simply states that traffic from incoming roads is distributed to outgoing roads according to the traffic distribution matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Condition (iii) is a mathematical formulation of the assumption made in [4], that respecting (ii), “drivers choose so as to maximize the total flux” through the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' One problem with the approach of [4] and [1] is that explicitly constructing the fluxes from Definition 8 requires the solution of a linear programming problem on the incoming fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is done in [4] for the purposes of constructing a Riemann solver at the junction and in [1] for the purposes of obtaining numerical fluxes at the junction in order to formulate the DG scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Closed-form solutions are provided in [1] in the special cases n = 1, m = 2 and n = 2, m = 1 and n = 2, m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In Section 4, we present an alternative construction of fluxes at the junction which has the advantage of a simple formulation for general n, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We will give an interpretation of our construction, which shows that it is more suited for certain situations, giving more realistic behavior of the drivers, than the approach from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We compare the two approaches in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1 One incoming and two outgoing roads This example is important to us, because it inspires us in the construction of the α-inside Godunov flux (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We use the method described in [1, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2] with our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this case, we have distribution coefficient α2,1 = α and α3,1 = 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then according to [4], we calculate the maximum possible inflow to the junction from the incoming road as H1(t) = min � fin(u− 1 (b1, t)), fout(u+ 2 (a2, t)) α , fout(u+ 3 (a3, t)) 1 − α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (15) The outflow from the junction to an outgoing road is calculated as H1 multiplied by the corresponding distribution coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' H2(t) = αH1(t) and H3(t) = (1 − α)H1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We note, that traffic congestion on one of the outgoing roads influences the traffic flow to the second outgoing road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For example, when fout(u+ 2 ) = 0, then H1 = H2 = H3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Therefore, the intersection is completely blocked, even in the case when the other outgoing road I3 is completely empty (u3 ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is caused by the assumption in Definition 8 that drivers strictly adhere to their original preferences for the choice of the outgoing road, even on a congested intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus the flows to the outgoing roads I2, I3 must always be in the ratio α to 1 − α, irrespective of the traffic situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2 α-outside Godunov flux In our previous paper [10], we based the construction of the numerical flux at the junction on the Lax- Friedrichs numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this paper we start from the Godunov numerical flux, which will have advantages that we will see in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In either case, we construct the junction fluxes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' At the junction, we consider an incoming road Ii and an outgoing road Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If these roads were the only roads at the junction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' if they were directly connected to each other, the (numerical) flux of traffic from Ii to Ij would simply be H � u− hi(bi, t), u+ hj(aj, t) � , where uhi and uhj are the DG solutions on Ii and Ij, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' From the traffic distribution matrix, we know the ratios of the traffic flow distribution to the outgoing roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus, we take the numerical flux Hj(t) at the left point of the outgoing road Ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' at the junction, at time t as Hj(t) := n � i=1 αj,iH � u− hi(bi, t), u+ hj(aj, t) � , (16) for j = n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The numerical flux Hj(t) can be viewed as the DG analogue of taking the combined traffic outflow �n i=1 αj,if � u− i (bi, t) � from all incoming roads and prescribing it as the inflow of traffic to the road Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Similarly, we take the numerical flux Hi(t) at the right point of the incoming road Ii, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' at the junction, at time t as Hi(t) := n+m � j=n+1 αj,iH � u− hi(bi, t), u+ hj(aj, t) � , (17) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Again, this can be viewed as an approximation of the traffic flow �n+m j=n+1 αj,if � u+ j (aj, t) � being prescribed as the outflow of traffic from Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Now we can formulate the DG method for the simple network with one junction using the numerical fluxes defined in (16) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then the case of general networks is a straightforward generalization, where the aforementioned construction of numerical fluxes at junctions is applied on each junction separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We consider the DG formulation (12) on every incoming and outgoing road represented by the intervals (ai, bi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n and (aj, bj), j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Since the DG method is applied on finite intervals, we replace the endpoints at ±∞ from Definition 1 by artificial inflow/outflow boundaries at finite points along with inflow Dirichlet data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For every interval (ak, bk), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m, we consider a partition Thk along with the corresponding discrete space Shk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We write the DG formulation directly for the case of LWR models (3) with unknown density u and flux f(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 9 (DG formulation on a simple network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We seek functions uhk ∈ C1 ([0, T] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Shk), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n+ m satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (i) Incoming roads: For all i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n and all ϕi ∈ Shi � bi ai (uhi)tϕi dx − � K∈Thi � K f(uhi)ϕ′ i dx + � x∈FI hi H � u− hi, u+ hi � [ϕi] +Hiϕ− i (bi) − H � uDi, u+ hi(ai) � ϕ+ i (ai) = 0, (18) where Hi = Hi(t) is the numerical flux defined in (17) and uDi is the Dirichlet datum corresponding to the left artificial inflow boundary point ai of (ai, bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (ii) Outgoing roads: For all j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m and all ϕj ∈ Shj � bj aj (uhj)tϕj dx − � K∈Thj � K f(uhj)ϕ′ j dx + � x∈FI hj H � u− hj, u+ hj � [ϕj] +H � u− hj(bj), uDj � ϕ− j (bj) − Hjϕ+ j (aj) = 0, (19) where Hj = Hj(t) is the numerical flux defined in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 10 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We note that the choice of the arguments in the numerical flux at the artificial boundary point bj in (19) corresponds to an outflow boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Typically, uDj = u− hj(bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This term could be rewritten using the original physical flux f(u− hj(bj)) due to consistency of the numerical flux H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definitions (16) and (17) are independent of the specific choice of the numerical flux H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the paper [10], we used the Lax-Friedrichs numerical flux, while in this paper we use Godunov’s flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Since the distribution coefficients αj,i are outside of the numerical flux H in (16) and (17), we call this construction the α-outside Godunov flux, as opposed to the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='3 α-inside Godunov flux When comparing the maximum possible traffic flow (15) with the α-outside Godunov flux (17), we find the main difference in the position of the distribution coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' While in (17) the traffic distribution coefficient is outside of the minimization (14) defining the Godunov flux, in (15) this coefficient is inside the minimization defining the Godunov-like flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This leads to the idea of moving the distribution coefficients in (16) and (17) inside the Godunov numerical fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' To this end we define an auxiliary Godunov-like flux with a third variable for the traffic distribution coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Definition 10 (Godunov numerical flux with three variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The Godunov numerical flux with three variables is defined as HGod � u−, u+, α � = min � αfin(u−), fout(u+) � , (20) where fin(u−) and fout(u+) are defined as in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The reason why we put the distribution coefficient in front of fin, is the representation of the real flow from the incoming road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Only αj,ifin(u− i (bi, t)) cars per time unit want to go from incoming road i to outgoing road j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' So it makes sense to use αj,ifin in the definition of the Godunov flux instead of fin as in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For simplicity, we drop the superscript “God” and define the numerical flux with three variables H(· , · , · ) as the flux (20) in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Now we can take the numerical flux Hj(t) with α-inside at the left point of the outgoing road Ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' at the junction, at time t as Hj(t) := n � i=1 H � u− hi(bi, t), u+ hj(aj, t), αj,i � , (21) for j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Similarly, we take the numerical flux Hi(t) with α-inside at the right point of the incoming road Ii, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' at the junction, as Hi(t) := n+m � j=n+1 H � u− hi(bi, t), u+ hj(aj, t), αj,i � , (22) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can use the same DG formulation as in Definition 9 with Hi defined as (22) and Hj defined as (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='4 One incoming and two outgoing roads We use the same example as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1 with one incoming and two outgoing road and compare all three approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Our aim is to identify and discuss the differences between them and describe their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For simplicity, we use the notation f (1) in := fin(u− 1 (b1, t)), f (2) out := fout(u+ 2 (a2, t)) and f (3) out := fout(u+ 3 (a3, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the case of the α-outside Godunov flux, we calculate the inflow to the junction from the incoming road as H1(t) = α min � f (1) in , f (2) out � + (1 − α) min � f (1) in , f (3) out � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (23) We compare this flux value to that obtained by the maximum possible flow (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' At first glance equations (15) and (23) seem completely different, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' if fin is less than both f (2) out and 11 Approach H1 H2 H3 Maximum min{f (1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (2) out α ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (3) out 1−α} α min{f (1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (2) out α ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (3) out 1−α} (1 − α) min{f (1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (2) out α ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (3) out 1−α} possible α-outside α min{f(1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f(2) out} + (1 − α) min{f(1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f(3) out} α min{f (1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (2) out} (1 − α) min{f (1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (3) out} Godunov α-inside min{αf(1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f(2) out} + min{(1 − α)f(1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f(3) out} min{αf (1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (2) out} min{(1 − α)f (1) in ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (3) out} Godunov Table 1: Comparison of the three approaches on the example with one incoming and two outgoing roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' f (3) out then equations (15) and (23) give the same value fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The outflows from the junction are the individual terms in the right hand side of (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Hence, H2(t) = α min{f (1) in , f (2) out} and H3(t) = (1 − α) min{f (1) in , f (3) out}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If we compare H2 and H3 from the maximum possible flow and α-outside Godunov flux, we can see the distribution coefficient α and 1 − α in front of the minimization in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Again, if fin is less than both f (2) out and f (3) out, both of the approaches give the same values of H2 and H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' But if the minimizer is f (2) out and f (2) out divided by the corresponding distribution coefficient, respectively, then the maximum possible flow gives us H2 = f (2) out while α-outside Godunov gives us H2 = αf (2) out, which is a lower flux value than the maximum possible traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can sum up these considerations informally in the statement that if the outgoing roads are emptier than the incoming road then the maximum possible traffic flow and the α-outside Godunov flux coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Once one of the outgoing roads is fuller than the incoming one, the two approaches differ, the latter one giving a smaller traffic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the case of the α-inside Godunov flux, we calculate the inflow to the junction from the incoming road as H1(t) = min � αf (1) in , f (2) out � + min � ((1 − α) f (1) in , f (3) out � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (24) This approach is somewhere between the α-outside Godunov flux and maximum possible flow, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Again, if fin is less than both f (2) out and f (3) out, equations (15), (23) and (24) give us same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' But if one of the fout is the minimizer, we typically get three different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Also in this case the outflows from the junction are the individual terms in (24), hence H2(t) = min{αf (1) in , f (2) out} and H3(t) = min{(1−α)f (1) in , f (3) out}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Here is the main difference between the α-outside and α-inside Godunov fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Of course, when fin is less than both f (2) out and f (3) out, we get the same values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' But if f (2) out is the minimizer, α-inside Godunov gives us the same H2 as the maximum possible flow (H1 and H3 are typically different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is why we say that α-inside Godunov lies between the other two approaches and takes positives from both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 5 Properties In this section, we look at the basic properties of the numerical fluxes at junctions that we considered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Namely, the Rankine-Hugoniot conditions and the satisfaction of the traffic distribution according to the coefficients in the traffic distribution matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1 Rankine-Hugoniot condition First, we show that our Godunov-based fluxes satisfy the discrete analogues of the Rankine-Hugoniot con- dition (5), which leads to conservativity of the resulting DG scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Lemma 11 (Discrete Rankine–Hugoniot condition for α-outside Godunov flux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The numerical fluxes (16) and (17) with α outside satisfy the discrete version of the Rankine–Hugoniot condition (5): n � i=1 Hi(t) = n+m � j=n+1 Hj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' From the definition of Hi and Hj, we immediately obtain n � i=1 Hi(t) = n � i=1 n+m � j=n+1 αj,iH � u− hi(bi, t), u+ hj(aj, t) � = n+m � j=n+1 n � i=1 αj,iH � u− hi(bi, t), u+ hj(aj, t) � = n+m � j=n+1 Hj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Lemma 12 (Discrete Rankine–Hugoniot condition for α-inside Godunov flux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The numerical fluxes (21) and (22) with α inside satisfy the discrete version of the Rankine–Hugoniot condition (5): n � i=1 Hi(t) = n+m � j=n+1 Hj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' From the definition of Hi and Hj, we immediately obtain n � i=1 Hi(t) = n � i=1 n+m � j=n+1 H � u− hi(bi, t), u+ hj(aj, t), αj,i � = n+m � j=n+1 n � i=1 H � u− hi(bi, t), u+ hj(aj, t), αj,i � = n+m � j=n+1 Hj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The previous two lemmas allow us to prove that for the DG scheme, the total number of vehicles in the network is conserved (modulo inlet and outlet boundary conditions), which is the basic property we expect from the conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Since the DG scheme is naturally conservative on each individual road, the question boils down to conservativity of the scheme at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Corollary 13 (Conservation property of the DG scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let uDj = u− hj(bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The DG scheme from Definition 9 conserves the total number of vehicles in the network in the sense that d dt n+m � k=1 � bk ak uhk dx = n � i=1 H � uDi, u+ hi(ai) � − n+m � j=n+1 H � u− hj(bj), uDj � for both α–outside and α–inside Godunov fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We set all test functions as ϕk ≡ 1 for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m and sum together all of the equations (18) and (19) for all i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We get d dt n+m � k=1 � bk ak uhk dx + n � i=1 Hi − n+m � j=n+1 Hj + n+m � j=n+1 H � u− hj(bj), uDj � − n � i=1 H � uDi, u+ hi(ai) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The second and third terms cancel one another since � i Hi − � j Hj = 0 due to Lemma 11 or 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2 Traffic distribution error The purpose of the traffic distribution coefficients αj,i from Definition 4 is to describe the preferences of the drivers from each incoming road for which outgoing road they want to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In other words, how the traffic from each incoming road is distributed among the outgoing roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the end this means we want to satisfy condition (ii) from Definition 8 in terms of the numerical fluxes at the junction: Hj(t) = n � i=1 αj,iHi(t), j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m (25) The maximum possible flux approach from [4] is based on maximizing the total flux through the junction while satisfying (25) exactly under any circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this section we show that in the case of our Godunov-based fluxes, this relation is satisfied if, loosely speaking, the junction is not congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In other words, whenever the outgoing roads can accept the incoming traffic, the drivers drive according to their original preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If any of the outgoing roads cannot accept the incoming traffic, relation (25) is no longer satisfied exactly, but with a small error (traffic distribution error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We interpret this as a natural behavior of human drivers – when one of the preferred outgoing roads is full, some drivers will change their original preferences and take a different route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this context, strictly adhering to (25) irrespective of the traffic situation would correspond to non-human drivers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' autonomous vehicles with a prescribed course which cannot be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Another interpretation is the presence of dedicated turning lanes in front of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' These allow other cars to pass the standing vehicles which want to go to a congested outgoing road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In a single-lane road, even one standing vehicle can block the whole road if it cannot proceed to its desired outgoing road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Such a vehicle blocks the way for other drivers, even those who could otherwise proceed since their desired outgoing roads are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This was discussed in Remark 2 for the maximum possible flux, where a single congested road blocks the entire junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this section, we analyze when the traffic distribution condition (25) is satisfied exactly for our Godunov- based fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the following lemmas, we express the traffic distribution error for these fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Lemma 14 (Traffic distribution error for α–outside Godunov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The numerical fluxes (16) and (17) with α outside satisfy Hj(t) = n � i=1 αj,iHi(t) + Ej(t) for all j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m, where the error term is Ej(t) = n � i=1 n+m � l=n+1 l̸=j αj,iαl,i � Hi,j(t) − Hi,l(t) � , (26) where Hi,j(t) := H � u− hi(bi, t), u+ hj(aj, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' By definition (16), Hj(t) = n � i=1 αj,iHi,j(t) = n � i=1 αj,iHi(t) + n � i=1 αj,i � Hi,j(t) − Hi(t) � � �� � Ej(t) , where Ej(t) is the error term which we will show has the form (26): By Definition (17), we have Ej(t) = n � i=1 αj,i � Hi,j(t) − n+m � l=n+1 αl,iHi,l(t) � = n � i=1 αj,i n+m � l=n+1 αl,i � Hi,j(t) − Hi,l(t) � = n � i=1 n+m � l=n+1 l̸=j αj,iαl,i � Hi,j(t) − Hi,l(t) � , 14 since �n+m l=n+1 αl,i = 1 due to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Lemma 15 (Traffic distribution error for α–inside Godunov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The numerical fluxes (21) and (22) with α inside satisfy Hj(t) = n � i=1 αj,iHi(t) + Ej(t) for all j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m, where the error term is Ej(t) = n � i=1 n+m � l=n+1 l̸=j (αl,iHi,j(t) − αj,iHi,l(t)) , (27) where Hi,j(t) := H � u− hi(bi, t), u+ hj(aj, t), αj,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' By definition (21), Hj(t) = n � i=1 Hi,j(t) = n � i=1 αj,iHi(t) + n � i=1 � Hi,j(t) − αj,iHi(t) � � �� � Ej(t) , where Ej(t) is the error term which we will show has the form (27): By Definition (22), we have Ej(t) = n � i=1 � Hi,j(t) − αj,i n+m � l=n+1 Hi,l(t) � = n � i=1 n+m � l=n+1 l̸=j � αl,iHi,j(t) − αj,iHi,l(t) � , since �n+m l=n+1 αl,i = 1 due to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Now we discuss general situations when the traffic distribution errors are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We note that we have also analyzed the traffic distribution errors in our previous paper [10], where the numerical fluxes at the junction were based on the Lax-Friedrichs flux instead of Godunov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In that case, the errors were in general always nonzero but small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We therefore view the Godunov-like approach of this paper as more natural, since the traffic distribution relation (25) is satisfied exactly in many situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Namely, in Theorem 1 we show that if the density on all the outgoing roads is smaller than u∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' the density where the traffic flow is maximal, then the traffic is distributed according to (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the following, we shall consider a fixed time t and omit the argument t from the functions uh, Hi,j, Ej and similar, in order to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Theorem 1 (Zero traffic distribution error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Let u+ hj(aj) ≤ u∗ for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then Ej = 0 for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m} for both the α–inside and α–outside Godunov fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For the α-outside flux, due to the form (26) of the error term, if Hi,j = Hi,l for all i, j, l, then Ej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This happens, in general, when the inflow term is the minimizer in (14) in both numerical fluxes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Hi,j = Hi,l = fin(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The simplest general case when this happens is if fout(u+ hj(aj)) = fout(u+ hl(al)) = f(u∗) ≥ fin(u− hi(bi)), or in other words if u+ hj(aj) ≤ u∗ and u+ hl(al) ≤ u∗ for all l, j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For the α-inside flux, due to the form (26) of the error term, if αl,iHi,j = αj,iHi,l for all i, j, l, then Ej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Again,if the inflow term is the minimizer in (20) in both numerical fluxes, we get αl,iHi,j = αl,iαj,ifin(u− hi(bi)) = αj,iHi,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This occurs if fout(u+ hj(aj)) = f(u∗) ≥ αj,ifin(u− hi(bi)), or in other words if u+ hj(aj) ≤ u∗ for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Theorem 1 states that the traffic is distributed exactly according to the original preferences if the outgoing roads are sufficiently free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The result is valid for both of the presented variants of the Godunov flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If we take into account the specific form of the flux, where f(0) = f(umax = 0, we can perform a finer analysis, which allows heavier traffic through the junction, while still satisfying (25) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 15 Theorem 2 (Zero traffic distribution error for α–outside).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Assume that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n} one of the following conditions is satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' u− hi(bi) ≥ u∗ and u+ hj(aj) ≤ u∗ for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' u− hi(bi) < u∗ and u+ hj(aj) ≤ ˜u for all j ∈ {n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n+m}, where ˜u > u∗ is such that f(˜u) = f(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then Ej = 0 for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As in the proof of Theorem 1, we wish to have Hi,j = Hi,l for all i, j, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' And again we achieve this by assuming the inflow term is the minimizer in both numerical fluxes in (14), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Hi,j = Hi,l = fin(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Consult Figure 4(a) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If u− hi(bi) ≥ u∗ then fin(u− hi(bi)) = f(u∗) and for Hi,j = Hi,l = fin(u∗) to hold, we must necessarily have also fout(u+ hj(aj)) = f(u∗), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' u+ hj(aj) ≤ u∗ for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Since u− hi(bi) < u∗ and since f is non-increasing on [u∗, ∞), there exists ˜u > u∗ such that f(˜u) = f(u− hi(bi)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then if u+ hj(aj) ≤ ˜u for all j ∈ {n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n+m}, we have fout(u+ hj(aj)) ≥ f(˜u) = f(u− hi(bi)) = fin(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Therefore, Hi,j = fin(u− hi(bi)) for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}, hence Hi,j = Hi,l for all j, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Theorem 3 (Zero traffic distribution error for α–inside).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Assume that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n} one of the following conditions is satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' u− hi(bi) ≥ u∗ and u+ hj(aj) ≤ ˜uj for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}, where ˜uj ≥ u∗ is such that f(˜uj) = αj,if(u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' u− hi(bi) < u∗ and u+ hj(aj) ≤ ˜uj for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}, where ˜uj > u∗ is such that f(˜uj) = αj,if(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Then Ej = 0 for all j ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' , n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Due to Lemma 15, we wish to have αl,iHi,j = αj,iHi,l for all i, j, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' And again we achieve this by assuming the inflow term is the minimizer in both numerical fluxes in the Godunov flux (20), since then we will have αl,iHi,j = αj,iHi,l = αj,iαl,ifin(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Consult Figure 4(b) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' ρ f ui (bi) u* u\uf02d fin fout (a) α–outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' ρ f ui (bi) u* u\uf02d α fin fout (b) α–inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Figure 4: Illustration of Theorems 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 16 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If u− hi(bi) ≥ u∗ then fin(u− hi(bi)) = f(u∗) and fout(u+ hj(aj)) ≥ f(˜uj) = αj,if(u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Hence Hi,j = αj,if(u∗) and similarly Hi,l = αl,if(u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Therefore αl,iHi,j = αj,iHi,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If u− hi(bi) < u∗ then fin(u− hi(bi)) = f(u− hi(bi)) and fout(u+ hj(aj)) ≥ f(˜uj) = αj,if(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Hence Hi,j = αj,if(u− hi(bi)) and similarly Hi,l = αl,if(u− hi(bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Therefore αl,iHi,j = αj,iHi,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We note that Theorems 2 and 3 are more general in that they allow larger densities on the outgoing roads than Theorem 1, while still satisfying the traffic distribution condition (25) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Theorem 1 requires u+ hj(aj) ≤ u∗ on all outgoing roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' However, Theorems 2 and 3 allow a weaker condition u+ hj(aj) ≤ ˜uj for some ˜uj > u∗ provided that u− hi(bi) < u∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' if the incoming roads are not too full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Moreover, the value of this allowed density ˜uj is higher for the α–inside Godunov than for α–outside, as can be seen when comparing the values of ˜u in Figures 4(a) and 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus the α–inside numerical flux allows for a larger traffic flow through the junction, while still satisfying the traffic distribution condition (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6 Numerical results Here we present numerical experiments comparing the three Godunov-like numerical fluxes considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As for the implementation, integrals over individual elements in (12) are evaluated using Gaussian quadrature rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Basis functions of the space Sh are taken as Legendre polynomials on individual elements, where the support of each basis function is a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' By writing uh in terms of basis functions in space and setting the test function ϕ to elements of the basis, equation (12) reduces to a system of ordinary differential equations which is solved by Adams–Bashforth methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The DG method is much less susceptible to the Gibbs phenomenon than the finite element method, however spurious oscillations can still occur locally in the vicinity of discontinuities or steep gradients in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' There are several approaches how to treat these local oscillations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' adding local artificial diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In our case, we apply limiters to the DG solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In our implementation, we use the modified minmod limiter from [2], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' also [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Often the solution of (7) is a physical quantity which satisfies some admissibility conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' the physical density must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If we obtain a solution which is not in the admissible interval, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' due to overshoots or undershoots, the problem can become ill-posed or even undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is our case, since the traffic density u must naturally satisfy u ∈ [0, umax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The DG method by itself does not guaranty such bounds are satisfied for the discrete solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Limiters usually prevent this from happening, however in traffic flows, it is natural that entire regions of the computational domain have u = 0 or u = umax and it is easy for the algorithm to produce e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' negative density due to round-off errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' To prevent this from happening, we use the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If the average density on an element K is in the admissible interval, we decrease the slope of our solution so that the modified density lies in [umin, umax] similarly as in the limiting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The important property is that the element average does not change after the application of the limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As further insurance, if the average density on an element K is not in the admissible interval [0, umax], then we change the solution such that u ≡ 0 or u ≡ umax on the whole element K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The latter case, when the average density on an element is not in the admissible interval is extremely rare and, for us, serves as an indicator that the time step is too large or the mesh is too coarse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Since in this case the described procedure does not conserve the total number of vehicles, we rather decrease the time step or increase the number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' For polynomials of higher degree, we can use the method described in the paper [12] by Zhang, Xia and Shu, which reduces to the procedure described above in the simplest case of piecewise linear approximations in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We consider a simple network with one incoming road (Road 1, colored red in Figures 5 and 6) and two outgoing roads, Road 2 (green) and Road 3 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The network will be closed at their endpoints (a1, b2 and b3) meaning that the inflow density at a1 is equal to zero and the outflow densities at b2 and b3 are maximal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus, we can check the total number of cars, because we have neither inflow nor outflow into the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We choose α2,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='75 and α3,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The length of all roads is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We use the combination of the explicit Euler method (step size τ = 10−4) and DG method (number of elements N = 150 on each 17 road).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We calculate the piecewise linear approximations of solutions and we use two Gaussian quadrature points in each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We use Greenshields model with vmax = 1 and ρmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the following section, we set the initial condition to see the differences between approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We choose congested examples to demonstrate the distribution error from Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In non–congested cases, the traffic distribution error is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1 α-outside vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' α-inside Godunov flux First, we compare the α-outside and α-inside Godunov fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This example shows that the traffic flow from the incoming road in the α-outside case isn’t as high as in the α-inside case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We also expect a distribution error in α-outside, which corresponds to Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We use the initial conditions u0,1(x) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5, u0,2(x) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='75, 0, u0,3(x) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='25, x ∈ [1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5], 0, x ∈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5, 2], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The total amount of vehicles is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5 on Road 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' These cars are distributed into Road 2 (it has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='375 cars already) and Road 3 (it has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='125 cars already) by the distribution coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' At the end, we can expect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='75 cars on Road 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='25 cars on Road 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can see the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can observe, that the α-outside Godunov numerical flux creates a traffic congestion at the end of Road 1, see Figure 5b and 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This effect is very subtle, but important – in Figures 5b and 5c, for the α-outside flux there is a slight increase in density at the end of Road 1 immediately before the bifurcation to Roads 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The key point is that the density there locally rises above u∗, which is, in a certain sense, the definition of a congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This effect does not occur for α-inside (the density remains beneath u∗) and is artificial in the α-outside case, as this congestion should not occur – Roads 2 and 3 are able to take in even the larger inflow which the α-inside Godunov flux prescribes there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' However, in this case the α-outside gives a smaller inflow to Roads 2 and 3, leading to the congestion at the end of Road 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' This is due to the position of the distribution coefficient and evaluation of the minimum in the calculation of the numerical flux between Road 1 and Road 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Because f (2) out < f (1) in and α2,1f (1) in < f (2) out, the α-inside case takes α2,1f (1) in as the minimizer in the flux definition, whereas the α-outside case takes f (2) out and multiplies it by α2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Thus, α-outside has lower inflow from Road 1 and makes a congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Once the traffic density on Road 2 decreases, the inflow is same in both cases, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Figure 5d and 5e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The final results are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The numerical flux with α–inside has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='75 cars on Road 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='25 on Road 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' there is no distribution error and traffic is distributed exactly according to the drivers’ preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The numerical flux with α–outside gives approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='7498 cars on Road 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2502 cars on Road 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this case, we have a small distribution error which is caused by a traffic congestion between Road 1 and Road 2, and therefore some drivers prefer Road 3 instead of their original preference of Road 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Note that this congestion is caused by the choice of the α–outside Godunov flux, not by the traffic situation in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2 α-inside Godunov flux vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Maximum possible traffic flow Second, we compare the α-inside Godunov flux and Maximum possible traffic flow flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We use initial conditions u0,1(x) = � 0, 1, u0,2(x) = � 1, 0, u0,3(x) = � 0, x ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5], 0, x ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5, 1], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The total amount of vehicles is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5 cars on Road 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' These cars are distributed into Road 2 (it has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5 cars already) and Road 3 according to the distribution coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' At the end, we can expect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='875 cars on Road 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='125 cars on Road 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We can see the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If we compare the flow through the junction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6b and 6c, we can see that our numerical flux allows flow between Road 1 and Road 3 while the Maximum possible flow 18 (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (c) t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (d) t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (e) t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (f) t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Figure 5: Comparison of α-inside (left) and α-outside (right) Godunov flux on network with Road 1, Road 2 and Road 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 19 p p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='0 F 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='0doesn’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' From the time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5, the congestion on Road 2 decreases and the traffic flows from Road 1 to Road 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The inflow from Road 1 in the case with maximum possible flow is still lower than in case with our numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In Figure 6d, the inflow is the same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' At t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='5, Road 1 is almost empty in the case of our Godunov flux (approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='0003 cars), whereas in the case of Maximum possible flow there are still approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='0414 cars, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Figure 6e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' It is obvious, that the movement of all cars finished earlier in the case of the Godunov flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The final results are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 6f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The maximum possible traffic flow has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='875 cars on Road 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='125 on Road 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' there is no distribution error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The Godunov numerical flux with α–inside has approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='8438 cars on Road 2 and approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='1562 cars on Road 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In this case, we have a non–zero distribution error which is caused by a traffic jam on Road 2, and therefore some drivers prefer rather Road 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We note that for both of the compared fluxes, the solution on Road 2 (green) is identical in Figures 6a–6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The difference is in the distribution of cars on Road 1 and Road 3 (blue), where the maximum possible flux gives a zero flow due to the traffic jam, unlike the α-inside flux which allows a small number of cars to enter Road 3 due to the traffic distribution error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The results on Road 2 start to differ in Figure 6e, since then all cars are evacuated from Road 1 (here the nonzero flux to Road 3 helped ‘drain’ Road 1 more quickly) in the α-inside Godunov case, unlike the maximum possible flux where there are still some cars left on Road 1 supplying an inflow to Roads 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' As can be seen from these two examples, the effect of the traffic distribution errors is rather subtle, but in our view leads to more realistic results, where human drivers tend to adapt to current traffic situations and change their original preferences in the presence of traffic jams, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' by taking alternative routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In the approach used in the maximum possible traffic flow, the drivers strictly adhere to their original preferences under all circumstances, even in extreme situations when one of the outgoing roads is completely jammed and the other is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We propose two interpretations of these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The first one is that the more flexible Godunov flux describes human drivers which adapt to current situations, while the Maximum flow describes e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' a fleet of communicating autonomous vehicles, which optimize (maximize) the total flow through the junction, while strictly adhering to the predetermined routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' On the other hand, a typical human driver does not care about maximizing the total flux through the junction, he simply wants to get through the junction to his desired outgoing road and does not really care what happens on the other roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Another possible interpretation is the presence of dedicated turning lanes in front of the junction (Go- dunov) and their absence (maximum flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' If dedicated turning lanes are not present, a traffic jam on one of the outgoing roads causes a congestion in the whole junction, as cars which want to go to another possibly empty road cannot do so, since they cannot overtake the standing vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' It is the presence of turning lanes that allows these cars to pass the other standing cars, resulting in a nonzero flow through the junction and a small violation of the predetermined traffic distribution coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Once again we remind that if the roads are sufficiently free (in the sense of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='2), the drivers strictly adhere to their original preferences even in the case of the Godunov flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' It is only in the presence of congestions that the flexibility of the Godunov approach manifests itself (in the form of the traffic distribution error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 7 Conclusion In this paper, we dealt with the construction and analysis of two new numerical fluxes for traffic flows on networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We use the discontinuous Galerkin method to discretize the governing equations in the form of first order nonlinear hyperbolic conservation laws describing the traffic flow on individual roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The main contribution is the construction of two new numerical fluxes at the network junctions that are a generalization of the Godunov numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' The construction is an extension of our previous work [10] which was based on a generic numerical flux, rather than Godunov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We prove basic properties of the two newly proposed Godunov-like fluxes, namely the conservativity of the resulting numerical method (via discrete Rankine-Hugoniot conditions) and analyze situations when the predetermined drivers’ preferences are satisfied or possibly violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Specifically, the drivers’ preferences at junctions are given by traffic distribution coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We show that if the junction is not congested, the traffic flows according to these preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Once the junction becomes congested, there can be a small traffic distribution error which we 21 interpret either as factoring of human behavior into the model or the existence of dedicated turning lanes in front of the junction, as opposed to single-lane roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' We demonstrate these phenomena numerically and compare with the approach to the construction of numerical fluxes taken in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' One of the advantages of our Godunov-like numerical fluxes is the simplicity of their explicit construction for all types of junctions, unlike the approach of [1] and [4], which requires the solution of a linear programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' In subsequent papers we will prove an entropy inequality for the scheme along with L2 stability and error estimates and analyze the behavior of limiters at junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' References [1] ˇCani´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Piccoli, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Qiu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Ren, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=': Runge-Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Journal of Scientific Computing 63, 233–255 (2015) [2] Cockburn, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Shu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' : TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Mathematics of Computation 52(186), 411–435 (1989) [3] Dolejˇs´ı, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Feistauer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=': Discontinuous Galerkin Method – Analysis and Applications to Compressible Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Springer, Heidelberg (2015) [4] Garavello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Piccoli, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=': Traffic Flow on Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006) [5] Greenshields, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' : A Study of Traffic Capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Highway Research Board 14, 448–477 (1935) [6] J¨ungel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=': Modeling and Numerical Approximation of Traffic Flow Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Universit¨at Mainz (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Available online: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='asc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='at/~juengel/scripts/trafficflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='pdf, accessed: 2020-08-14 [7] Kachroo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Sastry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=': Traffic Flow Theory: Mathematical Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' University of California Berkeley (2012) [8] Reed, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Hill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' : Triangular Mesh Methods for the Neutron Transport Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Los Alamos Scientific Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' (USA) (1973) [9] Shu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' : Discontinuous Galerkin Methods: General Approach and Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Numerical Solutions of Partial Differential Equations pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' 149–201 (2009) [10] Vacek, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Kuˇcera, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=': Discontinuous Galerkin method for macroscopic traffic flow models on networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Communications on Applied Mathematics and Computation 4(3), 986–1010 (2022) [11] van Wageningen-Kessels, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', van Lint, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Vuik, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Hoogendoorn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=': Genealogy of Traffic Flow Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' EURO Journal on Transportation and Logistics 4(4), 445–473 (2015) [12] Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=', Shu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' : Maximum–Principle–Satisfying and Positivity–Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} +page_content=' Journal of Scientific Computing 50, 29–62 (2012) 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQf0QWt/content/2301.03454v1.pdf'} diff --git a/l9FPT4oBgHgl3EQfIDTd/vector_store/index.faiss b/l9FPT4oBgHgl3EQfIDTd/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..67f8a16e88001e75366fec38aa6b3ba97d9de663 --- /dev/null +++ b/l9FPT4oBgHgl3EQfIDTd/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:920d9b76434e2888bacccf386d93e8396e1c4c193086150a4f16cdd9fe5b73bc +size 5898285 diff --git a/lNE0T4oBgHgl3EQf7wKH/content/2301.02780v1.pdf b/lNE0T4oBgHgl3EQf7wKH/content/2301.02780v1.pdf new file mode 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b/ltFJT4oBgHgl3EQfZSy-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e554079a753b4e00c7059127914b5f1640111f1f969ebdd6364eece60ab01a67 +size 4456493 diff --git a/mNE2T4oBgHgl3EQfzAhZ/content/tmp_files/2301.04126v1.pdf.txt b/mNE2T4oBgHgl3EQfzAhZ/content/tmp_files/2301.04126v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b88bb3b9005e5b9dbdcab462825db6e2ae93b69f --- /dev/null +++ b/mNE2T4oBgHgl3EQfzAhZ/content/tmp_files/2301.04126v1.pdf.txt @@ -0,0 +1,1335 @@ +Temporal Weights +Adam Kohan +akohan@cs.umass.edu +Edward A. Rietman +erietman@cs.umass.edu +Hava T. Siegelmann +hava@cs.umass.edu +Biologically Inspired Neural and Dynamical Systems +College of Information and Computer Sciences +University of Massachusetts Amherst +Amherst, MA 01003 +Abstract +In artificial neural networks, weights are a static representation of synapses. How- +ever, synapses are not static, they have their own interacting dynamics over time. +To instill weights with interacting dynamics, we use a model describing synchro- +nization that is capable of capturing core mechanisms of a range of neural and +general biological phenomena over time. An ideal fit for these Temporal Weights +(TW) are Neural ODEs, with continuous dynamics and a dependency on time. +The resulting recurrent neural networks efficiently model temporal dynamics by +computing on the ordering of sequences, and the length and scale of time. By +adding temporal weights to a model, we demonstrate better performance, smaller +models, and data efficiency on sparse, irregularly sampled time series datasets. +1 +Introduction +Time provides an opportunity for neural networks to change their computation. This change may +be in response to the evolving dynamics of the input distribution, possibly a change in trajectory +triggered by some sparse event, or may be to improve performance by taking another pass over the +same input, but with a different perspective. Static weights do not adjust for either of these situations +and perform the same computation regardless of the passage of time. A neural network with static +weights has only a form of short term, working memory with its internal hidden states (nodes) to +track time steps, but not change its computation (weights) on its inputs and memory. In contrast, +neural networks with temporal weights can change how they process their own memory and inputs +over time, not only keep track of time. +At each time step, a neural network with temporal weights changes its computation by constructing +weights W‘ from a fixed set of parameters W, which is considered to be a form of long term stored +memory, and the current time step t: W ‘ = S(W, t) +Here, the weights are a nonlinear function of parameters and time. As a result, the parameters +are shared across time, but the expressivity of the weight values is not limited to simple linear +combinations of weights or shifts in time. Instead, the weights themselves can capture some dynamics +of the input distribution and form their own trajectory over time. Given S is non-linear, we are +amplifying the effective amount of weights in the model by apply the same fixed number of parameters +differently at each time step. In contrast, static weights are only the parameters directly applied to the +inputs. They can only capture single values, not any dynamics that are separate from the layer or +network. +There are many options of temporal dynamics for S to model. We use a model of coupled oscillators +describing sychronization behaviors that captures core mechanisms of a range of neural and general +Preprint. Under review. +arXiv:2301.04126v1 [cs.NE] 13 Dec 2022 + +biological phenomena over time. Under this model, weights have an explicit dependence on time and +have changing interactions with each other over time. Details are in Section 2. +Static Weights +Temporal Weights +Latent Trajectories +Samples from Prior +Data Space +Latent Space +Data Space +Figure 1: A comparison between a neural ode model +with static weights or temporal weights. +To demonstrate the difference between +static and temporal weights in practice, we +graph in 1 three samples from a neural +ode with static weights (left) and temporal +weights (rights). The neural ode makes for +a strong baseline as it captures temporal tra- +jectories itself and is a state of the art model +for irregularly sampled data. Furthermore, +the effect of temporal weights would have +to be non-trivial to alter this trajectory and +significant to improve over it. To ensure +the practicality of a difficult problem, the +neural ode models are trained on a sparse, +irregularly sampled dataset (details of the +dataset are below). As shown, even a neu- +ral ode model with static weights is not as +expressive as the same model with tempo- +ral weights. Notice the well-defined kinks, +quasi-periodicity, and consistent alignment +of the temporal weights model, allowing +for a better fit for evolving dynamics and for sparse and irregularly sampled data. In comparison, the +same model with static weights is inconsistent across samples and has limited smooth periodicity, +which will have more difficulty fitting irregularities or changes in dynamics over time. We can see +that temporal weights successfully amplifies the parameters to increase the expressivity of the neural +ode model. +Sample 1 +Sample 2 +Sample 3 +Static Weights +Temporal Weights +Figure 2: Neural ODE model fit on a +sparse and irregular synthetic dataset, +with temporal weights or static weights. +The result of increasing the expressivity of the neural ode +model is a better fit on highly sparse and irregular data, +which the original model struggles to fit. In 2, we show +results of the neural ode models on a synthetic dataset of +periodic trajectories with variable frequency, amplitude, +noise, and discontinuities. To further increase the difficulty +of this demonstration, the missing points are not used for +training.Fitting a dataset without using the missing points +to learn from is an entirely more challenging task than +using them for training before evaluating the model with- +out them. It more accurately represents data in the wild, +such as the PhysioNet ICU dataset, where the points are +truly missing. So, the model will need to infer the missing +points by referencing other samples for examples of those +missing points, instead of training on them directly. We +also limit training to 50 iterations such that overfitting is +made difficult and the models must rely on their efficiency +in processing the limited data. +As shown in 2 (left), the neural ode model with static +weights struggles to fit the data, even though it is able to +capture the general trend. As mentioned above, this model +has a limited smooth periodicity. In contrast, the neural +ode with temporal weights, shown in 2 (right), is able to +efficiently capture the local details of the data. Through +time, this model can continuously perturb its own dynamics and adjust or change its trajectory. Given +the difficulty of this task, neither model perfectly fits the data. However, the neural ode with temporal +weights clearly performs better, demonstrating more variability across samples as it adapts to that +sample’s data points. +2 + +Sampleo(dataspace +Sample1 (dataspace) +Sample2(dataspace) +2.5 - +2.5 - +2.5 +2.0 +2.0 +2.0 +1.5 +1.5 +1.5 +X ++ +1.0 - +1.0 +1.0 +0.5 - +0.5 +0.5 +0.0 - +0.0 - +0.0 - +2 +3 +-1 +0 +0 +1 +2 +3 +4 +0 +3 +4 +Time +Time +Time +Latenttrajectoriesz(t)(latent space) +Sliceofvectorfield(latentspace) +Samplesfromprior(dataspace) +dim 1 +6 +1.0 - +5.0 - +dim 3 +dim 6 +2.5 +dim 8 +2 +0.5 - +dim 9 +N +0.0 +dim 10 +dim 12 +0.0 +-2.5 +dim 15 +dim 18 +-5.0 +dim 19 +-0.5 +0 +1 +-2 +3 +-6 +4 +2 +4 +-1 +2 +-4 +-2 +0 +3 +4 +Time +TimeSampleo(dataspace) +Sample1(dataspace) +Sample2(dataspace) +2.5 +2.5 - +2.5 - +2.0 +2.0 +2.0 +1.5 +1.5 +1.5 +X +1.0 - +1.0 +1.0 - +0.5 +0.5 +0.5 +0.0 - +0.0 - +0.0 - +0 +2 +3 +0 +2 +4 +0 +2 +3 +4 +Time +Time +Time +Latenttrajectoriesz(t)(latent space) +Sliceofvectorfield(latentspace) +Samplesfromprior(dataspace) +dim 0 +6 +1.5 - +5.0 +dim 1 +dim 2 +2.5 +dim 3 +1.0 +2 +0.0 - +dim 4 +dim 5 +0 +2.5 +dim 6 +dim 7 +5.0 +dim 8 +0.0 - +-7.5 +dim 9 +-1 +-2 +3 +.6 +-2 +0 +-1 +4 +-4 +2 +3 +4 +6 +4 +Time +TimeSampleo(dataspace +Sample1(dataspace) +Sample2(dataspace) +2.5 - +2.5 - +2.5 - +2.0 +2.0 - +2.0 +1.5 +1.5 +1.5 +X +X +1.0 - +1.0 +1.0 +0.5 - +0.5 +0.5 +0.0 - +0.0 - +0.0 - +4 +-1 +0 +2 +3 +0 +1 +2 +3 +4 +0 +3 +4 +Time +Time +Time +Latenttrajectoriesz(t)(latent space) +Sliceof vectorfield(latentspace) +Samplesfromprior(dataspace) +5.0 - +0.0 - +2.5 - +N +0.0 - +X +-0.5- +-2.5 - +1.0 - +-5.0 - +2 +-0 +1 +2 +3 +0 +0 +2 +1 +2 +-3 +-6 +-4 +-2 +4 +6 +4 +Time +TimeSampleo(dataspace +Sample1(dataspace) +Sample2(dataspace) +2.5 - +2.5 - +2.5 +2.0 +2.0 +2.0 +1.5 +5 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +0.0 - +0.0 - +0.0- +1 +2 +11 +0 +3 +4 +0 +2 +0 +1 +2 +3 +4 +Time +Time +Time +Latenttrajectoriesz(t)(latent space) +Sliceofvectorfield(latentspace) +Samplesfromprior(dataspace +6 +5.0 +1 +2.5 +2 +0.0 - +0 +2.5 +-5.0 - +2 +-7.5- +T- +2 +3 +6 +4 +-6 +-4 +0 +2 +4 +6 +0 +2 +3 +4 +Time +TimeSampleo(dataspace) +Sample1(dataspace) +Sample2(dataspace) +2.5 +2.5 - +2.5 - +2.0 +2.0 +2.0 +1.5 +1.5 +1.5 +X +1.0 - +1.0 +1.0 - +0.5 +0.5 +0.5 +0.0 - +0.0 - +0.0 - +0 +2 +3 +0 +2 +4 +0 +2 +3 +4 +Time +Time +Time +Latenttrajectoriesz(t)(latent space) +Sliceofvectorfield(latentspace) +Samplesfromprior(dataspace) +dim 0 +6 +1.5 - +5.0 +dim 1 +dim 2 +2.5 +dim 3 +1.0 +2 +0.0 - +dim 4 +dim 5 +0 +2.5 +dim 6 +dim 7 +5.0 +dim 8 +0.0 - +-7.5 +dim 9 +-1 +-2 +3 +.6 +-2 +0 +-1 +4 +-4 +2 +3 +4 +6 +4 +Time +TimeSampleo(dataspace +Sample1 (dataspace) +Sample2(dataspace) +2.5 - +2.5 - +2.5 +2.0 +2.0 +2.0 +1.5 +1.5 +1.5 +X ++ +1.0 - +1.0 +1.0 +0.5 - +0.5 +0.5 +0.0 - +0.0 - +0.0 - +2 +3 +-1 +0 +0 +1 +2 +3 +4 +0 +3 +4 +Time +Time +Time +Latenttrajectoriesz(t)(latent space) +Sliceofvectorfield(latentspace) +Samplesfromprior(dataspace) +dim 1 +6 +1.0 - +5.0 - +dim 3 +dim 6 +2.5 +dim 8 +2 +0.5 - +dim 9 +N +0.0 +dim 10 +dim 12 +0.0 +-2.5 +dim 15 +dim 18 +-5.0 +dim 19 +-0.5 +0 +1 +-2 +3 +-6 +4 +2 +4 +-1 +2 +-4 +-2 +0 +3 +4 +Time +Time2 +Methods +We developed a neural model whose weights depend explicitly on time and are . Our model increases +the capacity of the network by incorporating the natural phenomena of time into the parameters, +instead of solely increasing the number of parameters. In our approach, we use a biological model of +time dependent behavior in neurons as the basis for temporal weights. We formulate the biological +model as a weight scaling algorithm with oscillatory dynamics for time based reconfigurations of the +weights. +2.1 +Neural Synchronization +The scaling function used in our algorithm is based on models of mass neural synchronization, +which has been attributed to play a role in movement Cassidy et al. [2002] and memory Klimesch +[1996]. The dynamic modification of synaptic connections that is critical to neural synchronization +is recognized to be the basis of long-term memory in learning Abbott and Nelson [2000], Shimizu +et al. [2000]. These synaptic dynamics bring about adaptive development of network structure whose +trajectory is oriented by the dependencies of the inputs on underlying learning mechanisms. Herein, +we focus on time-based dependencies of the underlying learning mechanisms. +A scaling function S is learned alongside the parameters W of the network. Although W is fixed +after training, the network’s temporal weights W‘ = S(W, t) are different at each time-step due to +the time-dependence of the scaling function. We treat each weight w′ +i ∈ W‘ as individual units that +are controlled by the underlying mechanisms of mass neural synchronization. Our scaling function +models synchronization behaviors in natural dynamic systems: it varies the coupling and decoupling +of different subsets of weights given the relative time and the relationship between weights (Fig. 3). +Figure 3: Full and Partial Phase Locking +of Weights at Some Time Delta. +In partial phase locking, clusters of weights are each syn- +chronized to a different frequency. In full phase locking, +most or all of the weights are synchronized to the same fre- +quency. Our scaling function S taken over the weights W‘ +learns parameters W to adjust this phase locking behavior +at each time step. Our scaling function provides a time +based prior for the network. That is, separate weights at +each time step to process each input of a sequence are re- +lated to each other and are non-linear combinations of each +other. The network is thus capable of relating individual +weights together and constructing different combinations +of weights to focus on the properties of each input in a +sequence revealed by its point in time. It can isolate different subsets of weights (i.e. weight +configurations) to apply an input-dependent ordered series of functions to the model’s internal state. +2.2 +Neural ODE Model +We apply our scaling function to Neural Ordinary Differential Equation models Chen et al. [2018]. +Neural ODE models are time continuous models whose uniquely defined latent trajectory lets us +extrapolate predictions arbitrarily far forwards or backwards in time. In comparison, discrete models +require observations (frequent data points) to control the trajectory and hence are ill-defined with +missing data or irregularity. Neural ODEs are state of the art for irregularly sampled data Rubanova +et al. [2019]. Neural ODEs are a natural fit for temporal weights. While Neural ODEs are time- +dependent, their weights are static and only indirectly depend on time as part of the network’s internal +state. Our temporal weights makes weights dependent on time explicitly. +To generate the trajectories between inputs, the Neural ODE model makes calls to an ODE solver. +This solver breaks down the time interval into smaller subintervals and subsequently approximates +the solution to the ODE at the endpoints of these intervals. Given the solver must call the scaling +function when computing each of these intermediate values, it also has direct control of the number +of weight configurations produced by from a set of parameters. Depending on the complexity of +trajectory induced by the input, the model will apply a different number of functions to its internal +state. The model can effectively learn when to increase or decrease the use of its parameters. +3 + +Measureof PhaseLocking +Coherence +Phasecentroid +Partial PhaseLocking +Full PhaseLockingFigure 4: Computational Graph of Latent ODE and Temporal Weights. (Left) The trajectory of the +ODE model varies continuously with time, even at times between receiving inputs. On the other hand, +the state of RNN only changes in response to inputs. (Right) Temporal Weights construct weights at +each time step, which are used by the Latent ODE model. This includes time steps where there are +inputs or outputs (e.g. time steps 1, N, and M), and time steps where there is neither, but the ode +solver is making calls to the neural network, such as the time step between N and N + 1. A view into +the ODE solver, denoted by ODESOLVE in the figures, with temporal weights is shown in Figure 5. +e +0 +1 +2 +3 +!" +!# +!$ +!% +Input +0 +1 +2 +3 +&# +&' +&# +&' +&' +&# +&' +&# +&#% +&#% +&( +&( +&( +&#% +&#% +&(𝑊`# = 𝑆(𝑊#,t) +𝑊`* = 𝑆(𝑊*,t) +ℎ(𝑡-.#) +ℎ 𝑡- +𝑤ℎ𝑒𝑟𝑒 + ℎ 𝑡3 = 𝑧 +Time +1 +𝑁 +2 +𝑁 + 1 +𝑀 +ℎ`# = 𝑡𝑎𝑛ℎ ℎ(𝑡-)𝑊`# +ℎ`* = 𝑡𝑎𝑛ℎ ℎ`#𝑊`* +𝑑ℎ(𝑡-) +𝑑𝑡 += 𝑓> ℎ 𝑡- , 𝑡 +𝑂𝐷𝐸𝑆𝑂𝐿𝑉𝐸 +1 +𝑁 +2 +𝑁 + 1 +𝑀 +… +… +… +𝑓> ℎ 𝑡- , 𝑡 +ℎ 𝑡# +ℎ 𝑡D +ℎ 𝑡E +… +Inputs +• +Initial state z +• +Previous hidden state +Outputs are hidden states +(A) +(B) +(C) +Time +(D) +Figure 5: Internal Procedure of ODE Solver with Temporal +Weights. The ODE solver calls a multilayer fully connected +neural network internally fθ. (A) At each input time step, the +solver gives the time ti and the previous solver’s hidden state +h(ti) to the network fθ. (B) For each layer l, the network +constructs weights W‘l using the parameters Wl and the +time t. The weights W‘l are used by the layers to process the +network’s input h(ti) and the network’s previous hidden state +h‘l. (C) The network outputs the next hidden state h(ti+1), +which is given to the ODE solver. (D) If there is an output +time step at ti+1, h(ti+1) is also outputted from the solver. +The algorithm is provided in Fig 6. +The neural ODE is able to control the +use of temporal weights through the +ode solver. The ode solver chooses the +frequency and the time steps to call +the temporal weight scaling function. +So, the neural ODE automatically +adapts the usage temporal weights to +the input. In contrast, discrete neural +networks respond directly to observa- +tions and the observations set the fre- +quency and time steps with which to +call the temporal weight scaling func- +tion. As a result, the usage of tempo- +ral weights is limited by the number +of observations in discrete networks. +To more completely utilize the capac- +ity of temporal weights, we use neural +ODE models. +We apply temporal weights to the +Latent ODE and ODE RNN models +of Rubanova et al. [2019]. +Latent +ODEs and ODE RNNs are state of +the art models on data with arbitrary +time gaps between observations, irreg- +ularly sampled data. The equations +and internal procedures of the Neu- +ral ODE with temporal weights used +within these models are shown in Figures 4 and 5. Refer to Rubanova et al. [2019] for further details +of the base models. Unless otherwise stated in 3, we follow the training procedure and configuration +of hyperparameters from Rubanova et al. [2019] for consistency. We demonstrate that with temporal +weights, model size can be reduced and have similar or better performance. +2.3 +Temporal Scaling Function +The scaling function S drives a sinusoidal wave to scale the weights. The sine function can be used +to output a scalar between [0, 1] or [−1, 1] for each weight in the layer. Different weights will have +4 + +ODESOLVE +....ODESOLVEAlgorithm 1 Neural ODE with Temporal Weights +Input: Latent z, Output Times {ti}i=1...M +Provide: Network fθ with L Layers +call ODESOLV E(fθ, z, {ti}i=1...M) +for h (ti) , t given by the solver do +call fθ (h (ti) , t) +Let h‘0 = h (ti) +for layer l = 1 to L do +W‘l = S (Wl, t) +h‘l = tanh (h‘l−1W‘l) +end for +Output h (ti+1) = h‘L to the solver +end call +end for +Return {hi}i=1...M +end call +different magnitudes over time or even be inverted. The output of the sine function changes with time +t, given by the ODE solver. The time is shifting the phase of the sine wave. We let the network learn +parameters to control scale of time for each weight separately or jointly. That means the network has +the capacity to reuse the same learned dynamics repeating the period over time, or learn to split up +the period over different time intervals, or both. For example, the network may scale time such that +each period 2π repeated every dt time interval. Or, the network may split up the period 2π into π/2 +chunks spanning dt time intervals. Or, both. Regardless, the network is able to control the dynamics +of the network through the weights over time. The learned dynamics in the scaling algorithm are +a function of the parameters W and a comparison of each parameter to all the other weights. The +weights W‘ for a layer are a function of this comparison, not the actual weight itself, as shown in +Fig 4. The network’s weight matrix at each layer is of interactions between weights, instead of the +weights themselves. The resulting equation is: +w‘i = Si (Wl, t) +(1) += +#Wl +� +j=1 +K{ij} +l +#Wl +sin [fl(t) ∗ (wi − wj) + φl(t)] +(2) +where K{ij} +l +is the parameter matrix of coupling coefficients, (wi − wj) is the interaction of weight i +with other weights scaled and shifted by the current time t in the ODE solver, f(t) are the scaling +parameters, and φl(t) are the shifting parameters. All the parameters K{ij} +l +, fl(t), φl(t) are learned +with the parameters w ∈ Wl. +The scaling function takes the time t and parameter matrix Wl for layer l to construct a weight +w‘i ∈ W‘l. The matrix K contains learned coefficients that control the strength of coupling between +weights. The sine function is over the difference between each weight in a layer. The phase of this +function is controlled by t. The result is a periodic estimator of weight distances that evolves as a +function of time t. +The scaling function is taken for each layer and each time step as shown in Fig. 4. Each layer has +it’s neurons synchronized over time separately from the other layers. The comparison between all +weights in the scaling function can increase memory usage. Future work may explore a global neural +synchronization across the network if this issue is alleviated. +3 +Experiments +Ten experiments over five datasets with temporally dependent features show that models with temporal +weights more closely capture time dynamics. They have better performance in fewer epochs and +with less parameters than the same model with static weights. The datasets are sparse and irregularly +sampled to better ensure: (1) the applicability of our model in the wild where data may be missing, +5 + +difficult to acquire, or streamed on-line; and (2) that our model is learning the underlying distributions +over time, not memorizing. +We use the Pytorch library Paszke et al. [2019] and train networks on a single NVIDIA GeForce +GTX TITAN X GPU. We use the Adamax optimizer with learning rates 0.01, 0.04 and exponential +learning rate decays 0.999, 0.9999. The best performing model is reported. Individual details are +listed under each experiment. Refer to 2.2 for additional information. +3.1 +PhysioNet ICU +We evaluate TW on the PhysioNet Challenge 2012 Silva et al. [2012] dataset of 12,000 ICU stays. At +admission, a one-time set of general descriptors, such as age or gender, is collected one. During a +stay in the ICU, a sparse subsets of 37 measurements are taken, such as Lactate, Mg, and NA levels. +Measurements are from the at least the first 48 hours of an ICU stay. Each stay is labeled whether +the patient survives hospitalization or not. Some measurements are at regular intervals and others +are asynchronous and at irregular times, collected only when required. There are more than 4,600 +possible measurements per time series. To lower training time, we halved the number measurement +times by rounding to the nearest minute, leaving 2,880 possible measurements per time series. The +dataset is challenging: it is extremely sparse with a missing rate of around 80% and has highly +imbalanced class distributions with a prevalence ratio of around 14%. +Mortality Prediction We look to predict which patients survive hospitalization given the information +collected during the first 48 hours of an ICU stay. We use Area Under the Curve (AUC) as our +performance metric due to the class imbalance. Results are in 1a. The same model with TW converges +to a better performance in fewer epochs and with less than half the parameters, but is about 1.5x +slower per epoch (refer to 5). +Table 1: PhysioNet ICU +METHOD +AUC +EPOCHS +# PARAMS +L-ODE +0.857 (0.836) +80 (31) +163,972 +W/ TW +0.861 +31 +76,427 +(a) Mortality prediction on the PhysioNet ICU +dataset. Given the first 48 hours of an ICU stay, +predict in-hospital mortality. +METHOD +MSE X10-3 +EPOCHS +# PARAMS +L-ODE +2.280 (2.340) +59 (49) +67,071 +W/ TW +1.370 +49 +52,016 +L-ODE +2.208 (2.300) +91 (56) +67,071 +W/ TW +1.900 +56 +52,026 +(b) ICU measurements (Top) Interpolation. (Bot- +tom) Extrapolation. Given the first 24 hours of +measurements, predict the next 24 hours of mea- +surements. (Parenthesis compare the models at the +same epoch.) +Table 2: MuJoCo Physics Simulation +METHOD +MSE X10-3 +EPOCHS +# PARAMS +L-ODE +3.60 +94 +617,619 +W/ TW +3.00 +91 +112,739 +(a) Interpolation of hopper body position. We sub- +sample 10% of the time points and predict the other +90%. +METHOD +MSE X10-2 +EPOCHS +# PARAMS +L-ODE +1.190 +93 +617,619 +W/ TW +1.100 +99 +112,739 +L-ODE +1.480 +77 +617,619 +W/ TW +1.220 +68 +112,739 +(b) Extrapolation of hopper body positions. Given +the first half of the timeline, predict body position +in the second half. We subsample 10% of the +time points in the first half of the timeline at each +batch (top) or once when constructing the dataset +(bottom). +Interpolation and Extrapolation We also look at the task to model the PhysioNet Challenge +measurements data. In this task, we impute missing measurements in each sample to reconstruct the +full set of points in the time series. We report the mean squared error average over the reconstruction +of the dataset. Results are in 1b. In the extrapolation task, we provide the network with the first half +of the timeline and construct measurements in the second half of the timeline. Results are in 1b. The +latent ODE model with temporal weights converges to a significantly better performance in both the +extrapolation and interpolation tasks. +Architecture The Latent ODE is as follows: an encoder comprised of a GRU of 50 units, hidden +size of 40, and Neural ODE of size 50 with 3 layers; a latent size of 20; and a decoder comprised +6 + +of a Neural ODE of size 50 with 3 layers. The Latent ODE with temporal weights is as follows: an +encoder comprised of a GRU of 25 units, hidden size of 20, and Neural ODE of size 25 with 3 layers; +a latent size of 20; and a decoder comprised of a Neural ODE of size 25 with 3 layers. +3.2 +Human Activity +We evaluate TW on the Human Activity dataset Kaluža et al. [2010] of 5 individuals performing 7 +activities, such as sitting and walking, resulting in 25 sequences of 6,600 time points each on average. +6,554 sequences of 211 time points each. The dataset has 12 features of 3D positions from tags +attached to a belt, chest, and each ankle (3 dimensions x 4 tags). +Time-Point Classification The task is to classify each time point into one of the activities for a total +of 1,477 (7 x 211) outputs. The model with temporal weights has better performance, converges in +fewer epochs. Results are in 3. +Table 3: Human Activity. Classification at +each time point. (Parenthesis compare the +models at the same epoch.) +METHOD +ACC +EPOCHS +# PARAMS +L-ODE +0.846 (0.748) +52 (10) +1,696,763 +W/ TW +0.870 +10 +141,023 +Table 4: Climate prediction. +METHOD +MSE +NLL +EPOCHS +# PARAMS +GRUODE +0.43 +0.84 +74 +42,640 +W/ TW +0.40 +0.87 +68 +9,105 +Table 5: Sepsis prediction. At every hour, +predict whether the patient will have sepsis +within the next 6 to 12 hours. (Parenthesis +compare the models at the same epoch.) +METHOD +AUC +EPOCHS +# PARAMS +ODERNN +0.689 (0.669) +67 (24) +148,672 +ODERNN +0.765 (0.696) +86 (24) +680,462 +W/ TW +0.787 +24 +149,519 +NCDE +0.925 (0.905) +130 (110) +55,949 +NCDE +0.925 +110 +193,541 +W/ TW +0.931 (0.918) +180 (110) +58,553 +Architecture The Latent ODE is as follows: an encoder comprised of a GRU of 50 units, hidden size +of 100, and Neural ODE of size 500 with 4 layers; a latent size of 15; and a decoder comprised of +a Neural ODE of size 500 with 2 layers. The Latent ODE with temporal weights is as follows: an +encoder comprised of a GRU of 25 units, hidden size of 20, and Neural ODE of size 25 with 3 layers; +a latent size of 20; and a decoder comprised of a Neural ODE of size 25 with 3 layers. +3.3 +MuJoCo Physics Simulation +The physics simulation dataset Rubanova et al. [2019] is 10,000 sequences of 100 regularly sampled +14-dimensional time-points generated from the MuJoCo Todorov et al. [2012] simulator. The dataset +is of a bipedal model, called the Hopper model in the Deepmind Control Suite Tassa et al. [2018]. +The task is to learn to approximate Newtonian physics of the hopper rotating in the air and falling +to the ground starting from a randomly sampled position and velocity. The resulting deterministic +trajectories are dictated by their initial states. This should be straightforward for the Latent-ODE +model to fit as it matches the assumptions made by the Latent-ODE model, it is dictated by its initial +conditions. +In the interpolation task, we subsample 10% of the time points to simulate sparse observation times, +and predict the other 90% time points. Results are in 2a. In the extrapolation task, we provide the +network with the first half of the timeline, and then predict the second half. We subsampling 10% +of the time points from the first half of the timeline, but predict on all the points in the second half +of the timeline. We show results in for 2b two versions of this task. In the first version, shown in +the top rows of the table, we subsample 10% of the time points at every batch. This means that the +networks will through multiple epochs eventually see all the data points of an input since a different +7 + +10% of the time points are sampled each epoch. In the second version, shown in the bottom rows of +the table, we subsample 10% of the time points once when the dataset is constructed such that the +network will never see the missing the time points. The second version of this task is a more difficult +as the missing time points are never used for training. We see in the results that the performance gap +between the model with and without temporal weights doubles. In these tasks, we demonstrate that +the latent ODE models performs better with temporal weights, and has 5 times fewer parameters. Our +model has a similar capacity to the larger model, but with much fewer parameters. +Architecture The Latent ODE is as follows: an encoder comprised of a GRU of 100 units, hidden +size of 30, and Neural ODE of size 300 with 3 layers; a latent size of 15; and a decoder comprised of +a Neural ODE of size 300 with 3 layers. The Latent ODE with temporal weights is as follows: an +encoder comprised of a GRU of 50 units, hidden size of 40, and Neural ODE of size 50 with 3 layers; +a latent size of 15; and a decoder comprised of a Neural ODE of size 50 with 3 layers. +3.4 +PhysioNet Sepsis +The PhysioNet Sepsis dataset contains time series of observations from 40,000 ICU patients that were +aggregated from two different U.S. hospitals. Each patient has demographics as a general descriptor, +such as agent and gender. During a stay in the ICU, up to 40 measurements are taken, such as vital +signs and laboratory results. Measurements were recorded together every hour. Each hour is labeled +whether or not an onset of sepsis occurred. The dataset is sparse with a missing rate of around 74% +and has imbalanced class distributions with a prevalence ratio of around 7.3%. +Sepsis Prediction The task is to predict at every hour whether the patient will have sepsis within the +next 6 to 12 hours. Results are in 5. The model with temporal weights has better performance with a +similar number of parameters. +Architecture The smaller ODE RNN with and without TW is as follows: an encoder comprised of a +GRU of 50 units; a latent size of 32; a decoder comprised of an MLP of 100 units with 2 layers; and +a Neural ODE of size 10 with 1 layer. The larger ODE RNN without TW is as follows: an encoder +comprised of a GRU of 1024 units; a latent size of 32; a decoder comprised of an MLP of 100 units +with 2 layers; and a Neural ODE of size 20 with 1 layer. +3.5 +USHCN Daily Climate Data +The United State Historical Climatology Network (USHCN) daily dataset Menne et al. [2010] +contains measurements of 5 climate variables (temperatures, precipitation, and snow) over 150 years +for 1,218 meteorological stations. We use the processed data from De Brouwer et al. [2019]. It +contains a subset of 1,114 stations over an observation window of 4 years subsampled such that each +station has around 60 observations on average. The task is to predict the next 3 measurements after +the first 3 years of observation. +Architecture The smaller GRUODE with TW is as follows: A p_model with 2 layers of 25 units and +an input of 15 units. A classification network with 2 layers of 2 units and an input of 50 units. A gru +with 3 layers of 15 units and 2 layers of 45 units. A covariates map with 2 layers of 50 units and an +output of 15 units. +The larger GRUODE without TW is as follows: A p_model with 2 layers of 25 units and an input +of 50 units. A classification network with 2 layers of 2 units and an input of 50 units. A gru with 3 +layers of 50 units and 2 layers of 150 units. A covariates map with 2 layers of 50 units and an output +of 50 units. +4 +Related Works +Temporal weights are a model of dendrites and synapses. The common view is that memories are +encoded in the connection strength between neurons. In artificial neural networks, static weights +represent (are inspired by) synapses. However synapses are not static, they have their own dynamics, +in addition to the dynamics of the neurons they connect to. +In Poirazi and Mel [2001], the authors investigate the storage capacity of neurons that have synapses +with their own dynamics. The authors compare (1) synapses that are linearly summed across dendritic +8 + +arbors (contacts with neurons) with (2) synapses where the dendritic compartments are non-linear +functions. The results demonstrate a much larger capacity for neurons with non-linear synaptic and +dendritic subunits. This suggests that long term memory storage is an active process encoded in +the dynamics of synapses and dendrites. Our results support these findings: a neural network with +temporal weights has similar (or better) performance to a larger version of same network without +temporal weights. +Given the increased capacity from the non-linearity, in temporal weights, we makes this capacity +accessible to the neural network for processing inputs by modeling the behavior of synaptic plasticity +over time. Synaptic plasticity is commonly known for mediating learning and behavior by changing +synaptic strength given a local or global signal or an input. This mediation can be viewed as a form +of selective content addressing. We take a look at synaptic plasticity from this view, as a mechanism +to make information available to the network. Specifically, we take this view while looking at the +temporal behavior of synaptic plasticity in changing synaptic strength. Synaptic plasticity and in turn +changes in synaptic strength are usually looked at in direct response to some stimuli. However, we +look at similar response patterns to stimuli over time. +In Dobrunz and Stevens [1999], the authors stimulated synapses with natural patterns derived from in +vivo recordings. These natural patterns do not have constant frequency. They provide a better dataset +to judge the relative importance of short- (and long-) term synaptic plasticity for usual synaptic +function. The authors find that synaptic strength is being modulated by the timing of the stimulus. +Similar changes in synaptic strength occur at different time scales given input history. In temporal +weights, we capture this behavior in two approaches. First, the scaling function is based on models +of mass neural synchronization whose behavior oscillates over time. Second, we include additional +parameters in the scaling function for the network to learn a scaling and length of time for the dataset. +In total, the network is able to change its weights based on the timing of an observation in the input +sequence. +Temporal weights work similarly to synaptic plasticity mechanisms. Synaptic plasticity mechanisms +have been applied to neural networks for memory dependent tasks such as catastrophic forgetting and +more recently continual and lifelong learning Miconi et al. [2018], Ba et al. [2016]. In general, these +approaches use synaptic plasticity to help the network retain and consolidate memories from previous +data while training on new data, though the goal for this process varies amongst tasks. In contrast +to these works, temporal weights help the network learn the scaling and length of time underlying +the trends in the dataset, which is an active pattern shared across information, not the information +itself. Though, there are functional similarities. In these approaches and temporal weights, the +networks learns how accomplish their respective tasks using parameters, instead of specifying a fixed +behavior. Temporal weights has a similar formulation to Davis et al. [2020], Ha et al. [2016]. In +Davis et al. [2020] weight are explicitly dependent on time where the weights are linear functions +of time, however we model weights as synapses and dendrites, and are non-linear functions. In Ha +et al. [2016], the weights are non-linear functions as the weights are generated by additional networks +using the previous hidden state as input, and so are indirectly dependendent on time. +5 +Discussion +We begin by motivating discussion on the substance of temporal data, namely by distinguishing it +from other forms of data. Temporal data is different from sequential data or sequential data paired +with time stamps (pseudo-temporal data). Sequential data includes tasks such as copy, associative +recall, sorting, frequency, n-mnist, moving-mnist, and split-mnist that were made popular by the +neural memory, neural attention, neural ODE, and continual learning literatures (cite neural turing +machine work). Here are two examples. Under memory tasks, the moving-mnist is generated from +moving digits across a frame. Under attention related tasks, the n-mnist dataset is generated from +a sensor detecting a change in an image. With these descriptions of the tasks, it is clear that the +substance of these data is memory and attention. Furthermore, it is intented as such in their relevant +works. Yet, multiple works we cited use sequential data as a stand-in for temporal data. In contrast, +temporal data is generated from dynamic systems, that is a time dependence. Sequential data does +not have a time dependence. (Temporal causality.) +However, there is overlap when distinguishing between sequential and temporal data, and sequential +data can resemble temporal data. Sequential data and temporal data start to overlap when considering +9 + +event based data which resembles both temporal data and sequential data. We can allievate some of +the overlap by conditioning on attention based data, such as the n-mnist dataset. That is, event based +data that is generated from changes in attention on an underlying process is sequential data if the +underlying process is not a dynamical system. In other words, it depends on what we are paying +attention to. This recursive definition is intentional as it makes clear that event and attention based +data are themselves either temporal or sequential data. +Another and very similar point of overlap is in trajectory data. Well recognized properties of dynamical +systems are quasi-periodicity, almost periodicity, and regular periodicity. However, periodicity does +not imply a dynamical system. This is an important consideration for trajectory data where frequency +tasks, such as those using wave or spiral datasets, resemble temporal dyamical processes, but lack +the substance of dynamical systems. For example, learning to contruct periodic trajectories or +distinguishing between periodic trajectories are memory and attention tasks, respectively. They do +not have a time dependence and are not capturing dynamical systems. +We encourage the community to consider the following when naming data as temporal: (1) are +underlying dynamical systems generating the data, (2) has the temporal dimensionality of the data +been reduced such that it is no longer part of the dataset, (3) is the task online or offline, (4) if it +online, at what scale and duration is the input before making the prediction and the prediction itself. +These points distinguish temporal data from sequential data and provide context on the nature of time +in the temporal dataset. Of greater interest and importance, a direct acknowledgment of time bring us +into lifelong learning where consideration of time is essential for negotiating through a deep sea of +dynamical systems, where every entity is online except when it is protected. +Our temporal weights significantly increases the expressiveness of the Neural ODE model by learning +to dynamically reconfigure the weights over time using a model of neural synchronization. We +demonstrate that the resulting models have better performance, fewer parameters, and improved +data efficiency. The Neural ODE model with temporal weights performs much better on generative +prediction tasks, such as the PhysioNet ICU data imputation task. The neural synchronization +behavior seems be to able to better fit the underlying distribution of the data as it can learn the +temporal patterns in the data and change the weights accordingly at each time step. +In the Hopper and PhysioNet experiments, we demonstrate that temporal weights has a capacity +better than simply increasing the number of parameters. The model of neural synchronization better +captures the Newtonian physics of the hopper rotating in the air and falling to the ground, and the +underlying temporal distribution of the ICU measurements. +In the Human Activity and PhysioNet experiments, models with temporal weights have better or +similar performance than the state of the art models, but in fewer epochs and less parameters. That is, +there is better usage of the data during learning. However, the addition of nonlinear weights increases +the time of each training epoch by 1.2 − 2.0 times. This is partially due to the model of mass neural +synchronization requiring comparisons between all the weights, and then the backward pass needing +to compute the gradients for each of these comparisons. The temporal weights may also be straining +the ODE solver since the weights are no longer fixed every time the solver calls the network. In +general, Neural ODE models are slow to train due to the solver and it is a trade-off for using these +models. Discrete networks do not have this issue. We are working on expanding temporal weights to +discrete networks. +By introducing time as an explicit dependency of the weights, we have demonstrated that Neural +ODE models can better capture the temporal dynamics of a dataset. We have chosen smaller model +sizes and datasets with sparse, irregularly sampled time points to elucidate that model with temporal +weights are not memorizing, but have an improved capacity for learning. +References +Michael Cassidy, Paolo Mazzone, Antonio Oliviero, Angelo Insola, Pietro Tonali, Vincenzo Di Lazzaro, and +Peter Brown. Movement-related changes in synchronization in the human basal ganglia. Brain, 125(6): +1235–1246, 2002. +Wolfgang Klimesch. Memory processes, brain oscillations and eeg synchronization. International journal of +psychophysiology, 24(1-2):61–100, 1996. +Larry F Abbott and Sacha B Nelson. Synaptic plasticity: taming the beast. Nature neuroscience, 3(11): +1178–1183, 2000. +10 + +Eiji Shimizu, Ya-Ping Tang, Claire Rampon, and Joe Z Tsien. Nmda receptor-dependent synaptic reinforcement +as a crucial process for memory consolidation. Science, 290(5494):1170–1174, 2000. +Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations. +In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 6572– +6583, 2018. +Yulia Rubanova, Ricky TQ Chen, and David Duvenaud. Latent odes for irregularly-sampled time series. +In Proceedings of the 33rd International Conference on Neural Information Processing Systems, pages +5320–5330, 2019. +Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, +Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep +learning library. Advances in neural information processing systems, 32:8026–8037, 2019. +Ikaro Silva, George Moody, Daniel J Scott, Leo A Celi, and Roger G Mark. Predicting in-hospital mortality +of icu patients: The physionet/computing in cardiology challenge 2012. In 2012 Computing in Cardiology, +pages 245–248. IEEE, 2012. +Boštjan Kaluža, Violeta Mirchevska, Erik Dovgan, Mitja Luštrek, and Matjaž Gams. An agent-based approach +to care in independent living. In International joint conference on ambient intelligence, pages 177–186. +Springer, 2010. +Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 +IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 5026–5033. IEEE, 2012. +Yuval Tassa, Yotam Doron, Alistair Muldal, Tom Erez, Yazhe Li, Diego de Las Casas, David Budden, Abbas +Abdolmaleki, Josh Merel, Andrew Lefrancq, et al. Deepmind control suite. arXiv preprint arXiv:1801.00690, +2018. +MJ Menne, CN Williams Jr, and RS Vose. Long-term daily and monthly climate records from stations across +the contiguous united states. Website http://cdiac. ornl. gov/epubs/ndp/ushcn/access. html [accessed 23 +September 2010], 2010. +Edward De Brouwer, Jaak Simm, Adam Arany, and Yves Moreau. Gru-ode-bayes: Continuous modeling of +sporadically-observed time series. Advances in neural information processing systems, 32, 2019. +Panayiota Poirazi and Bartlett W Mel. Impact of active dendrites and structural plasticity on the memory capacity +of neural tissue. Neuron, 29(3):779–796, 2001. +Lynn E Dobrunz and Charles F Stevens. Response of hippocampal synapses to natural stimulation patterns. +Neuron, 22(1):157–166, 1999. +Thomas Miconi, Kenneth Stanley, and Jeff Clune. Differentiable plasticity: training plastic neural networks with +backpropagation. In International Conference on Machine Learning, pages 3559–3568. PMLR, 2018. +Jimmy Ba, Geoffrey E Hinton, Volodymyr Mnih, Joel Z Leibo, and Catalin Ionescu. Using fast weights to attend +to the recent past. Advances in Neural Information Processing Systems, 29:4331–4339, 2016. +Jared Quincy Davis, Krzysztof Choromanski, Jake Varley, Honglak Lee, Jean-Jacques Slotine, Valerii Likhos- +terov, Adrian Weller, Ameesh Makadia, and Vikas Sindhwani. Time dependence in non-autonomous neural +odes. arXiv preprint arXiv:2005.01906, 2020. +David Ha, Andrew Dai, and Quoc V Le. Hypernetworks. arXiv preprint arXiv:1609.09106, 2016. +Max Horn, Michael Moor, Christian Bock, Bastian Rieck, and Karsten Borgwardt. Set functions for time series. +In International Conference on Machine Learning, pages 4353–4363. PMLR, 2020. +Wenjie Du, David Côté, and Yan Liu. Saits: Self-attention-based imputation for time series. arXiv preprint +arXiv:2202.08516, 2022. +11 + +A +Appendix +We provide comparisons to additional models in the tables below. +Table 6: Classification PhysioNet ICU. +METHOD +AUC +LATENT ODE +0.857 +W/ TW +0.861 +SAITS +0.848 +BRITS +0.835 +GP-VAE +0.834 +M-RNN +0.822 +E2GAN +0.830 +GRUI-GAN +0.830 +GRU-D +0.863 +GRU-SIMPLE +0.808 +IP-NETS +0.860 +PHASED-LSTM +0.790 +SEFT-ATTN +0.851 +TRANSFORMER +0.863 +FROM HORN ET AL. [2020], DU ET AL. [2022] +Table 7: Classification on PhysioNet Sepsis +METHOD +AUC +ODERNN +0.765 +W/ TW +0.787 +NCDE +0.925 +NCDE +0.925 +W/ TW +0.931 +GRU-D +0.674 +GRU-SIMPLE +0.781 +IP-NETS +0.742 +PHASED-LSTM +0.754 +SELF-ATTN +0.768 +TRANSFORMER +0.658 +FROM HORN ET AL. [2020] +Table 8: Number of Parameters (Rounded) +METHOD +#PARAMS +L-ODE +67,071 +W/ TW +52,016 +M-RNN +70,000 +E2GAN +80,000 +GP-VAE +150,000 +GRUI-GAN +160,000 +BRITS +730,000 +TRANSFORMER +4,360,000 +SAITS +5,320,000 +12 + diff --git a/mNE2T4oBgHgl3EQfzAhZ/content/tmp_files/load_file.txt b/mNE2T4oBgHgl3EQfzAhZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..566b109277f7bae2bee7e9426f62c9b7e328c4b3 --- /dev/null +++ b/mNE2T4oBgHgl3EQfzAhZ/content/tmp_files/load_file.txt @@ -0,0 +1,689 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf,len=688 +page_content='Temporal Weights Adam Kohan akohan@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='umass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='edu Edward A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Rietman erietman@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='umass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='edu Hava T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Siegelmann hava@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='umass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='edu Biologically Inspired Neural and Dynamical Systems College of Information and Computer Sciences University of Massachusetts Amherst Amherst, MA 01003 Abstract In artificial neural networks, weights are a static representation of synapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' How- ever, synapses are not static, they have their own interacting dynamics over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' To instill weights with interacting dynamics, we use a model describing synchro- nization that is capable of capturing core mechanisms of a range of neural and general biological phenomena over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' An ideal fit for these Temporal Weights (TW) are Neural ODEs, with continuous dynamics and a dependency on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The resulting recurrent neural networks efficiently model temporal dynamics by computing on the ordering of sequences, and the length and scale of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' By adding temporal weights to a model, we demonstrate better performance, smaller models, and data efficiency on sparse, irregularly sampled time series datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 1 Introduction Time provides an opportunity for neural networks to change their computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This change may be in response to the evolving dynamics of the input distribution, possibly a change in trajectory triggered by some sparse event, or may be to improve performance by taking another pass over the same input, but with a different perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Static weights do not adjust for either of these situations and perform the same computation regardless of the passage of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A neural network with static weights has only a form of short term, working memory with its internal hidden states (nodes) to track time steps, but not change its computation (weights) on its inputs and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In contrast, neural networks with temporal weights can change how they process their own memory and inputs over time, not only keep track of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' At each time step, a neural network with temporal weights changes its computation by constructing weights W‘ from a fixed set of parameters W, which is considered to be a form of long term stored memory, and the current time step t: W ‘ = S(W, t) Here, the weights are a nonlinear function of parameters and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' As a result, the parameters are shared across time, but the expressivity of the weight values is not limited to simple linear combinations of weights or shifts in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Instead, the weights themselves can capture some dynamics of the input distribution and form their own trajectory over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Given S is non-linear, we are amplifying the effective amount of weights in the model by apply the same fixed number of parameters differently at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In contrast, static weights are only the parameters directly applied to the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' They can only capture single values, not any dynamics that are separate from the layer or network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' There are many options of temporal dynamics for S to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We use a model of coupled oscillators describing sychronization behaviors that captures core mechanisms of a range of neural and general Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='04126v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='NE] 13 Dec 2022 biological phenomena over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Under this model, weights have an explicit dependence on time and have changing interactions with each other over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Details are in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Static Weights Temporal Weights Latent Trajectories Samples from Prior Data Space Latent Space Data Space Figure 1: A comparison between a neural ode model with static weights or temporal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' To demonstrate the difference between static and temporal weights in practice, we graph in 1 three samples from a neural ode with static weights (left) and temporal weights (rights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The neural ode makes for a strong baseline as it captures temporal tra- jectories itself and is a state of the art model for irregularly sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Furthermore, the effect of temporal weights would have to be non-trivial to alter this trajectory and significant to improve over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' To ensure the practicality of a difficult problem, the neural ode models are trained on a sparse, irregularly sampled dataset (details of the dataset are below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' As shown, even a neu- ral ode model with static weights is not as expressive as the same model with tempo- ral weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Notice the well-defined kinks, quasi-periodicity, and consistent alignment of the temporal weights model, allowing for a better fit for evolving dynamics and for sparse and irregularly sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In comparison, the same model with static weights is inconsistent across samples and has limited smooth periodicity, which will have more difficulty fitting irregularities or changes in dynamics over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We can see that temporal weights successfully amplifies the parameters to increase the expressivity of the neural ode model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Sample 1 Sample 2 Sample 3 Static Weights Temporal Weights Figure 2: Neural ODE model fit on a sparse and irregular synthetic dataset, with temporal weights or static weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The result of increasing the expressivity of the neural ode model is a better fit on highly sparse and irregular data, which the original model struggles to fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In 2, we show results of the neural ode models on a synthetic dataset of periodic trajectories with variable frequency, amplitude, noise, and discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' To further increase the difficulty of this demonstration, the missing points are not used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='Fitting a dataset without using the missing points to learn from is an entirely more challenging task than using them for training before evaluating the model with- out them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' It more accurately represents data in the wild, such as the PhysioNet ICU dataset, where the points are truly missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' So, the model will need to infer the missing points by referencing other samples for examples of those missing points, instead of training on them directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We also limit training to 50 iterations such that overfitting is made difficult and the models must rely on their efficiency in processing the limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' As shown in 2 (left), the neural ode model with static weights struggles to fit the data, even though it is able to capture the general trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' As mentioned above, this model has a limited smooth periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In contrast, the neural ode with temporal weights, shown in 2 (right), is able to efficiently capture the local details of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Through time, this model can continuously perturb its own dynamics and adjust or change its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Given the difficulty of this task, neither model perfectly fits the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' However, the neural ode with temporal weights clearly performs better, demonstrating more variability across samples as it adapts to that sample’s data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 2 Sampleo(dataspace Sample1 (dataspace) Sample2(dataspace) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 - dim 3 dim 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 dim 8 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 - dim 9 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 dim 10 dim 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 dim 15 dim 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 dim 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 0 1 2 3 6 4 2 4 1 2 4 2 0 3 4 Time Time2 Methods We developed a neural model whose weights depend explicitly on time and are .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our model increases the capacity of the network by incorporating the natural phenomena of time into the parameters, instead of solely increasing the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In our approach, we use a biological model of time dependent behavior in neurons as the basis for temporal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We formulate the biological model as a weight scaling algorithm with oscillatory dynamics for time based reconfigurations of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='1 Neural Synchronization The scaling function used in our algorithm is based on models of mass neural synchronization, which has been attributed to play a role in movement Cassidy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2002] and memory Klimesch [1996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The dynamic modification of synaptic connections that is critical to neural synchronization is recognized to be the basis of long-term memory in learning Abbott and Nelson [2000], Shimizu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' These synaptic dynamics bring about adaptive development of network structure whose trajectory is oriented by the dependencies of the inputs on underlying learning mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Herein, we focus on time-based dependencies of the underlying learning mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A scaling function S is learned alongside the parameters W of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Although W is fixed after training, the network’s temporal weights W‘ = S(W, t) are different at each time-step due to the time-dependence of the scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We treat each weight w′ i ∈ W‘ as individual units that are controlled by the underlying mechanisms of mass neural synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our scaling function models synchronization behaviors in natural dynamic systems: it varies the coupling and decoupling of different subsets of weights given the relative time and the relationship between weights (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Figure 3: Full and Partial Phase Locking of Weights at Some Time Delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In partial phase locking, clusters of weights are each syn- chronized to a different frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In full phase locking, most or all of the weights are synchronized to the same fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our scaling function S taken over the weights W‘ learns parameters W to adjust this phase locking behavior at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our scaling function provides a time based prior for the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' That is, separate weights at each time step to process each input of a sequence are re- lated to each other and are non-linear combinations of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The network is thus capable of relating individual weights together and constructing different combinations of weights to focus on the properties of each input in a sequence revealed by its point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' It can isolate different subsets of weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' weight configurations) to apply an input-dependent ordered series of functions to the model’s internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='2 Neural ODE Model We apply our scaling function to Neural Ordinary Differential Equation models Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Neural ODE models are time continuous models whose uniquely defined latent trajectory lets us extrapolate predictions arbitrarily far forwards or backwards in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In comparison, discrete models require observations (frequent data points) to control the trajectory and hence are ill-defined with missing data or irregularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Neural ODEs are state of the art for irregularly sampled data Rubanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Neural ODEs are a natural fit for temporal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' While Neural ODEs are time- dependent, their weights are static and only indirectly depend on time as part of the network’s internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our temporal weights makes weights dependent on time explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' To generate the trajectories between inputs, the Neural ODE model makes calls to an ODE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This solver breaks down the time interval into smaller subintervals and subsequently approximates the solution to the ODE at the endpoints of these intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Given the solver must call the scaling function when computing each of these intermediate values, it also has direct control of the number of weight configurations produced by from a set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Depending on the complexity of trajectory induced by the input, the model will apply a different number of functions to its internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The model can effectively learn when to increase or decrease the use of its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3 Measureof PhaseLocking Coherence Phasecentroid Partial PhaseLocking Full PhaseLockingFigure 4: Computational Graph of Latent ODE and Temporal Weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (Left) The trajectory of the ODE model varies continuously with time, even at times between receiving inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' On the other hand, the state of RNN only changes in response to inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (Right) Temporal Weights construct weights at each time step, which are used by the Latent ODE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This includes time steps where there are inputs or outputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' time steps 1, N, and M), and time steps where there is neither, but the ode solver is making calls to the neural network, such as the time step between N and N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A view into the ODE solver, denoted by ODESOLVE in the figures, with temporal weights is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' e 0 1 2 3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='$ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content="% Input 0 1 2 3 &# &' &# &' &' &# &' &# &#% &#% &( &( &( &#% &#% &(𝑊`# = 𝑆(𝑊#,t) 𝑊`* = 𝑆(𝑊*,t) ℎ(𝑡-." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='#) ℎ 𝑡- 𝑤ℎ𝑒𝑟𝑒 ℎ 𝑡3 = 𝑧 Time 1 𝑁 2 𝑁 + 1 𝑀 ℎ`# = 𝑡𝑎𝑛ℎ ℎ(𝑡-)𝑊`# ℎ`* = 𝑡𝑎𝑛ℎ ℎ`#𝑊`* 𝑑ℎ(𝑡-) 𝑑𝑡 = 𝑓> ℎ 𝑡- , 𝑡 𝑂𝐷𝐸𝑆𝑂𝐿𝑉𝐸 1 𝑁 2 𝑁 + 1 𝑀 … … … 𝑓> ℎ 𝑡- , 𝑡 ℎ 𝑡# ℎ 𝑡D ℎ 𝑡E … Inputs Initial state z Previous hidden state Outputs are hidden states (A) (B) (C) Time (D) Figure 5: Internal Procedure of ODE Solver with Temporal Weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The ODE solver calls a multilayer fully connected neural network internally fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (A) At each input time step, the solver gives the time ti and the previous solver’s hidden state h(ti) to the network fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (B) For each layer l, the network constructs weights W‘l using the parameters Wl and the time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The weights W‘l are used by the layers to process the network’s input h(ti) and the network’s previous hidden state h‘l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (C) The network outputs the next hidden state h(ti+1), which is given to the ODE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (D) If there is an output time step at ti+1, h(ti+1) is also outputted from the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The algorithm is provided in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The neural ODE is able to control the use of temporal weights through the ode solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The ode solver chooses the frequency and the time steps to call the temporal weight scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' So, the neural ODE automatically adapts the usage temporal weights to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In contrast, discrete neural networks respond directly to observa- tions and the observations set the fre- quency and time steps with which to call the temporal weight scaling func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' As a result, the usage of tempo- ral weights is limited by the number of observations in discrete networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' To more completely utilize the capac- ity of temporal weights, we use neural ODE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We apply temporal weights to the Latent ODE and ODE RNN models of Rubanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Latent ODEs and ODE RNNs are state of the art models on data with arbitrary time gaps between observations, irreg- ularly sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The equations and internal procedures of the Neu- ral ODE with temporal weights used within these models are shown in Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Refer to Rubanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2019] for further details of the base models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Unless otherwise stated in 3, we follow the training procedure and configuration of hyperparameters from Rubanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2019] for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We demonstrate that with temporal weights, model size can be reduced and have similar or better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='3 Temporal Scaling Function The scaling function S drives a sinusoidal wave to scale the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The sine function can be used to output a scalar between [0, 1] or [−1, 1] for each weight in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Different weights will have 4 ODESOLVE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='.ODESOLVEAlgorithm 1 Neural ODE with Temporal Weights Input: Latent z, Output Times {ti}i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='M Provide: Network fθ with L Layers call ODESOLV E(fθ, z, {ti}i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='M) for h (ti) , t given by the solver do call fθ (h (ti) , t) Let h‘0 = h (ti) for layer l = 1 to L do W‘l = S (Wl, t) h‘l = tanh (h‘l−1W‘l) end for Output h (ti+1) = h‘L to the solver end call end for Return {hi}i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='M end call different magnitudes over time or even be inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The output of the sine function changes with time t, given by the ODE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The time is shifting the phase of the sine wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We let the network learn parameters to control scale of time for each weight separately or jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' That means the network has the capacity to reuse the same learned dynamics repeating the period over time, or learn to split up the period over different time intervals, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' For example, the network may scale time such that each period 2π repeated every dt time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Or, the network may split up the period 2π into π/2 chunks spanning dt time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Or, both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Regardless, the network is able to control the dynamics of the network through the weights over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The learned dynamics in the scaling algorithm are a function of the parameters W and a comparison of each parameter to all the other weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The weights W‘ for a layer are a function of this comparison, not the actual weight itself, as shown in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The network’s weight matrix at each layer is of interactions between weights, instead of the weights themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The resulting equation is: w‘i = Si (Wl, t) (1) = #Wl � j=1 K{ij} l #Wl sin [fl(t) ∗ (wi − wj) + φl(t)] (2) where K{ij} l is the parameter matrix of coupling coefficients, (wi − wj) is the interaction of weight i with other weights scaled and shifted by the current time t in the ODE solver, f(t) are the scaling parameters, and φl(t) are the shifting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' All the parameters K{ij} l , fl(t), φl(t) are learned with the parameters w ∈ Wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The scaling function takes the time t and parameter matrix Wl for layer l to construct a weight w‘i ∈ W‘l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The matrix K contains learned coefficients that control the strength of coupling between weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The sine function is over the difference between each weight in a layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The phase of this function is controlled by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The result is a periodic estimator of weight distances that evolves as a function of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The scaling function is taken for each layer and each time step as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Each layer has it’s neurons synchronized over time separately from the other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The comparison between all weights in the scaling function can increase memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Future work may explore a global neural synchronization across the network if this issue is alleviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3 Experiments Ten experiments over five datasets with temporally dependent features show that models with temporal weights more closely capture time dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' They have better performance in fewer epochs and with less parameters than the same model with static weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The datasets are sparse and irregularly sampled to better ensure: (1) the applicability of our model in the wild where data may be missing, 5 difficult to acquire, or streamed on-line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and (2) that our model is learning the underlying distributions over time, not memorizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We use the Pytorch library Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2019] and train networks on a single NVIDIA GeForce GTX TITAN X GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We use the Adamax optimizer with learning rates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='04 and exponential learning rate decays 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='9999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The best performing model is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Individual details are listed under each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Refer to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='2 for additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='1 PhysioNet ICU We evaluate TW on the PhysioNet Challenge 2012 Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2012] dataset of 12,000 ICU stays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' At admission, a one-time set of general descriptors, such as age or gender, is collected one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' During a stay in the ICU, a sparse subsets of 37 measurements are taken, such as Lactate, Mg, and NA levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Measurements are from the at least the first 48 hours of an ICU stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Each stay is labeled whether the patient survives hospitalization or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Some measurements are at regular intervals and others are asynchronous and at irregular times, collected only when required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' There are more than 4,600 possible measurements per time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' To lower training time, we halved the number measurement times by rounding to the nearest minute, leaving 2,880 possible measurements per time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The dataset is challenging: it is extremely sparse with a missing rate of around 80% and has highly imbalanced class distributions with a prevalence ratio of around 14%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Mortality Prediction We look to predict which patients survive hospitalization given the information collected during the first 48 hours of an ICU stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We use Area Under the Curve (AUC) as our performance metric due to the class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Results are in 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The same model with TW converges to a better performance in fewer epochs and with less than half the parameters, but is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5x slower per epoch (refer to 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Table 1: PhysioNet ICU METHOD AUC EPOCHS # PARAMS L-ODE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='857 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='836) 80 (31) 163,972 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='861 31 76,427 (a) Mortality prediction on the PhysioNet ICU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Given the first 48 hours of an ICU stay, predict in-hospital mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' METHOD MSE X10-3 EPOCHS # PARAMS L-ODE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='280 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='340) 59 (49) 67,071 W/ TW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='370 49 52,016 L-ODE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='208 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='300) 91 (56) 67,071 W/ TW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='900 56 52,026 (b) ICU measurements (Top) Interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (Bot- tom) Extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Given the first 24 hours of measurements, predict the next 24 hours of mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (Parenthesis compare the models at the same epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=') Table 2: MuJoCo Physics Simulation METHOD MSE X10-3 EPOCHS # PARAMS L-ODE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='60 94 617,619 W/ TW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='00 91 112,739 (a) Interpolation of hopper body position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We sub- sample 10% of the time points and predict the other 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' METHOD MSE X10-2 EPOCHS # PARAMS L-ODE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='190 93 617,619 W/ TW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='100 99 112,739 L-ODE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='480 77 617,619 W/ TW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='220 68 112,739 (b) Extrapolation of hopper body positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Given the first half of the timeline, predict body position in the second half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We subsample 10% of the time points in the first half of the timeline at each batch (top) or once when constructing the dataset (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Interpolation and Extrapolation We also look at the task to model the PhysioNet Challenge measurements data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In this task, we impute missing measurements in each sample to reconstruct the full set of points in the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We report the mean squared error average over the reconstruction of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Results are in 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In the extrapolation task, we provide the network with the first half of the timeline and construct measurements in the second half of the timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Results are in 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The latent ODE model with temporal weights converges to a significantly better performance in both the extrapolation and interpolation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Architecture The Latent ODE is as follows: an encoder comprised of a GRU of 50 units, hidden size of 40, and Neural ODE of size 50 with 3 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a decoder comprised 6 of a Neural ODE of size 50 with 3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The Latent ODE with temporal weights is as follows: an encoder comprised of a GRU of 25 units, hidden size of 20, and Neural ODE of size 25 with 3 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a decoder comprised of a Neural ODE of size 25 with 3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='2 Human Activity We evaluate TW on the Human Activity dataset Kaluža et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2010] of 5 individuals performing 7 activities, such as sitting and walking, resulting in 25 sequences of 6,600 time points each on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 6,554 sequences of 211 time points each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The dataset has 12 features of 3D positions from tags attached to a belt, chest, and each ankle (3 dimensions x 4 tags).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Time-Point Classification The task is to classify each time point into one of the activities for a total of 1,477 (7 x 211) outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The model with temporal weights has better performance, converges in fewer epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Results are in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Table 3: Human Activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Classification at each time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (Parenthesis compare the models at the same epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=') METHOD ACC EPOCHS # PARAMS L-ODE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='846 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='748) 52 (10) 1,696,763 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='870 10 141,023 Table 4: Climate prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' METHOD MSE NLL EPOCHS # PARAMS GRUODE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='84 74 42,640 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='87 68 9,105 Table 5: Sepsis prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' At every hour, predict whether the patient will have sepsis within the next 6 to 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (Parenthesis compare the models at the same epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=') METHOD AUC EPOCHS # PARAMS ODERNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='689 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='669) 67 (24) 148,672 ODERNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='765 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='696) 86 (24) 680,462 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='787 24 149,519 NCDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='925 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='905) 130 (110) 55,949 NCDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='925 110 193,541 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='931 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='918) 180 (110) 58,553 Architecture The Latent ODE is as follows: an encoder comprised of a GRU of 50 units, hidden size of 100, and Neural ODE of size 500 with 4 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a decoder comprised of a Neural ODE of size 500 with 2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The Latent ODE with temporal weights is as follows: an encoder comprised of a GRU of 25 units, hidden size of 20, and Neural ODE of size 25 with 3 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a decoder comprised of a Neural ODE of size 25 with 3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='3 MuJoCo Physics Simulation The physics simulation dataset Rubanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2019] is 10,000 sequences of 100 regularly sampled 14-dimensional time-points generated from the MuJoCo Todorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2012] simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The dataset is of a bipedal model, called the Hopper model in the Deepmind Control Suite Tassa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The task is to learn to approximate Newtonian physics of the hopper rotating in the air and falling to the ground starting from a randomly sampled position and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The resulting deterministic trajectories are dictated by their initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This should be straightforward for the Latent-ODE model to fit as it matches the assumptions made by the Latent-ODE model, it is dictated by its initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In the interpolation task, we subsample 10% of the time points to simulate sparse observation times, and predict the other 90% time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Results are in 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In the extrapolation task, we provide the network with the first half of the timeline, and then predict the second half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We subsampling 10% of the time points from the first half of the timeline, but predict on all the points in the second half of the timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We show results in for 2b two versions of this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In the first version, shown in the top rows of the table, we subsample 10% of the time points at every batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This means that the networks will through multiple epochs eventually see all the data points of an input since a different 7 10% of the time points are sampled each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In the second version, shown in the bottom rows of the table, we subsample 10% of the time points once when the dataset is constructed such that the network will never see the missing the time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The second version of this task is a more difficult as the missing time points are never used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We see in the results that the performance gap between the model with and without temporal weights doubles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In these tasks, we demonstrate that the latent ODE models performs better with temporal weights, and has 5 times fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our model has a similar capacity to the larger model, but with much fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Architecture The Latent ODE is as follows: an encoder comprised of a GRU of 100 units, hidden size of 30, and Neural ODE of size 300 with 3 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a decoder comprised of a Neural ODE of size 300 with 3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The Latent ODE with temporal weights is as follows: an encoder comprised of a GRU of 50 units, hidden size of 40, and Neural ODE of size 50 with 3 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a decoder comprised of a Neural ODE of size 50 with 3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='4 PhysioNet Sepsis The PhysioNet Sepsis dataset contains time series of observations from 40,000 ICU patients that were aggregated from two different U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' hospitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Each patient has demographics as a general descriptor, such as agent and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' During a stay in the ICU, up to 40 measurements are taken, such as vital signs and laboratory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Measurements were recorded together every hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Each hour is labeled whether or not an onset of sepsis occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The dataset is sparse with a missing rate of around 74% and has imbalanced class distributions with a prevalence ratio of around 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Sepsis Prediction The task is to predict at every hour whether the patient will have sepsis within the next 6 to 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Results are in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The model with temporal weights has better performance with a similar number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Architecture The smaller ODE RNN with and without TW is as follows: an encoder comprised of a GRU of 50 units;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a decoder comprised of an MLP of 100 units with 2 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a Neural ODE of size 10 with 1 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The larger ODE RNN without TW is as follows: an encoder comprised of a GRU of 1024 units;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a latent size of 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' a decoder comprised of an MLP of 100 units with 2 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' and a Neural ODE of size 20 with 1 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='5 USHCN Daily Climate Data The United State Historical Climatology Network (USHCN) daily dataset Menne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2010] contains measurements of 5 climate variables (temperatures, precipitation, and snow) over 150 years for 1,218 meteorological stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We use the processed data from De Brouwer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' It contains a subset of 1,114 stations over an observation window of 4 years subsampled such that each station has around 60 observations on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The task is to predict the next 3 measurements after the first 3 years of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Architecture The smaller GRUODE with TW is as follows: A p_model with 2 layers of 25 units and an input of 15 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A classification network with 2 layers of 2 units and an input of 50 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A gru with 3 layers of 15 units and 2 layers of 45 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A covariates map with 2 layers of 50 units and an output of 15 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The larger GRUODE without TW is as follows: A p_model with 2 layers of 25 units and an input of 50 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A classification network with 2 layers of 2 units and an input of 50 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A gru with 3 layers of 50 units and 2 layers of 150 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' A covariates map with 2 layers of 50 units and an output of 50 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 4 Related Works Temporal weights are a model of dendrites and synapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The common view is that memories are encoded in the connection strength between neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In artificial neural networks, static weights represent (are inspired by) synapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' However synapses are not static, they have their own dynamics, in addition to the dynamics of the neurons they connect to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In Poirazi and Mel [2001], the authors investigate the storage capacity of neurons that have synapses with their own dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The authors compare (1) synapses that are linearly summed across dendritic 8 arbors (contacts with neurons) with (2) synapses where the dendritic compartments are non-linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The results demonstrate a much larger capacity for neurons with non-linear synaptic and dendritic subunits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This suggests that long term memory storage is an active process encoded in the dynamics of synapses and dendrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our results support these findings: a neural network with temporal weights has similar (or better) performance to a larger version of same network without temporal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Given the increased capacity from the non-linearity, in temporal weights, we makes this capacity accessible to the neural network for processing inputs by modeling the behavior of synaptic plasticity over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Synaptic plasticity is commonly known for mediating learning and behavior by changing synaptic strength given a local or global signal or an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This mediation can be viewed as a form of selective content addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We take a look at synaptic plasticity from this view, as a mechanism to make information available to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Specifically, we take this view while looking at the temporal behavior of synaptic plasticity in changing synaptic strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Synaptic plasticity and in turn changes in synaptic strength are usually looked at in direct response to some stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' However, we look at similar response patterns to stimuli over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In Dobrunz and Stevens [1999], the authors stimulated synapses with natural patterns derived from in vivo recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' These natural patterns do not have constant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' They provide a better dataset to judge the relative importance of short- (and long-) term synaptic plasticity for usual synaptic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The authors find that synaptic strength is being modulated by the timing of the stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Similar changes in synaptic strength occur at different time scales given input history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In temporal weights, we capture this behavior in two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' First, the scaling function is based on models of mass neural synchronization whose behavior oscillates over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Second, we include additional parameters in the scaling function for the network to learn a scaling and length of time for the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In total, the network is able to change its weights based on the timing of an observation in the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Temporal weights work similarly to synaptic plasticity mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Synaptic plasticity mechanisms have been applied to neural networks for memory dependent tasks such as catastrophic forgetting and more recently continual and lifelong learning Miconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2018], Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In general, these approaches use synaptic plasticity to help the network retain and consolidate memories from previous data while training on new data, though the goal for this process varies amongst tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In contrast to these works, temporal weights help the network learn the scaling and length of time underlying the trends in the dataset, which is an active pattern shared across information, not the information itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Though, there are functional similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In these approaches and temporal weights, the networks learns how accomplish their respective tasks using parameters, instead of specifying a fixed behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Temporal weights has a similar formulation to Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2020], Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2020] weight are explicitly dependent on time where the weights are linear functions of time, however we model weights as synapses and dendrites, and are non-linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2016], the weights are non-linear functions as the weights are generated by additional networks using the previous hidden state as input, and so are indirectly dependendent on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 5 Discussion We begin by motivating discussion on the substance of temporal data, namely by distinguishing it from other forms of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Temporal data is different from sequential data or sequential data paired with time stamps (pseudo-temporal data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Sequential data includes tasks such as copy, associative recall, sorting, frequency, n-mnist, moving-mnist, and split-mnist that were made popular by the neural memory, neural attention, neural ODE, and continual learning literatures (cite neural turing machine work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Here are two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Under memory tasks, the moving-mnist is generated from moving digits across a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Under attention related tasks, the n-mnist dataset is generated from a sensor detecting a change in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' With these descriptions of the tasks, it is clear that the substance of these data is memory and attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Furthermore, it is intented as such in their relevant works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Yet, multiple works we cited use sequential data as a stand-in for temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In contrast, temporal data is generated from dynamic systems, that is a time dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Sequential data does not have a time dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' (Temporal causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=') However, there is overlap when distinguishing between sequential and temporal data, and sequential data can resemble temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Sequential data and temporal data start to overlap when considering 9 event based data which resembles both temporal data and sequential data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We can allievate some of the overlap by conditioning on attention based data, such as the n-mnist dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' That is, event based data that is generated from changes in attention on an underlying process is sequential data if the underlying process is not a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In other words, it depends on what we are paying attention to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This recursive definition is intentional as it makes clear that event and attention based data are themselves either temporal or sequential data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Another and very similar point of overlap is in trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Well recognized properties of dynamical systems are quasi-periodicity, almost periodicity, and regular periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' However, periodicity does not imply a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This is an important consideration for trajectory data where frequency tasks, such as those using wave or spiral datasets, resemble temporal dyamical processes, but lack the substance of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' For example, learning to contruct periodic trajectories or distinguishing between periodic trajectories are memory and attention tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' They do not have a time dependence and are not capturing dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We encourage the community to consider the following when naming data as temporal: (1) are underlying dynamical systems generating the data, (2) has the temporal dimensionality of the data been reduced such that it is no longer part of the dataset, (3) is the task online or offline, (4) if it online, at what scale and duration is the input before making the prediction and the prediction itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' These points distinguish temporal data from sequential data and provide context on the nature of time in the temporal dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Of greater interest and importance, a direct acknowledgment of time bring us into lifelong learning where consideration of time is essential for negotiating through a deep sea of dynamical systems, where every entity is online except when it is protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Our temporal weights significantly increases the expressiveness of the Neural ODE model by learning to dynamically reconfigure the weights over time using a model of neural synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We demonstrate that the resulting models have better performance, fewer parameters, and improved data efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The Neural ODE model with temporal weights performs much better on generative prediction tasks, such as the PhysioNet ICU data imputation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The neural synchronization behavior seems be to able to better fit the underlying distribution of the data as it can learn the temporal patterns in the data and change the weights accordingly at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In the Hopper and PhysioNet experiments, we demonstrate that temporal weights has a capacity better than simply increasing the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The model of neural synchronization better captures the Newtonian physics of the hopper rotating in the air and falling to the ground, and the underlying temporal distribution of the ICU measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In the Human Activity and PhysioNet experiments, models with temporal weights have better or similar performance than the state of the art models, but in fewer epochs and less parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' That is, there is better usage of the data during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' However, the addition of nonlinear weights increases the time of each training epoch by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='0 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' This is partially due to the model of mass neural synchronization requiring comparisons between all the weights, and then the backward pass needing to compute the gradients for each of these comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' The temporal weights may also be straining the ODE solver since the weights are no longer fixed every time the solver calls the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In general, Neural ODE models are slow to train due to the solver and it is a trade-off for using these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Discrete networks do not have this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We are working on expanding temporal weights to discrete networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' By introducing time as an explicit dependency of the weights, we have demonstrated that Neural ODE models can better capture the temporal dynamics of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' We have chosen smaller model sizes and datasets with sparse, irregularly sampled time points to elucidate that model with temporal weights are not memorizing, but have an improved capacity for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' References Michael Cassidy, Paolo Mazzone, Antonio Oliviero, Angelo Insola, Pietro Tonali, Vincenzo Di Lazzaro, and Peter Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Movement-related changes in synchronization in the human basal ganglia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Brain, 125(6): 1235–1246, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Wolfgang Klimesch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Memory processes, brain oscillations and eeg synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' International journal of psychophysiology, 24(1-2):61–100, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Larry F Abbott and Sacha B Nelson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Synaptic plasticity: taming the beast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Nature neuroscience, 3(11): 1178–1183, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 10 Eiji Shimizu, Ya-Ping Tang, Claire Rampon, and Joe Z Tsien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Nmda receptor-dependent synaptic reinforcement as a crucial process for memory consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Science, 290(5494):1170–1174, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Neural ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 6572– 6583, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Yulia Rubanova, Ricky TQ Chen, and David Duvenaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Latent odes for irregularly-sampled time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In Proceedings of the 33rd International Conference on Neural Information Processing Systems, pages 5320–5330, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Pytorch: An imperative style, high-performance deep learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Advances in neural information processing systems, 32:8026–8037, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Ikaro Silva, George Moody, Daniel J Scott, Leo A Celi, and Roger G Mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Predicting in-hospital mortality of icu patients: The physionet/computing in cardiology challenge 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In 2012 Computing in Cardiology, pages 245–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Boštjan Kaluža, Violeta Mirchevska, Erik Dovgan, Mitja Luštrek, and Matjaž Gams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' An agent-based approach to care in independent living.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In International joint conference on ambient intelligence, pages 177–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Springer, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Emanuel Todorov, Tom Erez, and Yuval Tassa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Mujoco: A physics engine for model-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 5026–5033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Yuval Tassa, Yotam Doron, Alistair Muldal, Tom Erez, Yazhe Li, Diego de Las Casas, David Budden, Abbas Abdolmaleki, Josh Merel, Andrew Lefrancq, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Deepmind control suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' arXiv preprint arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='00690, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' MJ Menne, CN Williams Jr, and RS Vose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Long-term daily and monthly climate records from stations across the contiguous united states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Website http://cdiac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' gov/epubs/ndp/ushcn/access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' html [accessed 23 September 2010], 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Edward De Brouwer, Jaak Simm, Adam Arany, and Yves Moreau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Gru-ode-bayes: Continuous modeling of sporadically-observed time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Advances in neural information processing systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Panayiota Poirazi and Bartlett W Mel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Impact of active dendrites and structural plasticity on the memory capacity of neural tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Neuron, 29(3):779–796, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Lynn E Dobrunz and Charles F Stevens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Response of hippocampal synapses to natural stimulation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Neuron, 22(1):157–166, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Thomas Miconi, Kenneth Stanley, and Jeff Clune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Differentiable plasticity: training plastic neural networks with backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 3559–3568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Jimmy Ba, Geoffrey E Hinton, Volodymyr Mnih, Joel Z Leibo, and Catalin Ionescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Using fast weights to attend to the recent past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 29:4331–4339, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Jared Quincy Davis, Krzysztof Choromanski, Jake Varley, Honglak Lee, Jean-Jacques Slotine, Valerii Likhos- terov, Adrian Weller, Ameesh Makadia, and Vikas Sindhwani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Time dependence in non-autonomous neural odes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='01906, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' David Ha, Andrew Dai, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Hypernetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' arXiv preprint arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='09106, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Max Horn, Michael Moor, Christian Bock, Bastian Rieck, and Karsten Borgwardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Set functions for time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 4353–4363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Wenjie Du, David Côté, and Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Saits: Self-attention-based imputation for time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='08516, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' 11 A Appendix We provide comparisons to additional models in the tables below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' Table 6: Classification PhysioNet ICU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' METHOD AUC LATENT ODE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='857 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='861 SAITS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='848 BRITS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='835 GP-VAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='834 M-RNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='822 E2GAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='830 GRUI-GAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='830 GRU-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='863 GRU-SIMPLE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='808 IP-NETS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='860 PHASED-LSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='790 SEFT-ATTN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='851 TRANSFORMER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='863 FROM HORN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2020], DU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2022] Table 7: Classification on PhysioNet Sepsis METHOD AUC ODERNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='765 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='787 NCDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='925 NCDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='925 W/ TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='931 GRU-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='674 GRU-SIMPLE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='781 IP-NETS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='742 PHASED-LSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='754 SELF-ATTN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='768 TRANSFORMER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content='658 FROM HORN ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} +page_content=' [2020] Table 8: Number of Parameters (Rounded) METHOD #PARAMS L-ODE 67,071 W/ TW 52,016 M-RNN 70,000 E2GAN 80,000 GP-VAE 150,000 GRUI-GAN 160,000 BRITS 730,000 TRANSFORMER 4,360,000 SAITS 5,320,000 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfzAhZ/content/2301.04126v1.pdf'} diff --git 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100644 index 0000000000000000000000000000000000000000..54355191c5ee43e12a2c3f1dc4c8118a4b9b4c84 --- /dev/null +++ b/n9AyT4oBgHgl3EQfY_cz/content/tmp_files/2301.00213v1.pdf.txt @@ -0,0 +1,1212 @@ +1 + +Electrically Sign-Reversible Topological Hall Effect +in a Top-Gated Topological Insulator (Bi,Sb)2Te3 on a +Ferrimagnetic Insulator Europium Iron Garnet +Jyun-Fong Wong,1○ Ko-Hsuan Mandy Chen,1○ Jui-Min Chia,1 Zih-Ping Huang,2 Sheng-Xin +Wang,1 Pei-Tze Chen,1 Lawrence Boyu Young,2 Yen-Hsun Glen Lin,2 Shang-Fan Lee,3 Chung-Yu +Mou,1,4 Minghwei Hong,2* and Jueinai Kwo1* +1Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan +2Graduate Institute of Applied Physics and Department of Physics, National Taiwan University, +Taipei 10617, Taiwan +3Institute of Physics, Academia Sinica, Taipei 11529, Taiwan +4Center for Quantum Technology, National Tsing Hua University, Hsinchu 30013, Taiwan +○J.-F. W. and K.-H. M. C. contributed equally to this work. +*Address +correspondence +to +J. +Kwo, +raynien@phys.nthu.edu.tw; +M. +Hong, +mhong@phys.ntu.edu.tw + + +2 + +ABSTRACT +Topological Hall effect (THE), an electrical transport signature of systems with chiral spin +textures like skyrmions, has been observed recently in topological insulator (TI)-based magnetic +heterostructures. The strong spin-orbit coupling and the broken spatial inversion symmetry in such +heterostructures could lead to a sizable interfacial Dzyaloshinsky–Moriya interaction, favorable +for skyrmion formation and pronounced THE. However, the intriguing interplay between the +topological surface state (TSS) and THE is yet to be fully understood. In this work, we report an +unprecedentedly large THE signal (4.0 µΩ‧cm at 2 K) with an electrically reversible sign in a top- +gated 4 nm TI (Bi0.3Sb0.7)2Te3 (BST) grown on a ferrimagnetic insulator (FI) europium iron garnet +(EuIG). Dependences of THE on temperature, external magnetic field angle, and gate bias were +investigated and are consistent with the prediction of a skyrmion-driven THE, amenable to +elucidate the origin of THE that occurred in TI-based heterostructures. Moreover, a sign change in +THE was discovered as the Fermi level was tuned electrically from the upper (electron-doped +region) to the lower parts (hole-doped region) of the gapped BST Dirac cone and vice versa. We +show that the exploitation of the TSS features has led to a sign-reversal of THE repeatedly in a +TI/FI top-gate stack. Ultimately, this discovery is anticipated to impact technological applications +in ultralow power skyrmion-based spintronics. + +3 + + +KEYWORDS +topological insulator, ferrimagnetic insulator, topological Hall effect, skyrmion, electric field effect + +INTRODUCTION + +Three-dimensional topological insulators (TIs), featured with their spin-momentum-locked +topological surface states (TSSs),1,2 have generated enormous interest in spintronics over the past +decade. The interplay between the spin-momentum-locked TSS and magnetism brings in novel +electrical transport phenomena.3 A well-known example is the achievement of quantum anomalous +Hall effect (QAHE) in a transition metal-doped TI (Bi,Sb)2Te3 (BST).4-6 Besides the magnetically +doped TIs, much attention is also given to TI heterostructures interfaced with magnetic materials +(MMs) to attain a long-range magnetic ordering via magnetic proximity effect (MPE).7-14 However, +most studies were focused on the discussion of the phenomena arising from the non-trivial Berry +curvature in reciprocal space. The investigation of the spin textures in real space and their related +electrical transport in TI heterostructures remains largely unexplored. +Topological Hall effect (THE), a Hall response that emerges from the deflection of charge +carriers flowing through non-trivial chiral spin textures, is a transport signature commonly used to + +4 + +identify these chiral spin textures, such as magnetic skyrmions.15,16 Initially, magnetic skyrmions +were reported in bulk magnetic crystals lacking spatial inversion symmetry (SIS), such as B20- +type chiral magnets.17-20 Recently, skyrmions have been observed in SIS-broken MM-based +heterostructures possessing strong spin-orbit coupling (SOC) materials such as heavy metals (HMs) +or TIs.21,22 The Dzyaloshinskii–Moriya interaction (DMI) at the interface plays a crucial role in +stabilizing skyrmions.23,24 In contrast to the high current density (1011–1012 A/m2) required for +magnetic domain wall motions, an ultralow current density (105–106 A/m2) has been accomplished +for skyrmion motions, thus potential for high-density information storage devices in memory and +computing technology with ultralow power consumption.16,20,23,25 The writing, deleting, and +processing of skyrmions are extensively investigated as well in the rapidly growing field of +skyrmion-electronics.24,26 THE, being an electrical transport phenomenon, is a promising method +to read skyrmions. +Among many THE studies, there have been debates about the exact origin of the reported +Hall responses.27-33 For example, because THE is commonly intertwined with anomalous Hall +effect (AHE), an alternative cause is the superposition of multi-AHE contributions.27-29,31-33 In this +scenario, one can conduct two- or three-AHE fits to decompose the signals, and ascertain if the +THE responses could indeed result from the overlapping of multi-AHE components.14,30,34,35 +Although several reports demonstrated pronounced THE in TI-based heterostructures,30,34,36-39 it is + +5 + +noteworthy that an in-depth discussion on the THE emerging from the carrier transport of spin- +momentum-locked TSS is still missing. A straightforward way to investigate the relationship +between THE and charge carriers in TSS is by implementing an electrical gate bias on TI. So far, +the gate-tunable THE has been reported in Mn-doped Bi2Te3 films and Cr-doped BST +heterostructures;37-39 however, the gate devices were achieved by using SrTiO3 (STO) as a +dielectric with a large gate bias of tens of volts,37-39 which are unfavorable in practical use. Finding +a suitable gate dielectric with excellent carrier tunability and reliability is thus an important issue +and needs to be addressed for both fundamental scientific studies and technological applications. +In this work, we report the observation of THE in TI BST/ferrimagnetic insulator (FI) +europium iron garnet (EuIG) bilayers. In particular, we demonstrate a successful manipulation of +THE by switching the charge carrier type using a top electrical gate within a few volts, which has +yet to be reported. By adopting a heterostructure of gate oxide/TI/FI, the current flow path and the +resulting Hall contribution can be limited to the TI layer, which is rather simple compared to other +magnetic TI (MTI)/TI/MTI or TI/MM heterostructures. Judging from the temperature, magnetic +field angle, and gate voltage dependences of Hall measurements, our findings were consistent with +the picture of a skyrmion-driven THE instead of the superposition of AHE loops. The largest THE +magnitude reached ~4.02 µΩ‧cm under an applied Vg of -0.6 V at 2 K. Moreover, the observation +of THE at zero fields suggested a stable skyrmion phase without an applied external magnetic field. + +6 + +Most importantly, we showed a pronounced and repeated THE sign reversal when the Fermi level +(EF) is tuned across the gapped Dirac point of BST. Hence, the exploration of THE in TI-based +heterostructures opens a new route in high-density and ultralow-power skyrmion-based devices in +spintronics. + +RESULTS AND DISCUSSION +Characterization of BST/EuIG heterostructures. The material growth of BST/EuIG +basically followed our previous work.14,40 Here, we chose to study the 4 nm BST thin films with a +simultaneously magnetized top and bottom TSS, of which the THE-like feature has been +repeatedly observed in many samples.14 In order to manipulate the EF close to the magnetic gap, +the Bi:Sb composition ratio was tuned to be 3:7. The streaky reflection high energy electron +diffraction (RHEED) pattern in Figure 1a manifested an atomically ordered and morphologically +smooth BST surface with excellent crystallinity grown on EuIG thin films. Ex-situ atomic force +microscopy (AFM) was performed after the deposition of 2 nm Y2O3 and 15 nm Al2O3; the flat +BST surface covered by the oxide layers with a small root-mean-square roughness of 0.633 nm is +presented in Figure 1b. +Figure 1c illustrates the device structure and the measurement setup. The electrical transport + +7 + +study of BST/EuIG was commenced by measuring longitudinal resistance (Rxx) as a function of +temperature, shown in Figure 1d. Rxx increased with decreasing temperatures from 300 K to 100 +K, revealing a semiconducting behavior caused by the reduction of charge carriers in BST bulk.41,42 +Then, Rxx reached a local maximum at ~100 K and decreased as the temperature was further +lowered from ~100 K to ~10 K, indicating a metallic behavior from the TSS of BST.41,42 At +temperatures below 10 K, Rxx increased with decreasing temperatures again, which could be +attributed to the electron-electron interaction (EEI).43 The longitudinal conductance (Gxx) was +proportional to the natural logarithm of T, as shown in the inset of Figure 1d, consistent with the +2D EEI of the TSS.44,45 As reported in the literature, the existence of the TSS could help generate +a strong interfacial DMI, giving rise to topological magnetic structures with a pronounced +THE.30,34 + +Identifying the topological Hall effect with electrical transport measurements. Before +reaching a definitive conclusion of having observed THE, we performed additional experimental +checks to rule out other possible causes. Figure 2a displays the Hall resistance of the ungated 4 nm +thick BST on EuIG at 2 K after the subtraction of the linear background from the ordinary Hall +effect (OHE). Besides the square AHE loop (black solid line in Figure 2a) originating from the +MPE,14 excessive Hall signals were clearly observed over a wide range of magnetic fields from - + +8 + +1.5 to 3 kOe as the magnetic field was swept from negative to positive. These excessive Hall +signals reached the largest value near the coercive field (Hc) and behaved like THE. Notice that it +is possible to have artificial THE-like responses from two overlapping AHE contributions.27-33 +Here, we adopted two separate methods, the minor loop approach and AHE curve fittings, to +resolve this issue. By utilizing the former method, the Hall traces coincided under successively +increasing sweeping magnetic fields as the expected behavior of THE responses.32 (see Figure S1 +in the Supporting Information) In addition, we analyzed the data by considering the superposition +of two AHE components. Although the Hall signals could be well fitted with two AHE +contributions mathematically, it is rather difficult to reconcile their dependences on temperatures +as well as gate biases with any reasonable or existing physical picture. (see Figure S2 in the +Supporting Information) Therefore, we inferred that these excessive Hall signals in BST/EuIG +were most likely to result from a genuine THE. +After excluding the Hall signals from the superposition of AHE loops, we proceeded to extract +the THE resistance (RTHE) by subtracting the AHE resistance (RAHE), as shown in Figure 2b. Here, +the maximum of the RTHE is denoted by RTHEMAX, and the H field corresponding to the RTHEMAX is +denoted by HT. A giant RTHEMAX ~7.64 Ω was observed with an applied gate voltage (Vg) of 0 V at +2 K, where the corresponding resistivity (ρTHEMAX) was ~3.06 µΩ‧cm. With the applied Vg of -0.6 +V, the ρTHEMAX reached the largest value of ~4.02 µΩ‧cm at 2 K. To the best of our knowledge, the + +9 + +highest record of ρTHEMAX reported in TI-based bi-layer systems was ~1.39 µΩ‧cm at 10 K;34 the +ρTHEMAX in this work was ~2.44 µΩ‧cm with Vg of 0 V at 8 K, which was nearly twice as large as +the previous record. The larger ρTHEMAX may be attributed to a higher density of chiral spin textures +because of the excellent interface between BST and EuIG. + +Temperature dependence of the topological Hall effect. The results of the temperature +dependence of THE in BST/EuIG are presented in Figure 2c,d with detailed THE loops shown in +Figure S3 in the Supporting Information. The THE gradually diminished with increasing +temperatures and disappeared at 75 K, which could be attributed to the thermal fluctuation or the +reduced DMI strength, consistent with the previous studies in TI-based systems.14,30,34,36-38 +Furthermore, shown in the color map of Figure 2c is the temperature dependence of HT that follows +closely with that of Hc. This observation is in accord with several reports on the THE driven by +the magnetization reversal process in systems hosting magnetic skyrmions.34,46 It is noteworthy +that our THE could be found at zero fields, suggesting a robust skyrmion phase without the support +of an external magnetic field, similar to the observation in FeGe.47,48 The RTHE at zero fields (RTHE0) +showed an akin temperature dependence to that of RTHEMAX and also vanished at 75 K, as +demonstrated in Figure 2d. + + +10 + +Angular dependence of the topological Hall effect. To deepen our understanding of the +THE in BST/EuIG, we further investigated the angular dependence of the THE with the +measurement geometry illustrated in Figure 3a. The angle θ is defined as the angle between the +external magnetic field H and the surface normal direction +z in the x-z plane. The Hall traces at +5 K after subtracting the linear OHE background from 10° to 80° are demonstrated in Figure 3b. +The AHE loops expanded with the increase in θ. The Hc enlarged and was proportional to 1/cosθ, +indicating that a larger external magnetic field was needed to flip the magnetization in BST to the +opposite direction as the θ increased. Furthermore, significant THE responses coexisting with +AHE were observed and sustained to 70°. The RTHE as a function of θ and H field is summarized +in the color map of Figure 3c; the H field was swept from negative to positive. Similar to the +temperature dependence of HT and Hc discussed in the previous section, the angular dependence +of HT also followed closely with that of Hc before the disappearance of THE. +We further extracted the RTHEMAX as a function of θ shown in Figure 3d. The RTHEMAX remained +almost the same from 10° to 70°, indicating that THE strength was not affected by a moderate in- +plane magnetic field. This angular dependence is reasonable because the topologically protected +skyrmions are robust against certain in-plane magnetic fields.30 While the θ further increased to +80°, the THE vanished, suggesting the collapse of the skyrmionic state.49-52 Similar angular +behavior of THE was reported in the Mn1-xFexSi, a well-known B20-type chiral magnet hosting + +11 + +magnetic skyrmions, with the disappearance of THE at 55°.49 Hence, our observation of angle- +dependent results in THE also supports the existence of skyrmions at the BST/EuIG interface. + +Manipulation of the topological Hall effect with an electric field. Next, manipulation of +THE was demonstrated by implementing a top gate electric field on our sample. Figure 4a shows +the gate dependence of Rxx and ordinary Hall coefficient (RH) derived from the linear background +of Hall traces. The EF was successfully tuned across the gapped Dirac point of BST via the gate +bias, as manifested by the rising behavior of Rxx in reaching a maximum and the sign change in RH +near the charge neutrality point (CNP) (the gray-shaded region in Figure 4a). The CNP here is +expected to locate at the center of the proximity-induced magnetic gap of TSS in BST. Moreover, +the small applied Vg for the CNP (VCNP) of ~0.6 V indicated that the EF of our un-gated BST was +fairly close to the center of the magnetic gap, providing an excellent starting point to alter the +carrier type and the carrier density of BST, thus to explore the systematic dependence of THE +under an external electric field. +Figure 4b displays the gate dependence of selected AHE loops together with the THE; the +scattered points are the measured RAHE + RTHE data, and the solid lines are the fitted AHE +contributions. In the p-type region (Vg < VCNP), a hump was found near the Hc of the measured data +as the H field was swept from negative to positive. (dark red arrows in Figure 4b) In the n-type + +12 + +region (Vg > VCNP), a dip was observed instead near the Hc of the measured data. (a green arrow in +Figure 4b) With the applied Vg ~VCNP, the measured data became a square hysteresis loop, a typical +feature of AHE without excessive Hall signals from THE. To extract the magnitude of THE, we +subtracted the AHE contributions (black solid lines) and plotted RTHE in Figure 4c. Positive THE +was identified in the p-type region of the up-sweep (red) curves, while negative THE was observed +in the n-type region. More gate-dependent THE loops with finer voltage steps are presented in +Figure S4 in the Supporting Information. The top-gate bias dependence of RTHEMAX is further +summarized in Figure 4d. + +Relationship between the topological Hall effect and the topological surface state. To +gain a better insight into the mechanism responsible for the sign reversal in THE, we examined +RTHE and its relations to other physical quantities. For systems hosting chiral spin textures of +skyrmions, an effective electromagnetic field Beff = nsk Φ0 will be generated by these structures, +where nsk is the skyrmion density and Φ0 is the magnetic flux quantum.15 The Hall resistance +resulting from THE is commonly approximated as: + RTHE ≈ RH P nsk Φ0, (1) +where P is the local spin polarization of the conduction carrier.15,34 When the EF is tuned from the +upper Dirac cone to the lower Dirac cone, the majority charge carriers will be altered from + +13 + +electrons to holes, leading to a sign change in RH. Furthermore, the electron spin polarization is +flipped to the opposite direction because of the spin-momentum locking of the gapped TSS. (see +the Fermi surface in Figure 5) Since the propagation direction of holes is opposite with respect to +that of electrons, the spin orientation for electrons and holes will thus be the same, giving rise to +the unchanged sign of P.53 Therefore, the sign of THE will be switched from negative to positive +when the EF is varied from the upper to the lower parts of the Dirac cone, as illustrated in Figure +5. The EF modulation by the gate bias in Figure 4 was roughly estimated to be from ~84 meV (Vg += +1.5 V) above the gapped Dirac point to ~128 meV (Vg = -1.5 V) below the gapped Dirac point, +as detailed in the Supporting Information. +In the vicinity of CNP, namely RH → 0, the vanishing RTHE could be attributed to the nearly +equal numbers of n- and p-type charge carriers. The absence of the THE signals with the EF near +the CNP was also observed in the Cr-doped BST sandwich heterostructures.38,39 However, in their +reports, RTHE did not show a sign reversal with the applied gate voltages; the majority carrier type +remained the same in both Vg < VCNP and Vg > VCNP regions, as suggested from the slopes of their +Hall traces. In contrast, our work demonstrated an effective manipulation of charge carriers and +THE via a top electrical gate. Moreover, our gate bias-dependent results lent strong support to the +picture of a skyrmion-driven THE in the BST/EuIG heterostructure, which can be well described +by Equation 1 for both positive and negative bias. Similar behavior of manipulating THE via the + +14 + +top-gate bias in another sample is also presented in Figure S5 in the Supporting Information. Given +these findings, inspecting the THE sign change when the EF is located above and below the CNP +could be another viable way to differentiate the genuine and artificial THE in TI-based +heterostructures. The electrically sign-reversible THE in TI/FI heterostructures could be a unique +feature directly associated with the spin-momentum-locking of the gapped TSS. This phenomenon +has not been observed and might well be absent in other non-TI skyrmion systems, such as bulk +chiral magnets or HM/FI heterostructures like Pt/TmIG. + +Repeated switching of the topological Hall effect with an electric field. To further examine +the reliability of the field-effect device and the reproducibility of the sign reversal in THE, we +performed the THE measurements with a series of selected gate biases. As displayed in Figure 6a, +the THE was reversibly switched. The RTHEMAX and HT values remained nearly identical to the +starting ones, as demonstrated in Figure 6b,c. Although the repeated switching of this “THE” +device is currently limited by the robustness of our gate oxides after applying excessive gate bias, +the achievement of THE switching within only a few volts demonstrates tremendous progress in +manipulating this phenomenon. The idea of an electrically tunable THE field-effect device may +open up a new avenue in skyrmion-based spintronic applications. + + +15 + +CONCLUSION + +In summary, we have demonstrated the electrically sign-reversible THE in a top-gated TI BST +on a FI EuIG. The magnetotransport behaviors on temperature, external magnetic field angle, and +gate bias were consistent with a picture of skyrmion-driven THE, as opposed to the alternative +mechanism of superimposed AHE loops. Moreover, the sign change in THE via an electrical gate +could be a distinctive feature related to the gapped TSS in TI/FI compared to other non-TI +skyrmion systems, such as bulk chiral magnets and HM/FI heterostructures. Our findings in this +work have provided a thorough understanding of the THE in a TI-based heterostructure. This +bilayer structure of BST/EuIG offers an excellent platform for studying the interplay among +magnetism, chiral spin textures, and topological band structures. Especially, the reproducible +electrical manipulation of THE within a few volts may be promising for ultralow-power skyrmion- +related applications. Further investigations on direct imaging of skyrmion spin structures in real +space will be essential, and experiments via scanning microscopy methods, such as spin-polarized +scanning tunneling microscopy and scanning transmission x-ray microscopy, are now underway. + +EXPERIMENTAL METHODS +Materials growth. 20 nm thick FI EuIG(001) films with perpendicular magnetic anisotropy + +16 + +(PMA) were grown on GGG(001) using an off-axis magnetron sputtering technique.40 4 nm thick +TI BST films were then deposited on EuIG thin films at ~260 °C by using MBE under a base +pressure below 4 × 10-10 Torr.14 The deposition of gate oxides was divided into two steps. In the +first step, the samples were transferred from the TI-MBE chamber to a multiple-chamber system +containing oxide-MBE and ALD chambers under ultra-high vacuum (UHV) to avoid +contaminations at the TI/oxide interface;54 a 2 nm thick e-beam evaporated Y2O3 (in the oxide +MBE chamber) followed by a 15 nm thick in-situ ALD Al2O3 was deposited on the pristine BST +surface, leading to an unpinned EF at the interface of gate oxide/BST, based on our extensive +expertise of high-κ dielectrics on semiconductor surfaces.55-59 To better protect samples for +transport measurements, we deposited a second Al2O3 layer with a thickness of 25 nm in another +ex-situ ALD system after the fabrication of Hall bars to prevent current leakage from the edges of +the Hall bar. +Electrical measurements. The samples were patterned into Hall bars (880 μm × 90 μm) +using photolithography and reactive ion etching. All electrical transport measurements were +conducted in the physical property measurement system (PPMS) connected with the following +instruments. A Keithley 2400 was used to provide a stable voltage source as the gate voltage. A +Keithley 6221 was used as a current source to generate the alternating current with a root-mean- +square amplitude of 100 nA. Two SR830 lock-in amplifiers were used as voltmeters to measure + +17 + +the Hall voltages and longitudinal voltages, where the reference signal was provided by Keithley +6221. The Hall effect data were antisymmetrized as a function of the magnetic field to eliminate +the Rxx component due to electrode misalignments. + +18 + +FIGURES + +Figure 1. Sample characterization and schematic illustration of a top-gated BST/EuIG device. (a) +RHEED pattern of the 4 nm BST(001) surface along the [100] axis. (b) Surface morphology of 15 +nm atomic layer deposition (ALD)-Al2O3/2 nm e-beam evaporated-Y2O3/4 nm molecular beam +epitaxy (MBE)-BST/20 nm sputtering-EuIG/GGG(001) in a 1×1 μm2 area by using AFM. (c) +Schematics of a cross-section view and a front view of our top-gated BST grown on EuIG, in +which a light blue vertical plane is to dissect the sample structure for the cross-section view. Also +shown are the Néel-type skyrmion spin texture at the BST/EuIG interface, and the optical + +200hm +Au~100nm +Ti~30nm +Al203~25nm +Al03~15nm +Y203~2nm +(BiSb)2Te3~4nm +EulG~20nm +200μm +GGG (001)19 + +microscope image of a Hall bar device with our measurement setup. (d) Rxx as a function of +temperature. The inset in (d) shows the Gxx as a function of temperature and the fit with a +logarithmic function of T. + + +Figure 2. Temperature-dependent THE properties of BST/EuIG. (a) Coexistence of AHE and THE +with an applied Vg of 0 V at 2 K. The scattered points are the Hall data after subtracting a linear +OHE background, and the solid line is the fitted AHE contribution. (b) The THE at Vg = 0 V. (c) +Temperature dependences of Hc and HT, and a color map of RTHE (color) as a function of +temperature and magnetic field with an applied Vg of 0 V; the H field was swept from negative to +positive. (d) The temperature dependences of RTHEMAX and RTHE0. + +H +V=0V20 + + + +Figure 3. Angular dependence of THE from 10° to 80° at 5 K of BST/EuIG. (a) A schematic of +the angle-dependent Hall measurements. (b) Angular dependence of the AHE loops plus the THE. +(c) Magnetic field angle dependences of Hc and HT, and a color map of RTHE (color) as a function +of magnetic field angle and H field; the H field was swept from negative to positive. (d) The +angular dependence of RTHEMAX. + + +H +BST +EulG +T= 5K21 + + +Figure 4. Gate-bias dependence of THE from -1.5 V to 1.5 V at 2 K of BST/EuIG. (a) Top-gate +voltage dependences of RH (left) and Rxx (right). Small RH and large Rxx occurred in the vicinity of +CNP (the gray-shaded region). (b) Concurrence of AHE and THE with selected gate voltages. The +dark red and green arrows in (b) indicate the hump and dip features near the Hc, respectively. (c) +THE with selected gate voltages. (d) Top-gate voltage dependence of RTHEMAX at 2 K. THE +disappeared in the vicinity of CNP (the gray-shaded region). + + +14.0 +8H +13.5 +113.0m +12.5 +CNPI22 + + +Figure 5. Illustration of the sign reversal in THE with varying gate bias. Schematics for gapped +TSS, the Fermi surface with the application of current in +x direction, and the corresponding THE +with (a) Vg > VCNP (n-type region, EF located at upper gapped TSS) and (b) Vg < VCNP (p-type +region, EF located at lower gapped TSS), respectively. + + +Nogative THE +Pccitive THE23 + + +Figure 6. Repeated switching of THE in BST/EuIG. (a) THE loops with applied Vg in the sequence +of 0 V, 1 V, 1.5 V, 1 V, and 0 V. The corresponding modulation of (b) RTHEMAX and (c) HT as a +function of Vg. + +ASSOCIATED CONTENT +Supporting Information. +Discussion on the THE, temperature- and gate-dependent THE loops, estimate of EF manipulated +by gate bias, and gate-dependent results of one additional field-effect device (PDF) + + +Vg=0V +1 v +1.5 V +1 v +ov ++H +++H ++H ++H ++H +H +H +0 V-1 V-1.5 V +0 V-1 V-→1.5 V +1.5 V1 VO V +1.5 V+1 V-0 V24 + +AUTHOR INFORMATION +Corresponding Authors +* Jueinai Kwo – Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan; +orcid.org/0000-0002-5088-6677; E-mail: raynien@phys.nthu.edu.tw +* Minghwei Hong – Graduate Institute of Applied Physics and Department of Physics, National +Taiwan +University, +Taipei +10617, +Taiwan; +orcid.org/0000-0003-4657-0933; +E-mail: +mhong@phys.ntu.edu.tw +Authors +Jyun-Fong Wong – Department of Physics, National Tsing Hua University, Hsinchu 30013, +Taiwan; orcid.org/0000-0001-5756-5801 +Ko-Hsuan Mandy Chen – Department of Physics, National Tsing Hua University, Hsinchu 30013, +Taiwan; orcid.org/0000-0002-4402-3541 +Jui-Min Chia – Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan; +orcid.org/0000-0003-1759-4725 +Zih-Ping Huang – Graduate Institute of Applied Physics and Department of Physics, National +Taiwan University, Taipei 10617, Taiwan; orcid.org/0000-0001-8585-8812 +Sheng-Xin Wang – Department of Physics, National Tsing Hua University, Hsinchu 30013, + +25 + +Taiwan; orcid.org/0000-0001-9725-2930 +Pei-Tze Chen – Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan; +orcid.org/0000-0002-6249-5446 +Lawrence Boyu Young – Graduate Institute of Applied Physics and Department of Physics, +National Taiwan University, Taipei 10617, Taiwan; orcid.org/0000-0003-2569-6094 +Yen-Hsun Glen Lin – Graduate Institute of Applied Physics and Department of Physics, National +Taiwan University, Taipei 10617, Taiwan; orcid.org/0000-0002-0757-4109 +Shang-Fan Lee – Institute of Physics, Academia Sinica, Taipei 11529, Taiwan; orcid.org/0000- +0001-5899-7200 +Chung-Yu Mou – Center for Quantum Technology and Department of Physics, National Tsing +Hua University, Hsinchu 30013, Taiwan; orcid.org/0000-0003-2694-0992 + +Author Contributions +○J.-F.W. and K.-H.M.C. contributed equally to this work. J.-F.W. and J.-M.C. fabricated the Hall +bar device, and collected and analyzed the transport data. Z.-P.H., S.-X.W., and P.-Z.C. produced +the BST/EuIG samples. L.B.Y. and Y.-H.G.L. deposited the in-situ top gate oxides. S.-F.L. and C.- +Y.M. provided scientific inputs. J.K. and M.H. supervised the project. J.-F.W., K.-H.M.C., and J.K. + +26 + +composed and wrote the manuscript with the comments of all the authors. +Notes +The authors declare no competing financial interest. + +ACKNOWLEDGMENTS +The authors would like to thank Professor Hsiu-Hau Lin, Professor Tay-Rong Chang, Keng-Yung +Lin, Wei-Nien Chen, Wei-Jhih Zou, and Hsuan-Ning Chen for helpful discussions. This work was +financially supported by the National Science and Technology Council (NSTC), Taiwan (NSTC +111-2112-M-007-043- and NSTC 111-2622-8-002-001-), and the Center for Quantum Technology +and Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan (NSTC 111- +2634-F-007-006-). The authors acknowledge resources and support from the Quantum Materials +Shared Facilities of Institute of Physics, Academia Sinica. The authors also thank the technical +support from Taiwan Semiconductor Research Institute (TSRI), Taiwan. + +REFERENCES +(1) Qi, X.-L.; Hughes, T. L.; Zhang, S.-C. Topological Field Theory of Time-Reversal Invariant + +27 + +Insulators. Phys. Rev. B 2008, 78, 195424. +(2) Hasan, M. Z.; Kane, C. L. Colloquium: Topological Insulators. Rev. Mod. Phys. 2010, 82, +3045. +(3) Tokura, Y.; Yasuda, K.; Tsukazaki, A. Magnetic Topological Insulators. Nat. Rev. Phys. 2019, +1, 126. +(4) Chang, C.-Z.; Zhang, J.; Feng, X.; Shen, J.; Zhang, Z.; Guo, M.; Li, K.; Ou, Y.; Wei, P.; +Wang, L.-L.; Ji, Z.-Q.; Feng, Y.; Ji, S.; Chen, X.; Jia, J.; Dai, X.; Fang, Z.; Zhang, S.-C.; He, K.; +Wang, Y.; et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic +Topological Insulator. Science 2013, 340, 167. +(5) Checkelsky, J. G.; Yoshimi, R.; Tsukazaki, A.; Takahashi, K. S.; Kozuka, Y.; Falson, J.; +Kawasaki, M.; Tokura, Y. Trajectory of the Anomalous Hall Effect towards the Quantized State in +a Ferromagnetic Topological Insulator. Nat. Phys. 2014, 10, 731. +(6) Kou, X.; Guo, S.-T.; Fan, Y.; Pan, L.; Lang, M.; Jiang, Y.; Shao, Q.; Nie, T.; Murata, K.; +Tang, J.; Wang, Y.; He, L.; Lee, T.-K.; Lee, W.-L.; Wang, K. L. Scale-Invariant Quantum +Anomalous Hall Effect in Magnetic Topological Insulators beyond the Two-Dimensional Limit. +Phys. Rev. Lett. 2014, 113, 137201. + +28 + +(7) Wei, P.; Katmis, F.; Assaf, B. A.; Steinberg, H.; Jarillo-Herrero, P.; Heiman, D.; Moodera, J. +S. Exchange-Coupling-Induced Symmetry Breaking in Topological Insulators. Phys. Rev. Lett. +2013, 110, 186807. +(8) Lang, M.; Montazeri, M.; Onbasli, M. C.; Kou, X.; Fan, Y.; Upadhyaya, P.; Yao, K.; Liu, F.; +Jiang, Y.; Jiang, W.; Wong, K. L.; Yu, G.; Tang, J.; Nie, T.; He, L.; Schwartz, R. N.; Wang, Y.; Ross, +C. A.; Wang, K. L. Proximity Induced High-Temperature Magnetic Order in Topological Insulator +- Ferrimagnetic Insulator Heterostructure. Nano Lett. 2014, 14, 3459. +(9) Katmis, F.; Lauter, V.; Nogueira, F. S.; Assaf, B. A.; Jamer, M. E.; Wei, P.; Satpati, B.; +Freeland, J. W.; Eremin, I.; Heiman, D.; Jarillo-Herrero, P.; Moodera, J. S. A High-Temperature +Ferromagnetic Topological Insulating Phase by Proximity Coupling. Nature 2016, 533, 513. +(10) Fanchiang, Y. T.; Chen, K. H. M.; Tseng, C. C.; Chen, C. C.; Cheng, C. K.; Yang, S. R.; +Wu, C. N.; Lee, S. F.; Hong, M.; Kwo, J. Strongly Exchange-Coupled and Surface-State- +Modulated Magnetization Dynamics in Bi2Se3/Yttrium Iron Garnet Heterostructures. Nat. +Commun. 2018, 9, 223. +(11) Watanabe, R.; Yoshimi, R.; Kawamura, M.; Mogi, M.; Tsukazaki, A.; Yu, X. Z.; Nakajima, +K.; Takahashi, K. S.; Kawasaki, M.; Tokura, Y. Quantum Anomalous Hall Effect Driven by + +29 + +Magnetic Proximity Coupling in All-Telluride Based Heterostructure. Appl. Phys. Lett. 2019, 115, +102403. +(12) Yang, S. R.; Fanchiang, Y. T.; Chen, C. C.; Tseng, C. C.; Liu, Y. C.; Guo, M. X.; Hong, M.; +Lee, S. F.; Kwo, J. Evidence for Exchange Dirac Gap in Magnetotransport of Topological +Insulator–Magnetic Insulator Heterostructures. Phys. Rev. B 2019, 100, 045138. +(13) Bhattacharyya, S.; Akhgar, G.; Gebert, M.; Karel, J.; Edmonds, M. T.; Fuhrer, M. S. Recent +Progress in Proximity Coupling of Magnetism to Topological Insulators. Adv. Mater. 2021, 33, +2007795. +(14) Zou, W.-J.; Guo, M.-X.; Wong, J.-F.; Huang, Z.-P.; Chia, J.-M.; Chen, W.-N.; Wang, S.-X.; +Lin, K.-Y.; Young, L. B.; Lin, Y.-H. G.; Yahyavi, M.; Wu, C.-T.; Jeng, H.-T.; Lee, S.-F.; Chang, T.- +R.; Hong, M.; Kwo, J. Enormous Berry-Curvature-Based Anomalous Hall Effect in Topological +Insulator (Bi,Sb)2Te3 on Ferrimagnetic Europium Iron Garnet beyond 400 K. ACS Nano 2022, 16, +2369. +(15) Neubauer, A.; Pfleiderer, C.; Binz, B.; Rosch, A.; Ritz, R.; Niklowitz, P. G.; Böni, P. +Topological Hall Effect in the A Phase of MnSi. Phys. Rev. Lett. 2009, 102, 186602. +(16) Nagaosa, N.; Tokura, Y. Topological Properties and Dynamics of Magnetic Skyrmions. Nat. + +30 + +Nanotechnol. 2013, 8, 899. +(17) Mühlbauer, S.; Binz, B.; Jonietz, F.; Pfleiderer, C.; Rosch, A.; Neubauer, A.; Georgii, R.; +Böni, P. Skyrmion Lattice in a Chiral Magnet. Science 2009, 323, 915. +(18) Yu, X. Z.; Kanazawa, N.; Onose, Y.; Kimoto, K.; Zhang, W. Z.; Ishiwata, S.; Matsui, Y.; +Tokura, Y. Near Room-Temperature Formation of a Skyrmion Crystal in Thin-Films of the +Helimagnet FeGe. Nat. Mater. 2011, 10, 106. +(19) Tonomura, A.; Yu, X.; Yanagisawa, K.; Matsuda, T.; Onose, Y.; Kanazawa, N.; Park, H. S.; +Tokura, Y. Real-Space Observation of Skyrmion Lattice in Helimagnet MnSi Thin Samples. Nano +Lett. 2012, 12, 1673. +(20) Yu, X. Z.; Kanazawa, N.; Zhang, W. Z.; Nagai, T.; Hara, T.; Kimoto, K.; Matsui, Y.; Onose, +Y.; Tokura, Y. Skyrmion Flow near Room Temperature in an Ultralow Current Density. Nat. +Commun. 2012, 3, 988. +(21) Wu, H.; Groß, F.; Dai, B.; Lujan, D.; Razavi, S. A.; Zhang, P.; Liu, Y.; Sobotkiewich, K.; +Förster, J.; Weigand, M.; Schütz, G.; Li, X.; Gräfe, J.; Wang, K. L. Ferrimagnetic Skyrmions in +Topological Insulator/Ferrimagnet Heterostructures. Adv. Mater. 2020, 32, 2003380. +(22) Vélez, S.; Ruiz-Gómez, S.; Schaab, J.; Gradauskaite, E.; Wörnle, M. S.; Welter, P.; Jacot, + +31 + +B. J.; Degen, C. L.; Trassin, M.; Fiebig, M.; Gambardella, P. Current-Driven Dynamics and Ratchet +Effect of Skyrmion Bubbles in a Ferrimagnetic Insulator. Nat. Nanotechnol. 2022, 17, 834. +(23) Fert, A.; Cros, V.; Sampaio, J. Skyrmions on the Track. Nat. Nanotechnol. 2013, 8, 152. +(24) Fert, A.; Reyren, N.; Cros, V. Magnetic Skyrmions: Advances in Physics and Potential +Applications. Nat. Rev. Mater. 2017, 2, 17031. +(25) Schulz, T.; Ritz, R.; Bauer, A.; Halder, M.; Wagner, M.; Franz, C.; Pfleiderer, C.; Everschor, +K.; Garst, M.; Rosch, A. Emergent Electrodynamics of Skyrmions in a Chiral Magnet. Nat. Phys. +2012, 8, 301. +(26) Zhang, X.; Zhou, Y.; Song, K. M.; Park, T.-E.; Xia, J.; Ezawa, M.; Liu, X.; Zhao, W.; Zhao, +G.; Woo, S. Skyrmion-Electronics: Writing, Deleting, Reading and Processing Magnetic +Skyrmions toward Spintronic Applications. J. Phys.: Condens. Matter 2020, 32, 143001. +(27) Gerber, A. Interpretation of Experimental Evidence of the Topological Hall Effect. Phys. +Rev. B 2018, 98, 214440. +(28) Fijalkowski, K. M.; Hartl, M.; Winnerlein, M.; Mandal, P.; Schreyeck, S.; Brunner, K.; +Gould, C.; Molenkamp, L. W. Coexistence of Surface and Bulk Ferromagnetism Mimics Skyrmion +Hall Effect in a Topological Insulator. Phys. Rev. X 2020, 10, 011012. + +32 + +(29) Wang, L.; Feng, Q.; Lee, H. G.; Ko, E. K.; Lu, Q.; Noh, T. W. Controllable Thickness +Inhomogeneity and Berry Curvature Engineering of Anomalous Hall Effect in SrRuO3 Ultrathin +Films. Nano Lett. 2020, 20, 2468. +(30) Li, P.; Ding, J.; Zhang, S. S.-L.; Kally, J.; Pillsbury, T.; Heinonen, O. G.; Rimal, G.; Bi, C.; +DeMann, A.; Field, S. B.; Wang, W.; Tang, J.; Jiang, J. S.; Hoffmann, A.; Samarth, N.; Wu, M. +Topological Hall Effect in a Topological Insulator Interfaced with a Magnetic Insulator. Nano Lett. +2021, 21, 84. +(31) Wang, F.; Wang, X.; Zhao, Y.-F.; Xiao, D.; Zhou, L.-J.; Liu, W.; Zhang, Z.; Zhao, W.; Chan, +M. H. W.; Samarth, N.; Liu, C.; Zhang, H.; Chang, C.-Z. Interface-Induced Sign Reversal of the +Anomalous Hall Effect in Magnetic Topological Insulator Heterostructures. Nat. Commun. 2021, +12, 79. +(32) Tai, L.; Dai, B.; Li, J.; Huang, H.; Chong, S. K.; Wong, K. L.; Zhang, H.; Zhang, P.; Deng, +P.; Eckberg, C.; Qiu, G.; He, H.; Wu, D.; Xu, S.; Davydov, A.; Wu, R.; Wang, K. L. Distinguishing +the Two-Component Anomalous Hall Effect from the Topological Hall Effect. ACS Nano 2022, +16, 17336. +(33) Kimbell, G.; Kim, C.; Wu, W.; Cuoco, M.; Robinson, J. W. A. Challenges in Identifying + +33 + +Chiral Spin Textures via the Topological Hall Effect. Commun. Mater. 2022, 3, 19. +(34) Zhang, X.; Ambhire, S. C.; Lu, Q.; Niu, W.; Cook, J.; Jiang, J. S.; Hong, D.; Alahmed, L.; +He, L.; Zhang, R.; Xu, Y.; Zhang, S. S.-L.; Li, P.; Bian, G. Giant Topological Hall Effect in van +der Waals Heterostructures of CrTe2/Bi2Te3. ACS Nano 2021, 15, 15710. +(35) Jeon, J. H.; Na, H. R.; Kim, H.; Lee, S.; Song, S.; Kim, J.; Park, S.; Kim, J.; Noh, H.; Kim, +G.; Jerng, S.-K.; Chun, S.-H. Emergent Topological Hall Effect from Exchange Coupling in +Ferromagnetic Cr2Te3/Noncoplanar Antiferromagnetic Cr2Se3 Bilayers. ACS Nano 2022, 16, 8974. +(36) Yasuda, K.; Wakatsuki, R.; Morimoto, T.; Yoshimi, R.; Tsukazaki, A.; Takahashi, K. S.; +Ezawa, M.; Kawasaki, M.; Nagaosa, N.; Tokura, Y. Geometric Hall Effects in Topological +Insulator Heterostructures. Nat. Phys. 2016, 12, 555. +(37) Liu, C.; Zang, Y.; Ruan, W.; Gong, Y.; He, K.; Ma, X.; Xue, Q.-K.; Wang, Y. Dimensional +Crossover-Induced Topological Hall Effect in a Magnetic Topological Insulator. Phys. Rev. Lett. +2017, 119, 176809. +(38) Jiang, J.; Xiao, D.; Wang, F.; Shin, J.-H.; Andreoli, D.; Zhang, J.; Xiao, R.; Zhao, Y.-F.; +Kayyalha, M.; Zhang, L.; Wang, K.; Zang, J.; Liu, C.; Samarth, N.; Chan, M. H. W.; Chang, C.-Z. +Concurrence of Quantum Anomalous Hall and Topological Hall Effects in Magnetic Topological + +34 + +Insulator Sandwich Heterostructures. Nat. Mater. 2020, 19, 732. +(39) Xiao, R.; Xiao, D.; Jiang, J.; Shin, J.-H.; Wang, F.; Zhao, Y.-F.; Zhang, R.-X.; Richardella, +A.; Wang, K.; Kayyalha, M.; Chan, M. H. W.; Liu, C.-X.; Chang, C.-Z.; Samarth, N. Mapping the +Phase Diagram of the Quantum Anomalous Hall and Topological Hall Effects in a Dual-Gated +Magnetic Topological Insulator Heterostructure. Phys. Rev. Res. 2021, 3, L032004. +(40) Guo, M. X.; Cheng, C. K.; Liu, Y. C.; Wu, C. N.; Chen, W. N.; Chen, T. Y.; Wu, C. T.; Hsu, +C. H.; Zhou, S. Q.; Chang, C. F.; Tjeng, L. H.; Lee, S. F.; Pai, C. F.; Hong, M.; Kwo, J. Single- +Crystal Epitaxial Europium Iron Garnet Films with Strain-Induced Perpendicular Magnetic +Anisotropy: Structural, Strain, Magnetic, and Spin Transport Properties. Phys. Rev. Mater. 2022, +6, 054412. +(41) Dutta, P.; Pariari, A.; Mandal, P. Prominent Metallic Surface Conduction and the Singular +Magnetic Response of Topological Dirac Fermion in Three-Dimensional Topological Insulator +Bi1.5Sb0.5Te1.7Se1.3. Sci. Rep. 2017, 7, 4883. +(42) Qin, L.-X.; Pan, X.-C.; Song, F.-Q.; Zhang, L.; Sun, Z.-H.; Li, M.-Q.; Gao, P.; Lin, B.-C.; +Huang, S.-M.; Zhu, R.; Xu, J.; Lin, F.; Lu, H.-Z.; Yu, D.; Liao, Z.-M. Confined-Path Interference +Suppressed Quantum Correction on Weak Antilocalization Effect in a BiSbTeSe2 Topological + +35 + +Insulator. Appl. Phys. Lett. 2018, 112, 032102. +(43) Lee, P. A.; Ramakrishnan, T. V. Disordered Electronic Systems. Rev. Mod. Phys. 1985, 57, +287. +(44) Wang, J.; DaSilva, A. M.; Chang, C.-Z.; He, K.; Jain, J. K.; Samarth, N.; Ma, X.-C.; Xue, +Q.-K.; Chan, M. H. W. Evidence for Electron-Electron Interaction in Topological Insulator Thin +Films. Phys. Rev. B 2011, 83, 245438. +(45) Li, P.; Kally, J.; Zhang, S. S.-L.; Pillsbury, T.; Ding, J.; Csaba, G.; Ding, J.; Jiang, J. S.; Liu, +Y.; Sinclair, R.; Bi, C.; DeMann, A.; Rimal, G.; Zhang, W.; Field, S. B.; Tang, J.; Wang, W.; +Heinonen, O. G.; Novosad, V.; Hoffmann, A.; et al. Magnetization Switching Using Topological +Surface States. Sci. Adv. 2019, 5, eaaw3415. +(46) Matsuno, J.; Ogawa, N.; Yasuda, K.; Kagawa, F.; Koshibae, W.; Nagaosa, N.; Tokura, Y.; +Kawasaki, M. Interface-Driven Topological Hall Effect in SrRuO3-SrIrO3 Bilayer. Sci. Adv. 2016, +2, e1600304. +(47) Huang, S. X.; Chien, C. L. Extended Skyrmion Phase in Epitaxial FeGe(111) Thin Films. +Phys. Rev. Lett. 2012, 108, 267201. +(48) Gallagher, J. C.; Meng, K. Y.; Brangham, J. T.; Wang, H. L.; Esser, B. D.; McComb, D. W.; + +36 + +Yang, F. Y. Robust Zero-Field Skyrmion Formation in FeGe Epitaxial Thin Films. Phys. Rev. Lett. +2017, 118, 027201. +(49) Yokouchi, T.; Kanazawa, N.; Tsukazaki, A.; Kozuka, Y.; Kawasaki, M.; Ichikawa, M.; +Kagawa, F.; Tokura, Y. Stability of Two-Dimensional Skyrmions in Thin Films of Mn1-xFexSi +Investigated by the Topological Hall Effect. Phys. Rev. B 2014, 89, 064416. +(50) Ohuchi, Y.; Kozuka, Y.; Uchida, M.; Ueno, K.; Tsukazaki, A.; Kawasaki, M. Topological +Hall Effect in Thin Films of the Heisenberg Ferromagnet EuO. Phys. Rev. B 2015, 91, 245115. +(51) Shao, Q.; Liu, Y.; Yu, G.; Kim, S. K.; Che, X.; Tang, C.; He, Q. L.; Tserkovnyak, Y.; Shi, +J.; Wang, K. L. Topological Hall Effect at above Room Temperature in Heterostructures Composed +of a Magnetic Insulator and a Heavy Metal. Nat. Electron. 2019, 2, 182. +(52) Sohn, B.; Kim, B.; Park, S. Y.; Choi, H. Y.; Moon, J. Y.; Choi, T.; Choi, Y. J.; Zhou, H.; +Choi, J. W.; Bombardi, A.; Porter, D. G.; Chang, S. H.; Han, J. H.; Kim, C. Stable Humplike Hall +Effect and Noncoplanar Spin Textures in SrRuO3 Ultrathin Films. Phys. Rev. Res. 2021, 3, 023232. +(53) Lee, J. S.; Richardella, A.; Hickey, D. R.; Mkhoyan, K. A.; Samarth, N. Mapping the +Chemical Potential Dependence of Current-Induced Spin Polarization in a Topological Insulator. +Phys. Rev. B 2015, 92, 155312. + +37 + +(54) Lin, K. Y.; Wan, H. W.; Chen, K. H. M.; Fanchiang, Y. T.; Chen, W. S.; Lin, Y. H.; Cheng, +Y. T.; Chen, C. C.; Lin, H. Y.; Young, L. B.; Cheng, C. P.; Pi, T. W.; Kwo, J.; Hong, M. Molecular +Beam Epitaxy, Atomic Layer Deposition, and Multiple Functions Connected via Ultra-High +Vacuum. J. Cryst. Growth 2019, 512, 223. +(55) Hong, M.; Passlack, M.; Mannaerts, J. P.; Kwo, J.; Chu, S. N. G.; Moriya, N.; Hou, S. Y.; +Fratello, V. J. Low Interface State Density Oxide‐GaAs Structures Fabricated by in situ Molecular +Beam Epitaxy. J. Vac. Sci. Technol. B 1996, 14, 2297. +(56) Hong, M.; Kwo, J.; Kortan, A. R.; Mannaerts, J. P.; Sergent, A. M. Epitaxial Cubic +Gadolinium Oxide as a Dielectric for Gallium Arsenide Passivation. Science 1999, 283, 1897. +(57) Kwo, J.; Hong, M.; Kortan, A. R.; Queeney, K. T.; Chabal, Y. J.; Mannaerts, J. P.; Boone, +T.; Krajewski, J. J.; Sergent, A. M.; Rosamilia, J. M. High ε Gate Dielectrics Gd2O3 and Y2O3 for +Silicon. Appl. Phys. Lett. 2000, 77, 130. +(58) Hong, M.; Kwo, J; Chu, S. N. G.; Mannaerts, J. P.; Kortan, A. R.; Ng, H. M.; Cho, A. Y.; +Anselm, K. A.; Lee, C. M.; Chyi, J. I. Single Crystal GaN/Gd2O3/GaN Heterostructure. J. Vac. Sci. +Technol. B 2002, 20, 1274. +(59) Lin, Y. H. G.; Wan, H. W.; Young, L. B.; Liu, J.; Cheng, Y. T.; Lin, K. Y.; Hong, Y. J.; Wu, + +38 + +C. T.; Kwo, J.; Hong, M. In situ Y2O3 on p-In0.53Ga0.47As—Attainment of Low Interfacial Trap +Density and Thermal Stability at High Temperatures. Appl. Phys. Lett. 2021, 118, 252104. + + +39 + + + + + +For Table of Contents Only + + + + +V.=0V +1V +1.5V +中 +H40 + +Supporting Information +Electrically Sign-Reversible Topological Hall Effect +in a Top-Gated Topological Insulator (Bi,Sb)2Te3 on a +Ferrimagnetic Insulator Europium Iron Garnet +Jyun-Fong Wong,1○ Ko-Hsuan Mandy Chen,1○ Jui-Min Chia,1 Zih-Ping Huang,2 Sheng-Xin +Wang,1 Pei-Tze Chen,1 Lawrence Boyu Young,2 Yen-Hsun Glen Lin,2 Shang-Fan Lee,3 Chung-Yu +Mou,1,4 Minghwei Hong,2* and Jueinai Kwo1* +1Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan +2Graduate Institute of Applied Physics and Department of Physics, National Taiwan University, +Taipei 10617, Taiwan +3Institute of Physics, Academia Sinica, Taipei 11529, Taiwan +4Center for Quantum Technology, National Tsing Hua University, Hsinchu 30013, Taiwan +○J.-F. W. and K.-H. M. C. contributed equally to this work. +*Address +correspondence +to +J. +Kwo, +raynien@phys.nthu.edu.tw; +M. +Hong, +mhong@phys.ntu.edu.tw + + +41 + +Hall measurements using the minor loop method +There has been debate if the chiral spin textures, such as skyrmion, can be identified via the +existence of THE.1 Alternatively, the so-called THE responses may arise from the overlapping of +two distinct AHE contributions, such as SrRuO3 (SRO) systems and TI systems.1-6 Note that in +these previously reported TI systems, the sign of AHE did not remain the same with varying gate +voltage and temperature;4-6 nevertheless, the AHE showed no sign change with gate voltage and +temperature in our top-gated BST on EuIG heterostructure. Thus, we have conducted the Hall +measurements using the minor loop method to clarify the origins of the THE responses.6 For the +artificial THE-like responses, which result from two overlapping AHE loops, the Hall traces with +different sweeping magnetic fields will not follow the trajectory of the full loop.6 For the genuine +THE signals, they will coincide within the full loop, which is the case of the results shown in +Figure S1. + +Figure S1. Minor loops at 5 K vs magnetic field with systematically increased sweeping range. +-2 +0 +2 +4 +6 +-1100 +-1050 +-1000 +-950 +-900 + 6 ~ -1.2 kOe + 6 ~ -1.4 kOe + 6 ~ -1.6 kOe + 6 ~ -1.8 kOe + 6 ~ -2.0 kOe +R () +H (kOe) + 6 ~ 0 kOe + 6 ~ -0.2 kOe + 6 ~ -0.4 kOe + 6 ~ -0.6 kOe + 6 ~ -0.8 kOe + 6 ~ -1.0 kOe +T = 5 K + +42 + +Discussion on the THE due to two distinct AHE contributions + +Given the debates discussed in Figure S1 in the Supporting Information, we further +demonstrate our experimental results and fitting curves in Figure S2a,b, where the fitting curves +are +composed +of +two +AHE +loops, +𝑅𝐴𝐻𝐸 = 𝑅𝐴𝐻𝐸1−𝑚𝑎𝑥 𝑡𝑎𝑛ℎ ( +𝐻 ± 𝐻𝑐1 +𝐻01 ) + +𝑅𝐴𝐻𝐸2−𝑚𝑎𝑥 𝑡𝑎𝑛ℎ( +𝐻 ± 𝐻𝑐2 +𝐻02 ). By applying a more positive gate voltage, the hysteretic contribution +with a positive sign (AHE2) disappeared at Vg ~VCNP, and then the sign of AHE2 changed to +negative at Vg > VCNP. As summarized in Figure S2e, AHE1 did not change the sign with the gate +voltage; however, AHE2 changed the sign with the gate voltage. In addition, we further analyzed +the temperature-dependent results. As shown in Figure S2f, the AHE2 gradually diminished with +increasing temperatures and disappeared at 75 K; in contrast, the AHE1 still existed up to 300 K +(not shown). + +The coercive field of the AHE1 remained nearly the same with varying gate voltage, which +was expected and was also demonstrated in our previous work.7 The coercive field only depends +on the thermal activation of the domain walls in EuIG, thus expected to be independent of the gate +bias, namely the EF of BST. However, the coercive field of AHE2 varied with gate voltages and +did not decrease with increasing temperatures. (see Figure S2g,h) Since the gate- and temperature- +dependent coercive field of AHE2 could not be explained physically, the excessive Hall signals on +the AHE loops in this work were less likely to result from the two distinct AHE contributions. + +43 + + +Figure S2. (a,b) The scattered points are the Hall data after subtracting the linear OHE background; +the solid lines are the fitted curves. The fitting curves in (a) and (b) are composed of (c) two AHE +loops with the opposite signs and (d) two AHE loops with the same sign, respectively. The gate + +44 + +and temperature dependences of the RAHE in AHE1 and AHE2 are shown in (e) and (f), respectively. +The gate and temperature dependences of the Hc in AHE1 and AHE2 are shown in (g) and (h), +respectively. + +Temperature-dependent THE with Vg of 0 V + +Figure S3. Temperature dependence of THE from 2 K to 85 K with Vg of 0 V. + + +-6 -3 +0 +3 +6 +-16 +-8 +0 +8 +16 +-6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 +-6 -3 +0 +3 +6 +-16 +-8 +0 +8 +16 +-6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 +-6 -3 +0 +3 +6 +-16 +-8 +0 +8 +16 +-6 -3 +0 +3 +6 +RTHE () +H (kOe) +T = 2 K +Vg = 0 V ++H +-H +T = 4 K +Vg = 0 V +H (kOe) +T = 6 K +Vg = 0 V +H (kOe) +T = 8 K +Vg = 0 V +H (kOe) +T = 15 K +Vg = 0 V +H (kOe) +T = 25 K +Vg = 0 V +RTHE () +H (kOe) +T = 35 K +Vg = 0 V +H (kOe) +T = 45 K +Vg = 0 V +H (kOe) +T = 55 K +Vg = 0 V +H (kOe) +T = 65 K +Vg = 0 V +H (kOe) +T = 75 K +Vg = 0 V +RTHE () +H (kOe) ++H +-H +-H ++H ++H +-H +-H ++H ++H +-H ++H +-H +-H ++H ++H +-H +-H ++H ++H ++H +-H +-H +T = 85 K +Vg = 0 V +H (kOe) + +45 + +Gate-dependent THE at 2 K + +Figure S4. Gate dependence of the THE at 2 K with systematically varying Vg. + + + + + +-6 -3 +0 +3 +6 +-16 +-8 +0 +8 +16 +-6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 +-6 -3 +0 +3 +6 +-16 +-8 +0 +8 +16 +-6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 +-6 -3 +0 +3 +6 +-16 +-8 +0 +8 +16 +-6 -3 +0 +3 +6 -6 -3 +0 +3 +6 -6 -3 +0 +3 +6 +Vg = -1.5 V +T = 2 K +RTHE () +H (kOe) + -1 V +T = 2 K +H (kOe) ++H +-H ++H +-H ++H +-H ++H +-H ++H +-H + -0.6 V +T = 2 K +H (kOe) ++H +-H +-H ++H +-H ++H +-H ++H ++H ++H +-H +-H ++H +-H ++H +-H +-H ++H +T = 2 K + -0.3 V +H (kOe) +T = 2 K + 0 V +H (kOe) +T = 2 K + 0.2 V +RTHE () +H (kOe) +T = 2 K + 0.5 V +H (kOe) +T = 2 K + 0.8 V +H (kOe) +T = 2 K + 1 V +H (kOe) +T = 2 K + 1.1 V +H (kOe) +T = 2 K + 1.2 V +RTHE () +H (kOe) +T = 2 K + 1.3 V +H (kOe) +T = 2 K + 1.4 V +H (kOe) +T = 2 K + 1.5 V +H (kOe) + +46 + +Estimate of EF manipulated by Vg + +By considering the conduction mainly from the TSS in the bulk-insulating BST, we calculated +the +modulation +of +the +sheet +carrier +density +(n2D) +per +volt +to +be +4.30 × 1012 cm-2 (n2D at Vg = -1.5 V) +|-1.5 V - 0.6 V (VCNP)| + ≈ 2.05 × 1012 cm-2V-1, where n2D is derived from the OHE background +using 𝑛2𝐷 = +1 +𝑅𝐻𝑒. Since the magnetic gap is small, we here assume the band dispersion of the +gapped Dirac cone is very similar to that of the gapless Dirac cone with nearly the same slope. In +a 2D system, the Fermi wavevector (kF) can be derived from the equation 𝑘𝐹 = √4𝜋𝑛2𝐷 . By +knowing the kF, the EF can be further extracted from the linear dispersion of the Dirac cone with +the equation 𝐸𝐹 = ℏ𝑣𝐷𝑘𝐹, where the Dirac velocity vD ~3.76 × 105 m/s is referred to the data of +(Bi0.25Sb0.75)2Te3 in the literature.8 Hence, we roughly estimated that the EF tuning via the gate bias +in Figure 4 was from 128 meV (Vg = -1.5 V) below the gapped Dirac point to 84 meV (Vg = +1.5 +V) above the gapped Dirac point. The tunability of EF might be overestimated because the n2D at +Vg = -1.5 V is at the margin of the ambipolar transport region. + + + + + + +47 + +Gate-dependent results of one additional field-effect device + +Figure S5. Additional data from another top-gated 4 nm (Bi,Sb)2Te3/20 nm EuIG at 2.5 K. Gate +dependences of (a) RH (left) and Rxx (right), and (b) RTHEMAX. + +REFERENCES +(1) Kimbell, G.; Kim, C.; Wu, W.; Cuoco, M.; Robinson, J. W. A. Challenges in Identifying +Chiral Spin Textures via the Topological Hall Effect. Commun. Mater. 2022, 3, 19. +(2) Gerber, A. Interpretation of Experimental Evidence of the Topological Hall Effect. Phys. Rev. +B 2018, 98, 214440. +(3) Wang, L.; Feng, Q.; Lee, H. G.; Ko, E. K.; Lu, Q.; Noh, T. W. Controllable Thickness +Inhomogeneity and Berry Curvature Engineering of Anomalous Hall Effect in SrRuO3 Ultrathin +Films. Nano Lett. 2020, 20, 2468. + +16 +15 +14 +13 +1248 + +(4) Fijalkowski, K. M.; Hartl, M.; Winnerlein, M.; Mandal, P.; Schreyeck, S.; Brunner, K.; Gould, +C.; Molenkamp, L. W. Coexistence of Surface and Bulk Ferromagnetism Mimics Skyrmion Hall +Effect in a Topological Insulator. Phys. Rev. X 2020, 10, 011012. +(5) Wang, F.; Wang, X.; Zhao, Y.-F.; Xiao, D.; Zhou, L.-J.; Liu, W.; Zhang, Z.; Zhao, W.; Chan, +M. H. W.; Samarth, N.; Liu, C.; Zhang, H.; Chang, C.-Z. Interface-Induced Sign Reversal of the +Anomalous Hall Effect in Magnetic Topological Insulator Heterostructures. Nat. Commun. 2021, +12, 79. +(6) Tai, L.; Dai, B.; Li, J.; Huang, H.; Chong, S. K.; Wong, K. L.; Zhang, H.; Zhang, P.; Deng, +P.; Eckberg, C.; Qiu, G.; He, H.; Wu, D.; Xu, S.; Davydov, A.; Wu, R.; Wang, K. L. Distinguishing +the Two-Component Anomalous Hall Effect from the Topological Hall Effect. ACS Nano 2022, +16, 17336. +(7) Zou, W.-J.; Guo, M.-X.; Wong, J.-F.; Huang, Z.-P.; Chia, J.-M.; Chen, W.-N.; Wang, S.-X.; +Lin, K.-Y.; Young, L. B.; Lin, Y.-H. G.; Yahyavi, M.; Wu, C.-T.; Jeng, H.-T.; Lee, S.-F.; Chang, T.- +R.; Hong, M.; Kwo, J. Enormous Berry-Curvature-Based Anomalous Hall Effect in Topological +Insulator (Bi,Sb)2Te3 on Ferrimagnetic Europium Iron Garnet beyond 400 K. ACS Nano 2022, 16, +2369. + +49 + +(8) Zhang, J.; Chang, C.-Z.; Zhang, Z.; Wen, J.; Feng, X.; Li, K.; Liu, M.; He, K.; Wang, L.; +Chen, X.; Xue, Q.-K.; Ma, X.; Wang, Y. Band Structure Engineering in (Bi1−xSbx)2Te3 Ternary +Topological Insulators. Nat. Commun. 2011, 2, 574. + + diff --git a/n9AyT4oBgHgl3EQfY_cz/content/tmp_files/load_file.txt b/n9AyT4oBgHgl3EQfY_cz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..335b42ae0ecbbf026482a27a7c969d91ceff04bb --- /dev/null +++ b/n9AyT4oBgHgl3EQfY_cz/content/tmp_files/load_file.txt @@ -0,0 +1,2247 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf,len=2246 +page_content='1 Electrically Sign-Reversible Topological Hall Effect in a Top-Gated Topological Insulator (Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='Sb)2Te3 on a Ferrimagnetic Insulator Europium Iron Garnet Jyun-Fong Wong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1○ Ko-Hsuan Mandy Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1○ Jui-Min Chia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 Zih-Ping Huang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 Sheng-Xin Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 Pei-Tze Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 Lawrence Boyu Young,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 Yen-Hsun Glen Lin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 Shang-Fan Lee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 Chung-Yu Mou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='4 Minghwei Hong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2* and Jueinai Kwo1* 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' National Tsing Hua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hsinchu 30013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan 2Graduate Institute of Applied Physics and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' National Taiwan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taipei 10617,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan 3Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Academia Sinica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taipei 11529,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan 4Center for Quantum Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' National Tsing Hua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hsinchu 30013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan ○J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' *Address correspondence to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, raynien@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, mhong@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='tw 2 ABSTRACT Topological Hall effect (THE), an electrical transport signature of systems with chiral spin textures like skyrmions, has been observed recently in topological insulator (TI)-based magnetic heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The strong spin-orbit coupling and the broken spatial inversion symmetry in such heterostructures could lead to a sizable interfacial Dzyaloshinsky–Moriya interaction, favorable for skyrmion formation and pronounced THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' However, the intriguing interplay between the topological surface state (TSS) and THE is yet to be fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In this work, we report an unprecedentedly large THE signal (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='0 µΩ‧cm at 2 K) with an electrically reversible sign in a top- gated 4 nm TI (Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='7)2Te3 (BST) grown on a ferrimagnetic insulator (FI) europium iron garnet (EuIG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Dependences of THE on temperature, external magnetic field angle, and gate bias were investigated and are consistent with the prediction of a skyrmion-driven THE, amenable to elucidate the origin of THE that occurred in TI-based heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moreover, a sign change in THE was discovered as the Fermi level was tuned electrically from the upper (electron-doped region) to the lower parts (hole-doped region) of the gapped BST Dirac cone and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' We show that the exploitation of the TSS features has led to a sign-reversal of THE repeatedly in a TI/FI top-gate stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ultimately, this discovery is anticipated to impact technological applications in ultralow power skyrmion-based spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 3 KEYWORDS topological insulator, ferrimagnetic insulator, topological Hall effect, skyrmion, electric field effect INTRODUCTION Three-dimensional topological insulators (TIs), featured with their spin-momentum-locked topological surface states (TSSs),1,2 have generated enormous interest in spintronics over the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The interplay between the spin-momentum-locked TSS and magnetism brings in novel electrical transport phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 A well-known example is the achievement of quantum anomalous Hall effect (QAHE) in a transition metal-doped TI (Bi,Sb)2Te3 (BST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='4-6 Besides the magnetically doped TIs, much attention is also given to TI heterostructures interfaced with magnetic materials (MMs) to attain a long-range magnetic ordering via magnetic proximity effect (MPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='7-14 However, most studies were focused on the discussion of the phenomena arising from the non-trivial Berry curvature in reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The investigation of the spin textures in real space and their related electrical transport in TI heterostructures remains largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Topological Hall effect (THE), a Hall response that emerges from the deflection of charge carriers flowing through non-trivial chiral spin textures, is a transport signature commonly used to 4 identify these chiral spin textures, such as magnetic skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='15,16 Initially, magnetic skyrmions were reported in bulk magnetic crystals lacking spatial inversion symmetry (SIS), such as B20- type chiral magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='17-20 Recently, skyrmions have been observed in SIS-broken MM-based heterostructures possessing strong spin-orbit coupling (SOC) materials such as heavy metals (HMs) or TIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='21,22 The Dzyaloshinskii–Moriya interaction (DMI) at the interface plays a crucial role in stabilizing skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='23,24 In contrast to the high current density (1011–1012 A/m2) required for magnetic domain wall motions, an ultralow current density (105–106 A/m2) has been accomplished for skyrmion motions, thus potential for high-density information storage devices in memory and computing technology with ultralow power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='16,20,23,25 The writing, deleting, and processing of skyrmions are extensively investigated as well in the rapidly growing field of skyrmion-electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='24,26 THE, being an electrical transport phenomenon, is a promising method to read skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Among many THE studies, there have been debates about the exact origin of the reported Hall responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='27-33 For example, because THE is commonly intertwined with anomalous Hall effect (AHE), an alternative cause is the superposition of multi-AHE contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='27-29,31-33 In this scenario, one can conduct two- or three-AHE fits to decompose the signals, and ascertain if the THE responses could indeed result from the overlapping of multi-AHE components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='14,30,34,35 Although several reports demonstrated pronounced THE in TI-based heterostructures,30,34,36-39 it is 5 noteworthy that an in-depth discussion on the THE emerging from the carrier transport of spin- momentum-locked TSS is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A straightforward way to investigate the relationship between THE and charge carriers in TSS is by implementing an electrical gate bias on TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' So far, the gate-tunable THE has been reported in Mn-doped Bi2Te3 films and Cr-doped BST heterostructures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='37-39 however, the gate devices were achieved by using SrTiO3 (STO) as a dielectric with a large gate bias of tens of volts,37-39 which are unfavorable in practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Finding a suitable gate dielectric with excellent carrier tunability and reliability is thus an important issue and needs to be addressed for both fundamental scientific studies and technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In this work, we report the observation of THE in TI BST/ferrimagnetic insulator (FI) europium iron garnet (EuIG) bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In particular, we demonstrate a successful manipulation of THE by switching the charge carrier type using a top electrical gate within a few volts, which has yet to be reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' By adopting a heterostructure of gate oxide/TI/FI, the current flow path and the resulting Hall contribution can be limited to the TI layer, which is rather simple compared to other magnetic TI (MTI)/TI/MTI or TI/MM heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Judging from the temperature, magnetic field angle, and gate voltage dependences of Hall measurements, our findings were consistent with the picture of a skyrmion-driven THE instead of the superposition of AHE loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The largest THE magnitude reached ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='02 µΩ‧cm under an applied Vg of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 V at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moreover, the observation of THE at zero fields suggested a stable skyrmion phase without an applied external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 6 Most importantly, we showed a pronounced and repeated THE sign reversal when the Fermi level (EF) is tuned across the gapped Dirac point of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hence, the exploration of THE in TI-based heterostructures opens a new route in high-density and ultralow-power skyrmion-based devices in spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' RESULTS AND DISCUSSION Characterization of BST/EuIG heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The material growth of BST/EuIG basically followed our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='14,40 Here, we chose to study the 4 nm BST thin films with a simultaneously magnetized top and bottom TSS, of which the THE-like feature has been repeatedly observed in many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='14 In order to manipulate the EF close to the magnetic gap, the Bi:Sb composition ratio was tuned to be 3:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The streaky reflection high energy electron diffraction (RHEED) pattern in Figure 1a manifested an atomically ordered and morphologically smooth BST surface with excellent crystallinity grown on EuIG thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ex-situ atomic force microscopy (AFM) was performed after the deposition of 2 nm Y2O3 and 15 nm Al2O3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' the flat BST surface covered by the oxide layers with a small root-mean-square roughness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='633 nm is presented in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Figure 1c illustrates the device structure and the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The electrical transport 7 study of BST/EuIG was commenced by measuring longitudinal resistance (Rxx) as a function of temperature, shown in Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rxx increased with decreasing temperatures from 300 K to 100 K, revealing a semiconducting behavior caused by the reduction of charge carriers in BST bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='41,42 Then, Rxx reached a local maximum at ~100 K and decreased as the temperature was further lowered from ~100 K to ~10 K, indicating a metallic behavior from the TSS of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='41,42 At temperatures below 10 K, Rxx increased with decreasing temperatures again, which could be attributed to the electron-electron interaction (EEI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='43 The longitudinal conductance (Gxx) was proportional to the natural logarithm of T, as shown in the inset of Figure 1d, consistent with the 2D EEI of the TSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='44,45 As reported in the literature, the existence of the TSS could help generate a strong interfacial DMI, giving rise to topological magnetic structures with a pronounced THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='30,34 Identifying the topological Hall effect with electrical transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Before reaching a definitive conclusion of having observed THE, we performed additional experimental checks to rule out other possible causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Figure 2a displays the Hall resistance of the ungated 4 nm thick BST on EuIG at 2 K after the subtraction of the linear background from the ordinary Hall effect (OHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Besides the square AHE loop (black solid line in Figure 2a) originating from the MPE,14 excessive Hall signals were clearly observed over a wide range of magnetic fields from - 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 to 3 kOe as the magnetic field was swept from negative to positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' These excessive Hall signals reached the largest value near the coercive field (Hc) and behaved like THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Notice that it is possible to have artificial THE-like responses from two overlapping AHE contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='27-33 Here, we adopted two separate methods, the minor loop approach and AHE curve fittings, to resolve this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' By utilizing the former method, the Hall traces coincided under successively increasing sweeping magnetic fields as the expected behavior of THE responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='32 (see Figure S1 in the Supporting Information) In addition, we analyzed the data by considering the superposition of two AHE components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Although the Hall signals could be well fitted with two AHE contributions mathematically, it is rather difficult to reconcile their dependences on temperatures as well as gate biases with any reasonable or existing physical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (see Figure S2 in the Supporting Information) Therefore, we inferred that these excessive Hall signals in BST/EuIG were most likely to result from a genuine THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' After excluding the Hall signals from the superposition of AHE loops, we proceeded to extract the THE resistance (RTHE) by subtracting the AHE resistance (RAHE), as shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Here, the maximum of the RTHE is denoted by RTHEMAX, and the H field corresponding to the RTHEMAX is denoted by HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A giant RTHEMAX ~7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='64 Ω was observed with an applied gate voltage (Vg) of 0 V at 2 K, where the corresponding resistivity (ρTHEMAX) was ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='06 µΩ‧cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' With the applied Vg of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 V, the ρTHEMAX reached the largest value of ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='02 µΩ‧cm at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' To the best of our knowledge, the 9 highest record of ρTHEMAX reported in TI-based bi-layer systems was ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='39 µΩ‧cm at 10 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='34 the ρTHEMAX in this work was ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='44 µΩ‧cm with Vg of 0 V at 8 K, which was nearly twice as large as the previous record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The larger ρTHEMAX may be attributed to a higher density of chiral spin textures because of the excellent interface between BST and EuIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Temperature dependence of the topological Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The results of the temperature dependence of THE in BST/EuIG are presented in Figure 2c,d with detailed THE loops shown in Figure S3 in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The THE gradually diminished with increasing temperatures and disappeared at 75 K, which could be attributed to the thermal fluctuation or the reduced DMI strength, consistent with the previous studies in TI-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='14,30,34,36-38 Furthermore, shown in the color map of Figure 2c is the temperature dependence of HT that follows closely with that of Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' This observation is in accord with several reports on the THE driven by the magnetization reversal process in systems hosting magnetic skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='34,46 It is noteworthy that our THE could be found at zero fields, suggesting a robust skyrmion phase without the support of an external magnetic field, similar to the observation in FeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='47,48 The RTHE at zero fields (RTHE0) showed an akin temperature dependence to that of RTHEMAX and also vanished at 75 K, as demonstrated in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 10 Angular dependence of the topological Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' To deepen our understanding of the THE in BST/EuIG, we further investigated the angular dependence of the THE with the measurement geometry illustrated in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The angle θ is defined as the angle between the external magnetic field H and the surface normal direction +z in the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The Hall traces at 5 K after subtracting the linear OHE background from 10° to 80° are demonstrated in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The AHE loops expanded with the increase in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The Hc enlarged and was proportional to 1/cosθ, indicating that a larger external magnetic field was needed to flip the magnetization in BST to the opposite direction as the θ increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Furthermore, significant THE responses coexisting with AHE were observed and sustained to 70°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The RTHE as a function of θ and H field is summarized in the color map of Figure 3c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' the H field was swept from negative to positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Similar to the temperature dependence of HT and Hc discussed in the previous section, the angular dependence of HT also followed closely with that of Hc before the disappearance of THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' We further extracted the RTHEMAX as a function of θ shown in Figure 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The RTHEMAX remained almost the same from 10° to 70°, indicating that THE strength was not affected by a moderate in- plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' This angular dependence is reasonable because the topologically protected skyrmions are robust against certain in-plane magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='30 While the θ further increased to 80°, the THE vanished, suggesting the collapse of the skyrmionic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='49-52 Similar angular behavior of THE was reported in the Mn1-xFexSi, a well-known B20-type chiral magnet hosting 11 magnetic skyrmions, with the disappearance of THE at 55°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='49 Hence, our observation of angle- dependent results in THE also supports the existence of skyrmions at the BST/EuIG interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Manipulation of the topological Hall effect with an electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Next, manipulation of THE was demonstrated by implementing a top gate electric field on our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Figure 4a shows the gate dependence of Rxx and ordinary Hall coefficient (RH) derived from the linear background of Hall traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The EF was successfully tuned across the gapped Dirac point of BST via the gate bias, as manifested by the rising behavior of Rxx in reaching a maximum and the sign change in RH near the charge neutrality point (CNP) (the gray-shaded region in Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The CNP here is expected to locate at the center of the proximity-induced magnetic gap of TSS in BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moreover, the small applied Vg for the CNP (VCNP) of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 V indicated that the EF of our un-gated BST was fairly close to the center of the magnetic gap, providing an excellent starting point to alter the carrier type and the carrier density of BST, thus to explore the systematic dependence of THE under an external electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Figure 4b displays the gate dependence of selected AHE loops together with the THE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' the scattered points are the measured RAHE + RTHE data, and the solid lines are the fitted AHE contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In the p-type region (Vg < VCNP), a hump was found near the Hc of the measured data as the H field was swept from negative to positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (dark red arrows in Figure 4b) In the n-type 12 region (Vg > VCNP), a dip was observed instead near the Hc of the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (a green arrow in Figure 4b) With the applied Vg ~VCNP, the measured data became a square hysteresis loop, a typical feature of AHE without excessive Hall signals from THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' To extract the magnitude of THE, we subtracted the AHE contributions (black solid lines) and plotted RTHE in Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Positive THE was identified in the p-type region of the up-sweep (red) curves, while negative THE was observed in the n-type region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' More gate-dependent THE loops with finer voltage steps are presented in Figure S4 in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The top-gate bias dependence of RTHEMAX is further summarized in Figure 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Relationship between the topological Hall effect and the topological surface state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' To gain a better insight into the mechanism responsible for the sign reversal in THE, we examined RTHE and its relations to other physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' For systems hosting chiral spin textures of skyrmions, an effective electromagnetic field Beff = nsk Φ0 will be generated by these structures, where nsk is the skyrmion density and Φ0 is the magnetic flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='15 The Hall resistance resulting from THE is commonly approximated as: RTHE ≈ RH P nsk Φ0, (1) where P is the local spin polarization of the conduction carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='15,34 When the EF is tuned from the upper Dirac cone to the lower Dirac cone, the majority charge carriers will be altered from 13 electrons to holes, leading to a sign change in RH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Furthermore, the electron spin polarization is flipped to the opposite direction because of the spin-momentum locking of the gapped TSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (see the Fermi surface in Figure 5) Since the propagation direction of holes is opposite with respect to that of electrons, the spin orientation for electrons and holes will thus be the same, giving rise to the unchanged sign of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='53 Therefore, the sign of THE will be switched from negative to positive when the EF is varied from the upper to the lower parts of the Dirac cone, as illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The EF modulation by the gate bias in Figure 4 was roughly estimated to be from ~84 meV (Vg = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V) above the gapped Dirac point to ~128 meV (Vg = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V) below the gapped Dirac point, as detailed in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In the vicinity of CNP, namely RH → 0, the vanishing RTHE could be attributed to the nearly equal numbers of n- and p-type charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The absence of the THE signals with the EF near the CNP was also observed in the Cr-doped BST sandwich heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='38,39 However, in their reports, RTHE did not show a sign reversal with the applied gate voltages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' the majority carrier type remained the same in both Vg < VCNP and Vg > VCNP regions, as suggested from the slopes of their Hall traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In contrast, our work demonstrated an effective manipulation of charge carriers and THE via a top electrical gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moreover, our gate bias-dependent results lent strong support to the picture of a skyrmion-driven THE in the BST/EuIG heterostructure, which can be well described by Equation 1 for both positive and negative bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Similar behavior of manipulating THE via the 14 top-gate bias in another sample is also presented in Figure S5 in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Given these findings, inspecting the THE sign change when the EF is located above and below the CNP could be another viable way to differentiate the genuine and artificial THE in TI-based heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The electrically sign-reversible THE in TI/FI heterostructures could be a unique feature directly associated with the spin-momentum-locking of the gapped TSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' This phenomenon has not been observed and might well be absent in other non-TI skyrmion systems, such as bulk chiral magnets or HM/FI heterostructures like Pt/TmIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Repeated switching of the topological Hall effect with an electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' To further examine the reliability of the field-effect device and the reproducibility of the sign reversal in THE, we performed the THE measurements with a series of selected gate biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' As displayed in Figure 6a, the THE was reversibly switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The RTHEMAX and HT values remained nearly identical to the starting ones, as demonstrated in Figure 6b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Although the repeated switching of this “THE” device is currently limited by the robustness of our gate oxides after applying excessive gate bias, the achievement of THE switching within only a few volts demonstrates tremendous progress in manipulating this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The idea of an electrically tunable THE field-effect device may open up a new avenue in skyrmion-based spintronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 15 CONCLUSION In summary, we have demonstrated the electrically sign-reversible THE in a top-gated TI BST on a FI EuIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The magnetotransport behaviors on temperature, external magnetic field angle, and gate bias were consistent with a picture of skyrmion-driven THE, as opposed to the alternative mechanism of superimposed AHE loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moreover, the sign change in THE via an electrical gate could be a distinctive feature related to the gapped TSS in TI/FI compared to other non-TI skyrmion systems, such as bulk chiral magnets and HM/FI heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Our findings in this work have provided a thorough understanding of the THE in a TI-based heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' This bilayer structure of BST/EuIG offers an excellent platform for studying the interplay among magnetism, chiral spin textures, and topological band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Especially, the reproducible electrical manipulation of THE within a few volts may be promising for ultralow-power skyrmion- related applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Further investigations on direct imaging of skyrmion spin structures in real space will be essential, and experiments via scanning microscopy methods, such as spin-polarized scanning tunneling microscopy and scanning transmission x-ray microscopy, are now underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' EXPERIMENTAL METHODS Materials growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 20 nm thick FI EuIG(001) films with perpendicular magnetic anisotropy 16 (PMA) were grown on GGG(001) using an off-axis magnetron sputtering technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='40 4 nm thick TI BST films were then deposited on EuIG thin films at ~260 °C by using MBE under a base pressure below 4 × 10-10 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='14 The deposition of gate oxides was divided into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In the first step, the samples were transferred from the TI-MBE chamber to a multiple-chamber system containing oxide-MBE and ALD chambers under ultra-high vacuum (UHV) to avoid contaminations at the TI/oxide interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='54 a 2 nm thick e-beam evaporated Y2O3 (in the oxide MBE chamber) followed by a 15 nm thick in-situ ALD Al2O3 was deposited on the pristine BST surface, leading to an unpinned EF at the interface of gate oxide/BST, based on our extensive expertise of high-κ dielectrics on semiconductor surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='55-59 To better protect samples for transport measurements, we deposited a second Al2O3 layer with a thickness of 25 nm in another ex-situ ALD system after the fabrication of Hall bars to prevent current leakage from the edges of the Hall bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Electrical measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The samples were patterned into Hall bars (880 μm × 90 μm) using photolithography and reactive ion etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' All electrical transport measurements were conducted in the physical property measurement system (PPMS) connected with the following instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A Keithley 2400 was used to provide a stable voltage source as the gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A Keithley 6221 was used as a current source to generate the alternating current with a root-mean- square amplitude of 100 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Two SR830 lock-in amplifiers were used as voltmeters to measure 17 the Hall voltages and longitudinal voltages, where the reference signal was provided by Keithley 6221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The Hall effect data were antisymmetrized as a function of the magnetic field to eliminate the Rxx component due to electrode misalignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 18 FIGURES Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sample characterization and schematic illustration of a top-gated BST/EuIG device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (a) RHEED pattern of the 4 nm BST(001) surface along the [100] axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (b) Surface morphology of 15 nm atomic layer deposition (ALD)-Al2O3/2 nm e-beam evaporated-Y2O3/4 nm molecular beam epitaxy (MBE)-BST/20 nm sputtering-EuIG/GGG(001) in a 1×1 μm2 area by using AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (c) Schematics of a cross-section view and a front view of our top-gated BST grown on EuIG, in which a light blue vertical plane is to dissect the sample structure for the cross-section view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Also shown are the Néel-type skyrmion spin texture at the BST/EuIG interface, and the optical 200hm Au~100nm Ti~30nm Al203~25nm Al03~15nm Y203~2nm (BiSb)2Te3~4nm EulG~20nm 200μm GGG (001)19 microscope image of a Hall bar device with our measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (d) Rxx as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The inset in (d) shows the Gxx as a function of temperature and the fit with a logarithmic function of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Temperature-dependent THE properties of BST/EuIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (a) Coexistence of AHE and THE with an applied Vg of 0 V at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The scattered points are the Hall data after subtracting a linear OHE background, and the solid line is the fitted AHE contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (b) The THE at Vg = 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (c) Temperature dependences of Hc and HT, and a color map of RTHE (color) as a function of temperature and magnetic field with an applied Vg of 0 V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' the H field was swept from negative to positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (d) The temperature dependences of RTHEMAX and RTHE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H V=0V20 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Angular dependence of THE from 10° to 80° at 5 K of BST/EuIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (a) A schematic of the angle-dependent Hall measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (b) Angular dependence of the AHE loops plus the THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (c) Magnetic field angle dependences of Hc and HT, and a color map of RTHE (color) as a function of magnetic field angle and H field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' the H field was swept from negative to positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (d) The angular dependence of RTHEMAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H BST EulG T= 5K21 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gate-bias dependence of THE from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V at 2 K of BST/EuIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (a) Top-gate voltage dependences of RH (left) and Rxx (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Small RH and large Rxx occurred in the vicinity of CNP (the gray-shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (b) Concurrence of AHE and THE with selected gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The dark red and green arrows in (b) indicate the hump and dip features near the Hc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (c) THE with selected gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (d) Top-gate voltage dependence of RTHEMAX at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' THE disappeared in the vicinity of CNP (the gray-shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='0 8H 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='0m 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 CNPI22 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Illustration of the sign reversal in THE with varying gate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Schematics for gapped TSS, the Fermi surface with the application of current in +x direction, and the corresponding THE with (a) Vg > VCNP (n-type region, EF located at upper gapped TSS) and (b) Vg < VCNP (p-type region, EF located at lower gapped TSS), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nogative THE Pccitive THE23 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Repeated switching of THE in BST/EuIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (a) THE loops with applied Vg in the sequence of 0 V, 1 V, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V, 1 V, and 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The corresponding modulation of (b) RTHEMAX and (c) HT as a function of Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ASSOCIATED CONTENT Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Discussion on the THE, temperature- and gate-dependent THE loops, estimate of EF manipulated by gate bias, and gate-dependent results of one additional field-effect device (PDF) Vg=0V 1 v 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V 1 v ov +H ++H +H +H +H H H 0 V 1 V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V 0 V 1 V →1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V1 VO V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V+1 V 0 V24 AUTHOR INFORMATION Corresponding Authors * Jueinai Kwo – Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='org/0000-0002-5088-6677;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' E-mail: raynien@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='tw * Minghwei Hong – Graduate Institute of Applied Physics and Department of Physics, National Taiwan University, Taipei 10617, Taiwan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='org/0000-0003-4657-0933;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' E-mail: mhong@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='tw Authors Jyun-Fong Wong – Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} 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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' fabricated the Hall bar device, and collected and analyzed the transport data.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 26 composed and wrote the manuscript with the comments of all the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Notes The authors declare no competing financial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors would like to thank Professor Hsiu-Hau Lin, Professor Tay-Rong Chang, Keng-Yung Lin, Wei-Nien Chen, Wei-Jhih Zou, and Hsuan-Ning Chen for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' This work was financially supported by the National Science and Technology Council (NSTC), Taiwan (NSTC 111-2112-M-007-043- and NSTC 111-2622-8-002-001-), and the Center for Quantum Technology and Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan (NSTC 111- 2634-F-007-006-).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The authors acknowledge resources and support from the Quantum Materials Shared Facilities of Institute of Physics, Academia Sinica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The authors also thank the technical support from Taiwan Semiconductor Research Institute (TSRI), Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' REFERENCES (1) Qi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hughes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Topological Field Theory of Time-Reversal Invariant 27 Insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2008, 78, 195424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (2) Hasan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kane, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Colloquium: Topological Insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2010, 82, 3045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (3) Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yasuda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tsukazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Magnetic Topological Insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2019, 1, 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (4) Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Feng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Shen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ji, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ji, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Dai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Science 2013, 340, 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (5) Checkelsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yoshimi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tsukazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Takahashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kozuka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Falson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Trajectory of the Anomalous Hall Effect towards the Quantized State in a Ferromagnetic Topological Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2014, 10, 731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (6) Kou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Guo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Shao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Murata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Scale-Invariant Quantum Anomalous Hall Effect in Magnetic Topological Insulators beyond the Two-Dimensional Limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2014, 113, 137201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 28 (7) Wei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Katmis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Assaf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Steinberg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jarillo-Herrero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Heiman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moodera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Exchange-Coupling-Induced Symmetry Breaking in Topological Insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2013, 110, 186807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (8) Lang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Montazeri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Onbasli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Upadhyaya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jiang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Schwartz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ross, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Proximity Induced High-Temperature Magnetic Order in Topological Insulator - Ferrimagnetic Insulator Heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2014, 14, 3459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (9) Katmis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lauter, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nogueira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Assaf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jamer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Satpati, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Freeland, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Eremin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Heiman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jarillo-Herrero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moodera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A High-Temperature Ferromagnetic Topological Insulating Phase by Proximity Coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nature 2016, 533, 513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (10) Fanchiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tseng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Strongly Exchange-Coupled and Surface-State- Modulated Magnetization Dynamics in Bi2Se3/Yttrium Iron Garnet Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2018, 9, 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (11) Watanabe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yoshimi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kawamura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tsukazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nakajima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Takahashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Quantum Anomalous Hall Effect Driven by 29 Magnetic Proximity Coupling in All-Telluride Based Heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2019, 115, 102403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (12) Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fanchiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tseng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Evidence for Exchange Dirac Gap in Magnetotransport of Topological Insulator–Magnetic Insulator Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2019, 100, 045138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (13) Bhattacharyya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Akhgar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gebert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Karel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Edmonds, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fuhrer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Recent Progress in Proximity Coupling of Magnetism to Topological Insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2021, 33, 2007795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (14) Zou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Young, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yahyavi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jeng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Enormous Berry-Curvature-Based Anomalous Hall Effect in Topological Insulator (Bi,Sb)2Te3 on Ferrimagnetic Europium Iron Garnet beyond 400 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ACS Nano 2022, 16, 2369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (15) Neubauer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pfleiderer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Binz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rosch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ritz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Niklowitz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': 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+page_content=' (16) Nagaosa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Topological Properties and Dynamics of Magnetic Skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 30 Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2013, 8, 899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (17) Mühlbauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Binz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jonietz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pfleiderer, C.' metadata={'source': 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+page_content=' Kanazawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Onose, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kimoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ishiwata, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Matsui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Near Room-Temperature Formation of a Skyrmion Crystal in Thin-Films of the Helimagnet FeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2011, 10, 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (19) Tonomura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yanagisawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Matsuda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Onose, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kanazawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Real-Space Observation of Skyrmion Lattice in Helimagnet MnSi Thin Samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2012, 12, 1673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (20) Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kanazawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nagai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kimoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Matsui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Onose, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Skyrmion Flow near Room Temperature in an Ultralow Current Density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2012, 3, 988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (21) Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Groß, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Dai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lujan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Razavi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sobotkiewich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Förster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Weigand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Schütz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gräfe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ferrimagnetic Skyrmions in Topological Insulator/Ferrimagnet Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2020, 32, 2003380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (22) Vélez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ruiz-Gómez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Schaab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gradauskaite, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wörnle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Welter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jacot, 31 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Degen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Trassin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fiebig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gambardella, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Current-Driven Dynamics and Ratchet Effect of Skyrmion Bubbles in a Ferrimagnetic Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2022, 17, 834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (23) Fert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cros, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sampaio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Skyrmions on the Track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2013, 8, 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (24) Fert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Reyren, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cros, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Magnetic Skyrmions: Advances in Physics and Potential Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2017, 2, 17031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (25) Schulz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ritz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Bauer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Halder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wagner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Franz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pfleiderer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Everschor, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Garst, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rosch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Emergent Electrodynamics of Skyrmions in a Chiral Magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2012, 8, 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (26) Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Song, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Park, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ezawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Woo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Skyrmion-Electronics: Writing, Deleting, Reading and Processing Magnetic Skyrmions toward Spintronic Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Matter 2020, 32, 143001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (27) Gerber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Interpretation of Experimental Evidence of the Topological Hall Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2018, 98, 214440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (28) Fijalkowski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hartl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Winnerlein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mandal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Schreyeck, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Brunner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gould, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Molenkamp, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Coexistence of Surface and Bulk Ferromagnetism Mimics Skyrmion Hall Effect in a Topological Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' X 2020, 10, 011012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 32 (29) Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Feng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Noh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Controllable Thickness Inhomogeneity and Berry Curvature Engineering of Anomalous Hall Effect in SrRuO3 Ultrathin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2020, 20, 2468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (30) Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kally, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pillsbury, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Heinonen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rimal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Bi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' DeMann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Field, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hoffmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Samarth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Topological Hall Effect in a Topological Insulator Interfaced with a Magnetic Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2021, 21, 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (31) Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Samarth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Interface-Induced Sign Reversal of the Anomalous Hall Effect in Magnetic Topological Insulator Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2021, 12, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (32) Tai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Dai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Deng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Eckberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Qiu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Davydov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Distinguishing the Two-Component Anomalous Hall Effect from the Topological Hall Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ACS Nano 2022, 16, 17336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (33) Kimbell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cuoco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Challenges in Identifying 33 Chiral Spin Textures via the Topological Hall Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2022, 3, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (34) Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ambhire, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Niu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cook, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Alahmed, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Bian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Giant Topological Hall Effect in van der Waals Heterostructures of CrTe2/Bi2Te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ACS Nano 2021, 15, 15710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (35) Jeon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Na, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Noh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jerng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Emergent Topological Hall Effect from Exchange Coupling in Ferromagnetic Cr2Te3/Noncoplanar Antiferromagnetic Cr2Se3 Bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ACS Nano 2022, 16, 8974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (36) Yasuda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wakatsuki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Morimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yoshimi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tsukazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Takahashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ezawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nagaosa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Geometric Hall Effects in Topological Insulator Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2016, 12, 555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (37) Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ruan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xue, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Dimensional Crossover-Induced Topological Hall Effect in a Magnetic Topological Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2017, 119, 176809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (38) Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Shin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Andreoli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xiao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kayyalha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Samarth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Concurrence of Quantum Anomalous Hall and Topological Hall Effects in Magnetic Topological 34 Insulator Sandwich Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2020, 19, 732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (39) Xiao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Shin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Richardella, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kayyalha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Samarth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mapping the Phase Diagram of the Quantum Anomalous Hall and Topological Hall Effects in a Dual-Gated Magnetic Topological Insulator Heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2021, 3, L032004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (40) Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hsu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tjeng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Single- Crystal Epitaxial Europium Iron Garnet Films with Strain-Induced Perpendicular Magnetic Anisotropy: Structural, Strain, Magnetic, and Spin Transport Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2022, 6, 054412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (41) Dutta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pariari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mandal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Prominent Metallic Surface Conduction and the Singular Magnetic Response of Topological Dirac Fermion in Three-Dimensional Topological Insulator Bi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5Te1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='7Se1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2017, 7, 4883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (42) Qin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Confined-Path Interference Suppressed Quantum Correction on Weak Antilocalization Effect in a BiSbTeSe2 Topological 35 Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2018, 112, 032102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (43) Lee, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ramakrishnan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Disordered Electronic Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 1985, 57, 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (44) Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' DaSilva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Samarth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xue, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Evidence for Electron-Electron Interaction in Topological Insulator Thin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2011, 83, 245438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (45) Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kally, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pillsbury, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Csaba, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sinclair, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Bi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' DeMann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rimal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Field, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Heinonen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Novosad, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hoffmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Magnetization Switching Using Topological Surface States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2019, 5, eaaw3415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (46) Matsuno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ogawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yasuda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kagawa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Koshibae, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nagaosa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Interface-Driven Topological Hall Effect in SrRuO3-SrIrO3 Bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2016, 2, e1600304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (47) Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chien, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Extended Skyrmion Phase in Epitaxial FeGe(111) Thin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2012, 108, 267201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (48) Gallagher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Meng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Brangham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Esser, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' McComb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 36 Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Robust Zero-Field Skyrmion Formation in FeGe Epitaxial Thin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2017, 118, 027201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (49) Yokouchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kanazawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tsukazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kozuka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ichikawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kagawa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tokura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Stability of Two-Dimensional Skyrmions in Thin Films of Mn1-xFexSi Investigated by the Topological Hall Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2014, 89, 064416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (50) Ohuchi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kozuka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Uchida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ueno, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tsukazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kawasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Topological Hall Effect in Thin Films of the Heisenberg Ferromagnet EuO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2015, 91, 245115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (51) Shao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Che, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Tserkovnyak, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Topological Hall Effect at above Room Temperature in Heterostructures Composed of a Magnetic Insulator and a Heavy Metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2019, 2, 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (52) Sohn, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Choi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Choi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Choi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Choi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Bombardi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Porter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Stable Humplike Hall Effect and Noncoplanar Spin Textures in SrRuO3 Ultrathin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2021, 3, 023232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (53) Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Richardella, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hickey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mkhoyan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Samarth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mapping the Chemical Potential Dependence of Current-Induced Spin Polarization in a Topological Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2015, 92, 155312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 37 (54) Lin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fanchiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Young, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Pi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Molecular Beam Epitaxy, Atomic Layer Deposition, and Multiple Functions Connected via Ultra-High Vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Growth 2019, 512, 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (55) Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Passlack, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mannaerts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Moriya, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Fratello, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Low Interface State Density Oxide‐GaAs Structures Fabricated by in situ Molecular Beam Epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 1996, 14, 2297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (56) Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kortan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mannaerts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sergent, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Epitaxial Cubic Gadolinium Oxide as a Dielectric for Gallium Arsenide Passivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Science 1999, 283, 1897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (57) Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kortan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Queeney, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chabal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mannaerts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Boone, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Krajewski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sergent, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rosamilia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' High ε Gate Dielectrics Gd2O3 and Y2O3 for Silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2000, 77, 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (58) Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mannaerts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kortan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cho, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Anselm, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chyi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Single Crystal GaN/Gd2O3/GaN Heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2002, 20, 1274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (59) Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Young, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, 38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In situ Y2O3 on p-In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='53Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='47As—Attainment of Low Interfacial Trap Density and Thermal Stability at High Temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2021, 118, 252104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 39 For Table of Contents Only V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='=0V 1V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5V 中 H40 Supporting Information Electrically Sign-Reversible Topological Hall Effect in a Top-Gated Topological Insulator (Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='Sb)2Te3 on a Ferrimagnetic Insulator Europium Iron Garnet Jyun-Fong Wong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1○ Ko-Hsuan Mandy Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1○ Jui-Min Chia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 Zih-Ping Huang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 Sheng-Xin Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 Pei-Tze Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 Lawrence Boyu Young,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 Yen-Hsun Glen Lin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 Shang-Fan Lee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 Chung-Yu Mou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='4 Minghwei Hong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2* and Jueinai Kwo1* 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' National Tsing Hua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hsinchu 30013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan 2Graduate Institute of Applied Physics and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' National Taiwan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taipei 10617,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan 3Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Academia Sinica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taipei 11529,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan 4Center for Quantum Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' National Tsing Hua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hsinchu 30013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Taiwan ○J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' *Address correspondence to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, raynien@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, mhong@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='tw 41 Hall measurements using the minor loop method There has been debate if the chiral spin textures, such as skyrmion, can be identified via the existence of THE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 Alternatively, the so-called THE responses may arise from the overlapping of two distinct AHE contributions, such as SrRuO3 (SRO) systems and TI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1-6 Note that in these previously reported TI systems, the sign of AHE did not remain the same with varying gate voltage and temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='4-6 nevertheless, the AHE showed no sign change with gate voltage and temperature in our top-gated BST on EuIG heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Thus, we have conducted the Hall measurements using the minor loop method to clarify the origins of the THE responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 For the artificial THE-like responses, which result from two overlapping AHE loops, the Hall traces with different sweeping magnetic fields will not follow the trajectory of the full loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 For the genuine THE signals, they will coincide within the full loop, which is the case of the results shown in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Minor loops at 5 K vs magnetic field with systematically increased sweeping range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' -2 0 2 4 6 -1100 -1050 -1000 -950 -900 6 ~ -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 kOe 6 ~ -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='4 kOe 6 ~ -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 kOe 6 ~ -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='8 kOe 6 ~ -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='0 kOe R (\uf057) H (kOe) 6 ~ 0 kOe 6 ~ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 kOe 6 ~ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='4 kOe 6 ~ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 kOe 6 ~ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='8 kOe 6 ~ -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='0 kOe T = 5 K 42 Discussion on the THE due to two distinct AHE contributions Given the debates discussed in Figure S1 in the Supporting Information, we further demonstrate our experimental results and fitting curves in Figure S2a,b, where the fitting curves are composed of two AHE loops, 𝑅𝐴𝐻𝐸 = 𝑅𝐴𝐻𝐸1−𝑚𝑎𝑥 𝑡𝑎𝑛ℎ ( 𝐻 ± 𝐻𝑐1 𝐻01 ) + 𝑅𝐴𝐻𝐸2−𝑚𝑎𝑥 𝑡𝑎𝑛ℎ( 𝐻 ± 𝐻𝑐2 𝐻02 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' By applying a more positive gate voltage, the hysteretic contribution with a positive sign (AHE2) disappeared at Vg ~VCNP, and then the sign of AHE2 changed to negative at Vg > VCNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' As summarized in Figure S2e, AHE1 did not change the sign with the gate voltage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' however, AHE2 changed the sign with the gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In addition, we further analyzed the temperature-dependent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' As shown in Figure S2f, the AHE2 gradually diminished with increasing temperatures and disappeared at 75 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' in contrast, the AHE1 still existed up to 300 K (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The coercive field of the AHE1 remained nearly the same with varying gate voltage, which was expected and was also demonstrated in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='7 The coercive field only depends on the thermal activation of the domain walls in EuIG, thus expected to be independent of the gate bias, namely the EF of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' However, the coercive field of AHE2 varied with gate voltages and did not decrease with increasing temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (see Figure S2g,h) Since the gate- and temperature- dependent coercive field of AHE2 could not be explained physically, the excessive Hall signals on the AHE loops in this work were less likely to result from the two distinct AHE contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 43 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (a,b) The scattered points are the Hall data after subtracting the linear OHE background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' the solid lines are the fitted curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The fitting curves in (a) and (b) are composed of (c) two AHE loops with the opposite signs and (d) two AHE loops with the same sign, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The gate 44 and temperature dependences of the RAHE in AHE1 and AHE2 are shown in (e) and (f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The gate and temperature dependences of the Hc in AHE1 and AHE2 are shown in (g) and (h), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Temperature-dependent THE with Vg of 0 V Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Temperature dependence of THE from 2 K to 85 K with Vg of 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 0 3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='8 0 8 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 0 3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 0 3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 0 3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 0 3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='8 0 8 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 0 3 6 RTHE (\uf057) H (kOe) T = 2 K Vg = 0 V +H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H T = 4 K Vg = 0 V H (kOe) T = 6 K Vg = 0 V H (kOe) T = 8 K Vg = 0 V H (kOe) T = 15 K Vg = 0 V H (kOe) T = 25 K Vg = 0 V RTHE (\uf057) H (kOe) T = 35 K Vg = 0 V H (kOe) T = 45 K Vg = 0 V H (kOe) T = 55 K Vg = 0 V H (kOe) T = 65 K Vg = 0 V H (kOe) T = 75 K Vg = 0 V RTHE (\uf057) H (kOe) +H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H +H +H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H +H +H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H +H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H +H +H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H +H +H +H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='H T = 85 K Vg = 0 V H (kOe) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='Gate-dependent THE at 2 K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gate dependence of the THE at 2 K with systematically varying Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 6 3 0 3 6 16 8 0 8 16 6 3 0 3 6 6 3 0 3 6 6 3 0 3 6 6 3 0 3 6 6 3 0 3 6 16 8 0 8 16 6 3 0 3 6 6 3 0 3 6 6 3 0 3 6 6 3 0 3 6 6 3 0 3 6 16 8 0 8 16 6 3 0 3 6 6 3 0 3 6 6 3 0 3 6 Vg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V T = 2 K RTHE (\uf057) H (kOe) 1 V T = 2 K H (kOe) +H H +H H +H H +H H +H H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 V T = 2 K H (kOe) +H H H +H H +H H +H +H +H H H +H H +H H H +H T = 2 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 V H (kOe) T = 2 K 0 V H (kOe) T = 2 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 V RTHE (\uf057) H (kOe) T = 2 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V H (kOe) T = 2 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='8 V H (kOe) T = 2 K 1 V H (kOe) T = 2 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='1 V H (kOe) T = 2 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='2 V RTHE (\uf057) H (kOe) T = 2 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='3 V H (kOe) T = 2 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='4 V H (kOe) T = 2 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V H (kOe) 46 Estimate of EF manipulated by Vg By considering the conduction mainly from the TSS in the bulk-insulating BST, we calculated the modulation of the sheet carrier density (n2D) per volt to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='30 × 1012 cm-2 (n2D at Vg = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V) |-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='6 V (VCNP)| ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='05 × 1012 cm-2V-1, where n2D is derived from the OHE background using 𝑛2𝐷 = 1 𝑅𝐻𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Since the magnetic gap is small, we here assume the band dispersion of the gapped Dirac cone is very similar to that of the gapless Dirac cone with nearly the same slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' In a 2D system, the Fermi wavevector (kF) can be derived from the equation 𝑘𝐹 = √4𝜋𝑛2𝐷 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' By knowing the kF, the EF can be further extracted from the linear dispersion of the Dirac cone with the equation 𝐸𝐹 = ℏ𝑣𝐷𝑘𝐹, where the Dirac velocity vD ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='76 × 105 m/s is referred to the data of (Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='25Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='75)2Te3 in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='8 Hence, we roughly estimated that the EF tuning via the gate bias in Figure 4 was from 128 meV (Vg = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V) below the gapped Dirac point to 84 meV (Vg = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V) above the gapped Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' The tunability of EF might be overestimated because the n2D at Vg = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 V is at the margin of the ambipolar transport region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 47 Gate-dependent results of one additional field-effect device Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Additional data from another top-gated 4 nm (Bi,Sb)2Te3/20 nm EuIG at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gate dependences of (a) RH (left) and Rxx (right), and (b) RTHEMAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' REFERENCES (1) Kimbell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Cuoco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Challenges in Identifying Chiral Spin Textures via the Topological Hall Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2022, 3, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (2) Gerber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Interpretation of Experimental Evidence of the Topological Hall Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B 2018, 98, 214440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (3) Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Feng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Noh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Controllable Thickness Inhomogeneity and Berry Curvature Engineering of Anomalous Hall Effect in SrRuO3 Ultrathin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2020, 20, 2468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 16 15 14 13 1248 (4) Fijalkowski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hartl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Winnerlein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Mandal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Schreyeck, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Brunner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Gould, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Molenkamp, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Coexistence of Surface and Bulk Ferromagnetism Mimics Skyrmion Hall Effect in a Topological Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' X 2020, 10, 011012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (5) Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Samarth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Interface-Induced Sign Reversal of the Anomalous Hall Effect in Magnetic Topological Insulator Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2021, 12, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (6) Tai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Dai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Deng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Eckberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Qiu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Davydov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Distinguishing the Two-Component Anomalous Hall Effect from the Topological Hall Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ACS Nano 2022, 16, 17336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' (7) Zou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Young, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Yahyavi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Jeng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Kwo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Enormous Berry-Curvature-Based Anomalous Hall Effect in Topological Insulator (Bi,Sb)2Te3 on Ferrimagnetic Europium Iron Garnet beyond 400 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ACS Nano 2022, 16, 2369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 49 (8) Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Feng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Xue, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Band Structure Engineering in (Bi1−xSbx)2Te3 Ternary Topological Insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} +page_content=' 2011, 2, 574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AyT4oBgHgl3EQfY_cz/content/2301.00213v1.pdf'} diff --git a/ndE2T4oBgHgl3EQfewdN/vector_store/index.faiss b/ndE2T4oBgHgl3EQfewdN/vector_store/index.faiss new file mode 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B. W. McQuinn,1 Yao-Yuan Mao,1, 2 Matthew R. Buckley,1 David Shih,1 Roger E. Cohen,1 and +Andrew E. Dolphin3, 4 +1Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Rd, Piscataway, NJ 08854, USA +2Department of Physics and Astronomy, University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA +3Raytheon Technologies, 1151 E. Hermans Road, Tucson, AZ 85756, USA +4University of Arizona, Steward Observatory, 933 North Cherry Avenue, Tucson, AZ 85721, USA +ABSTRACT +We report the discovery of an ultrafaint dwarf (UFD) galaxy, Pegasus W, located on the far side +of the Milky Way-M31 system and outside the virial radius of M31. The distance to the galaxy is +915+60 +−91 kpc, measured using the luminosity of horizontal branch (HB) stars identified in Hubble Space +Telescope optical imaging. The galaxy has a half-light radius (rh) of 100+11 +−13 pc, MV = −7.20+0.17 +−0.16 mag, +and a present-day stellar mass of 6.5+1.1 +−1.4×104 M⊙. We identify sources in the color-magnitude diagram +(CMD) that may be younger than ∼ 500 Myr suggesting late-time star formation in the UFD galaxy, +although further study is needed to confirm these are bona fide young stars in the galaxy. Based on +fitting the CMD with stellar evolution libraries, Pegasus W shows an extended star formation history +(SFH). Using the τ90 metric (defined as the timescale by which the galaxy formed 90% of its stellar +mass), the galaxy was quenched only 7.4+2.2 +−2.6 Gyr ago, which is similar to the quenching timescale of +a number of UFD satellites of M31 but significantly more recent than the UFD satellites of the Milky +Way. Such late-time quenching is inconsistent with the more rapid timescale expected by reionization +and suggests that, while not currently a satellite of M31, Pegasus W was nonetheless slowly quenched +by environmental processes. +Keywords: stars: color-magnitude diagrams − galaxies: Local Group − galaxies: dwarf +1. INTRODUCTION +Very low-mass (M∗ ≲ 105 M⊙) galaxies are expected +to be numerous in the present-day universe, based on a +Λ Cold Dark Matter cosmology (e.g., Kauffmann et al. +1993; Moore et al. 1999; Bullock & Boylan-Kolchin 2017, +and references therein). +Such low-mass systems have +correspondingly low-luminosities (MV ≲ −7.7 mag) and +are referred to as ultrafaint dwarfs (UFDs; see, e.g., Si- +mon 2019, for a recent definition). The shallow potential +wells of UFD galaxies make them extremely sensitive to +both external perturbations (such as heating from meta- +galactic UV background radiation and local environmen- +tal conditions; e.g., Bullock et al. 2000; Benson et al. +2002; Somerville 2002; Geha et al. 2012; Brown et al. +2014; Wetzel et al. 2015; Rey et al. 2020; Akins et al. +2021; Pan et al. 2022) and internal perturbations (such +kristen.mcquinn@rutgers.edu +as stellar feedback; e.g., McQuinn et al. 2015a,b; Ap- +plebaum et al. 2021). Thus, these small galaxies make +excellent laboratories in which to test physical models +and galaxy formation theories, and constrain the im- +pact of reionization on the growth of low-mass halos in +the early universe. They are also critical components +for tests of the hierarchical structure formation theories +and dark matter (e.g., Nadler et al. 2021). +UFD galaxies went mostly undetected due to their low +surface brightness, faint luminosities, and small physical +sizes until ∼ 20 years ago. With the advent of wide-field, +deeper optical surveys, such as the Sloan Digital Sky +Survey (SDSS; York et al. 2000), the Panoramic Survey +Telescope & Rapid Response System (Pan-STARRS; +Chambers et al. 2016), the Pan-Andromeda Archeolog- +ical Survey (PAndAS; McConnachie et al. 2009), the +Dark Energy Survey (DES; Drlica-Wagner et al. 2020), +the DECam Local Volume Exploration (DELVE; Cerny +et al. 2022a,b), and the DESI Legacy Imaging Surveys +(Dey et al. 2019), searches for very low-mass galaxies +arXiv:2301.04157v1 [astro-ph.GA] 10 Jan 2023 + +2 +McQuinn et al. +have been incredibly fruitful, evidenced by the growing +number of UFD galaxies now known (e.g., Simon 2019, +and references therein). The majority of the galaxies dis- +covered lie in close proximity to the Milky Way (MW) +and the Large Magellanic Cloud (LMC), as intrinsically +faint systems are more readily detected at closer dis- +tances. A smaller number have been identified as part +of the M31 satellite system, found via the targeted PAn- +dAS observations (McConnachie et al. 2009). +Here, we report the discovery of an UFD galaxy in +the Local Group (LG) named Pegasus W. The galaxy +was identified in the DESI Legacy Imaging Surveys data +as an over-density in the photometric stellar catalog, +similar to previous discoveries (e.g., Bechtol et al. 2015; +Drlica-Wagner et al. 2015; Koposov et al. 2015; Collins +et al. 2022), but the system is much farther than other +known UFD galaxies in the LG. Hubble Space Telescope +(HST) optical imaging of Pegasus W enables a robust +distance measurement which places the galaxy on the +far side of M31 but outside M31’s virial radius. Thus, +Pegasus W offers a unique opportunity to measure the +properties and study the evolution of an UFD galaxy +that is not a satellite galaxy but that is still in close +enough proximity for detailed observations. +The paper is organized as follows. Section 2 describes +the HST observations, photometry, and data process- +ing. Section 3 presents the structural parameters of Pe- +gasus W. Section 4 explores the CMD of the galaxy. +Section 5 includes the distance measurement to the +galaxy based on horizontal branch (HB) stars and in- +vestigates the LG environment around Pegasus W. Sec- +tion 6 presents the SFH based on fitting stellar evo- +lution libraries to the CMD, calculates star formation +timescales, and measures the integrated luminosity and +stellar mass of the galaxy. Section 7 summaries the over- +all properties of Pegasus W in the context of other Local +Group UFDs and discusses the implications of an UFD +galaxy that was not quenched by reionization. +2. OBSERVATIONS AND DATA PROCESSING +2.1. Observations +HST observations of Pegasus W were obtained using +the Wide Field Camera (WFC) of the Advanced Camera +for Surveys (ACS) instrument (Ford et al. 1998) on 2022 +June 27 in the F606W and F814W filters as part of HST- +GO-16916. All the HST data used in this paper can be +found in MAST: 10.17909/x8qj-bn51. Two observations +were made per filter during one orbit, with a 5×5 pixel +dither (acs wfc dither line pattern #14) between +exposures to help reject cosmic rays and to mitigate de- +tector cosmetic defects. The ACS pointing was centered +on J2000 RA = 23 : 53 : 14.229, Dec = +22 : 05 : 35.54, +Table 1. Pegasus W Properties +Property +Value +RA (J2000) +358.31248167◦±1′′ +Dec (J2000) +22.10197022◦±1′′ +Position angle θ (◦ E of N) +92±3 +ellipticity (ϵ = 1 − b +a) +0.17+0.07 +−0.08 +rh (′′) +23±2 +rh (pc) +100+11 +−13 +MV (mag) +−7.20+0.17 +−0.16 +M∗ (M⊙) +6.5+1.1 +−1.5 × 104 +HB mV,0 (mag) +25.30+0.12 +−0.20 +µ (mag) +24.81+0.14 +−0.22 +Σb (arcmin−2) +16.8+4.1 +−15.4 +Distance (kpc) +915+60 +−91 +[M/H] (dex) +−1.9 ± 0.1 +τ90 (Gyr) +7.4+2.2 +−2.6 +AV (mag) +0.315 +AF 606W (mag) +0.284 +AF 814W (mag) +0.175 +Note—The properties of Peg W were measured in this work, +with the exception of the foreground extinction which is +from Schlegel et al. (1998) with recalibration from Schlafly +& Finkbeiner (2011). +which placed the galaxy slightly offset from the center +of the ACS field of view. +This selected pointing en- +sured maximum coverage of the stellar disk while avoid- +ing nearby bright stars. The exposure time was evenly +split between the two ACS/WFC filters with a final in- +tegration time of 1140 s per filter. +A second field was simultaneously imaged in parallel +using the Wide Field Camera 3 (WFC3) UVIS instru- +ment in the same bandpasses, enabling an investigation +of potential background and foreground contamination +in a field near to the galaxy. The WFC3 pointing was +centered on RA = 23 : 52 : 51.255, Dec = +22 : 07 : 48.48 +with integration times of 1020 and 1045 s in F606W and +F814W filters, respectively. +Figure 1 presents 3-color images of the HST ACS ob- +servations (left), a zoom-in on the region with Pega- +sus W (center), and the parallel field from the WFC3 +observations (right). The 3-color images were made us- +ing the F606W, F814W, and (F606W+F814W)/2 mo- +saics created from the charge transfer efficiency cor- +rected (flc.fits) files with the HST drizzlepac v3.0 +python package (Hack et al. 2013; Avila et al. 2015). +Despite being low-luminosity, the galaxy is visible in the +center image as an over-density of point sources. + +Pegasus W +3 +Figure 1. +Three-color images of the observations. +Left: The ACS full field of view with an ellipse encircling the stellar +component of Pegasus W out to 2 rh based on the best-fitting structural parameters. Middle: A zoom-in on the region of the +galaxy. Right: The WFC3 parallel field which can help quantify potential foreground and background contamination. The +three-color images were created using F606W for red, and average of F606W and F814W images for green, and F814W for blue. +2.2. Photometry +Photometry was performed on the flc.fits images +using the point spread function (PSF) fitting software +DOLPHOT (Dolphin 2000, 2016), with includes specific +ACS/WFC and WFC3/UVIS modules. +The DOLPHOT +photometric parameters were set according to the val- +ues recommended in Williams et al. (2014, 2021). +The photometric output was filtered for well-recovered +stars using a combination of quality metrics returned +from DOLPHOT that help to characterize each source. +Specifically, we selected sources with an output error +flag < 4, object type ≤ 2, signal-to-noise ratio ≥ 5 +in both filters, sharp2 +F 606W + sharp2 +F 814W < 0.075, and +crowdF 606W + crowdF 814W < 0.1. +The sharpness pa- +rameter is a measure of how peaked or broad a source is +relative to the PSF and helps to reject cosmic rays and +background galaxies, respectively. +We chose to apply +strict sharpness cuts to limit contamination from faint, +unresolved background galaxies in the stellar catalogs. +The crowding parameter measures how much brighter a +source would have been if stars nearby on the sky had +not been fit simultaneously. While an important quality +metric to consider, strict crowding cuts are not as criti- +cal in creating a high-fidelity catalog given the spareness +of the field. +Artificial stars tests were performed on both the ACS +and WFC3 data to measure the observational uncertain- +ties and completeness of the images using the same pho- +tometric software. Approximately 500k artificial stars +were injected into each individual image following to +the spatial distribution of all sources identified in the +photometry (i.e., the pre-filtered DOLPHOT output). The +sources were then recovered photometrically and the +same quality cuts used for the photometry were applied +to the output. The field was sufficiently uncrowded that +we detected no significant trends of incompleteness with +distance from the center of the galaxy. +3. STRUCTURAL PARAMETERS +We determined the structural parameters of Pega- +sus W, including orientation on the sky (position angle, +θ), shape (semi-major axis, a; ellipticity, ϵ = 1 − b +a), +and half-light radius (rh). These parameters help char- +acterize Pegasus W and provide a way to compare to +the properties of Pegasus W to other UFD galaxies (see +Section 7). +They are also used to apply spatial cuts +to the photometry to create our final stellar catalog for +Pegasus W. +The structural parameters were determined using an +unbinned maximum likelihood approach. +Specifically, +we perform a Markov Chain Monte Carlo (MCMC) fit +of an exponential density profile (allowing for non-zero +ellipticity) to the spatial distribution of observed sources +over the full ACS/WFC field of view. +To avoid any +impact of incompleteness while maximizing the contri- +bution of galaxy members, we consider only sources +with F606W < 26.6, F814W < 25.7 (1.5 mag bright- +ward of the 50% completeness limit in both filters, +ascertained from the artificial star tests) and a color +(F606W−F814W) < 1.5. We verified that the resulting +structural parameters presented below are quite robust +to these cuts, such that either eliminating the color cut +entirely and/or moving the magnitude cuts faintward +yielded results that were consistent to within their 1σ +uncertainties. +Operationally, we fit for six free parameters: +The +tangent plane coordinates x0 and y0, which are the +location of the center relative to an arbitrary posi- + +22°08' +ACS/WFC Primary Field +07' +06' +Dec +05' +04' +23h53m25s +20s +15s +10s +05s +RA22°07'00" +Pegasus W +06'30" +Dec +00" +05'30" +100 pc +23h53m18s +16s +14s +12s +RAWFC3/UViS Parallel Field +22°09' +Dec +08' +07' +23h52m56s +52s +48s +44s +RA4 +McQuinn et al. +tional zeropoint1, the half-light radius rh, the ellipticity +ϵ = (1 − b/a) where b/a is the ratio of minor to major +axis lengths, the position angle θ in degrees east of north, +and N⋆, which is the total number of stars in the galaxy +(within the aforementioned CMD limits). Note that N⋆ +is an extrapolation from the density profile rather than +the number of stars we observe within the ACS/WFC +field of view, which is Nobs = 314 after applying our +CMD cuts. +To determine the best-fit values of the structural pa- +rameters and their uncertainties, we search for the set +of parameters for which the data are most likely. +In +other words, for parameters (p1, p2, ..., p6) we maximize +the likelihood function: +L(p1, p2, ..., p6) = +� +i +ℓi(p1, p2, ..., p6), +(1) +where ℓi(p1, p2, ..., p6) is the probability of finding the ith +datapoint given parameters (p1, p2, ..., p6). This proba- +bility is calculated following Martin et al. (2008, 2016), +assuming the target galaxy has an exponential density +profile ρgal(r) as a function of elliptical radius r: +ρgal(r) = +1.682 +2πr2 +h(1 − ϵ)N⋆ exp(−1.68r/rh), +(2) +where the elliptical radius r is related to the tangent +plane coordinates x and y as +r = +�� +1 +1 − ϵ((x − x0) cos θ − (y − y0) sin θ) +�2 ++ +((x − x0) sin θ + (y − y0) cos θ)2 +�1/2 +. +(3) +We also allow for a spatially constant background den- +sity Σb. The background density in each MCMC itera- +tion is set by requiring that the number of background +sources is equal to the difference between the model- +predicted number of sources and the observed number +of sources in the full field of view, with area A: +Σb = +� +Nobs − +� +A +ρgal dA +� +/A. +(4) +In practice, the integration of Eq. 2 over the field of +view and the calculation of A are performed numerically +by generating a spatial grid of pixels with an area of 1 +arcsec2. +With Σb in hand for each iteration, the full +density profile for a given set of trial parameters is: +ρmodel(r) = ρgal(r) + Σb. +(5) +1 The positional zeropoint was set to a rough guess center of +(RA,Dec) = (358.312732◦,22.102465◦) based on the distribution +of sources recovered photometrically. +x0 ["] = +0.85+1.48 +1.41 +5.0 +2.5 +0.0 +2.5 +y0 ["] +y0 ["] = +1.81+1.14 +1.13 +16 +20 +24 +28 +32 +Rh ["] +Rh ["] = 22.58+2.04 +1.79 +0.1 +0.2 +0.3 +0.4 + = 0.17+0.07 +0.08 +80 +85 +90 +95 +100 + [ ] + [ ] = 91.67+2.66 +2.63 +6 +3 +0 +3 +6 +x0 ["] +240 +300 +360 +420 +N +5.0 +2.5 +0.0 +2.5 +y0 ["] +16 +20 +24 +28 +32 +Rh ["] +0.1 +0.2 +0.3 +0.4 +80 +85 +90 +95 +100 + [ ] +240 +300 +360 +420 +N +N = 277.34+31.47 +27.28 +Figure 2. Full posterior distributions of Pegasus W struc- +tural parameters over 10,000 post-burn-in MCMC iterations. +The black contour lines correspond to 1,2 and 3σ. +Best- +fitting values are listed in Table 1. +Combining Eqs. 1-5, the log likelihood is then: +ln L = +Nobs +� +i +ρmodel(ri) − Nobs. +(6) +We impose broad, flat, astrophysically motivated pri- +ors on the free parameters: +• |x0|, |y0| < 100′′, essentially requiring the center +of the galaxy to lie in our field of view. +• rh > 0. +• 0 < ϵ ≤ 1. +• 0 < θ ≤ π. +• N⋆ > 0. +• Σb ≥ 0. +We sample the posterior distributions of the param- +eters using the emcee package (Foreman-Mackey et al. +2013), running 50 affine-invariant walkers for 5000 burn- +in iterations followed by 10000 production iterations +(more than sufficient given autocorrelation lengths of +<100 steps for all parameters). The best-fit parameter +values we report are the medians over all post-burn-in + +Pegasus W +5 +0 +1 +2 +3 +r [arcmin] +0 +100 +200 +300 +400 +500 +600 +700 +800 +Density [arcmin +2] +10 +1 +100 +r [arcmin] +101 +102 +103 +Density [arcmin +2] +Figure 3. Binned radial density profile of Pegasus W, shown +on linear (left) and logarithmic (right) scales. Black points +are the binned observed data with errorbars indicating Pois- +sonian uncertainties. The red line is the best-fit exponential +profile from our maximum likelihood analysis, and the grey +lines represent 100 random draws from the posterior distri- +butions of the profile parameters. +iterations, with uncertainties corresponding to the 16th +and 84th percentiles. +Figure 2 shows the full posterior distributions of our +fits and their correlations. As an additional check, we +calculate a binned radial density profile in Figure 3, +where black points are the binned observed data and ver- +tical errorbars represent Poissonian uncertainties. Over- +plotted on the binned data is the maximum-likelihood +solution in red, with 100 random individual draws from +the posterior distributions shown in grey. +The final structural parameters including center RA, +Dec coordinates of Pegasus W, rh, ellipticity, and posi- +tion angle, as well as the value of Σb, are listed in Ta- +ble 1. We also overplot an ellipse based on these param- +eters and encompassing 2 rh on the 3-color ACS images +in Figure 1. +4. COLOR-MAGNITUDE DIAGRAM +4.1. Spatial Selection of Sources +Figure 4 presents the spatial distribution of all sources +that pass our photometric quality cuts in X-Y coordi- +nates. Two ellipses are overplotted based on our struc- +tural parameters with semi-major axes of 2 rh and 3 rh, +respectively. The overdensity of sources from Pegasus W +is readily apparent in the smaller ellipse. +Outside of +this region there are still a few hundred sources that +pass our quality cuts. While some of the sources may +be bona fide members of Pegasus W, the majority are +likely foreground stars or unresolved background galax- +ies. +Thus, to create a high-fidelity photometric catalog of +Pegasus W with minimal contamination, we used the +structural parameters and applied spatial cuts to the +data. Our final stellar catalog for Pegasus W includes +all sources contained within the ellipse extending to +2 rh, shown as cyan stars in Figure 4, and includes 377 +sources. To quantify the potential contamination to the +Pegasus W catalog, we use the sources outside the larger +ellipse that extends to 3 rh and create a field catalog. +These sources are shown as orange circles and include +a total of 465 sources from an area ∼ 7× larger than +that used for Pegasus W. This field catalog is used to +estimate contamination in the CMD in Section 4 and to +model the contamination in the SFH fitting in Section 6. +The sources in between 2 − 3rh are not used in either +catalog and are shown as black circles. +Note that the WFC3 UVIS parallel field was origi- +nally intended to quantify the potential contamination +of the final Pegasus W stellar catalog. We examined the +WFC3 data and found a somewhat smaller number of +sources (241) passed the photometric quality cuts from +the full UVIS field of view. We opt to use the field cata- +log from the ACS data as representative of the potential +contamination as the field is closer spatially to Pega- +sus W, avoids transforming from WFC3 UVIS to ACS +magnitudes, and also avoids any systematics caused by +slightly different spatial resolution and throughputs. +4.2. CMD of Final Stellar Catalog +Figure 5 shows the CMD from the final photometric +catalog for Pegasus W in the left panel, a 2D histogram +representation of the CMD (i.e., a Hess diagram) of the +field catalog scaled to the same area as Pegasus W in the +middle panel, and a residual Hess diagram of the field +CMD subtracted from the Pegasus W CMD. The stellar +density of the Pegasus W catalog is 0.072 sources per +arcsec2, compared with 0.013 in the field catalog. The +data are plotted to the 50% completeness limits and +representative photometric uncertainties per magnitude +bin are shown in the Pegasus W CMD; no corrections +for completeness have been applied to the stellar density +values. +The CMD for Pegasus W shows well-defined RGB, +a blue and red horizontal branch (HB), and red clump +(RC) features, which are largely absent in the field Hess +diagram. The differences between the CMDs of Pega- +sus W and the field region can be seen more quantita- +tively in the residual Hess diagram in the right panel of +Figure 5. The red color represents higher star counts +from Pegasus W over the field region and includes the +RGB and HB features in the CMD. The blue color in- +dicates that contamination dominates the source counts +and is seen predominantly at fainter magnitudes. +There are also two groups of sources in the CMD that +warrant additional attention. +There are four sources +blue-ward of the RGB and above the HB feature (i.e., + +6 +McQuinn et al. +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +X +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Y +2rh 3rh +Figure 4. Distribution of point sources passing the photo- +metric quality cuts in the X–Y coordinates of the ACS field +of view. Pegasus W is clearly identifiable as an over den- +sity of point sources. +The smaller ellipse is based on our +measured structural parameters of Pegasus W extends to +2 rh, and encircles the point sources (cyan stars) used in +our analysis of the galaxy. The larger ellipse reaches 3 rh; +sources outside this area (orange circles) quantify the po- +tential contamination of the final Pegasus W stellar catalog. +The sources located between 2 < rh < 3 (black points) are +not used in our analysis. +F606W−F814W < 0.8; F606W <24.8) which have op- +tical colors and magnitudes consistent with blue helium +burning (BHeB) stars with ages of < 500 Myr (McQuinn +et al. 2011). There is one source in a similar region of +CMD-space in the catalog from the off-galaxy field area +that is ∼ 7× larger. +There is also a grouping of thirty blue sources fainter +than the HB with colors ≲ 0.1 mag that could be the +bright end of what is referred to as the ‘blue plume’. +The blue plume is often detected in the CMDs of globu- +lar clusters and UFDs that reach the old main sequence +turn-off (oMSTO; e.g., Momany et al. 2007; Mapelli +et al. 2007, 2009; Sand et al. 2010; Monelli et al. 2012; +Okamoto et al. 2012; Santana et al. 2013). The nature of +these sources is unclear. They may be young (≲ 3 Gyr) +main sequence stars (possibly with unresolved binaries), +indicating that a galaxy has hosted late-time star forma- +tion. They could also be blue stragglers (BSs), which +are older, hot, blue stellar sources with optical colors +and magnitudes similar to younger main sequence stars +(Mateo et al. 1995). BSs are thought to be a product of +stellar binary systems, with a collisional or mass-transfer +origin. +In the case of Pegasus W, the population of faint, blue +stars lie at the bright end of what is typically consid- +ered the blue plume region identified in CMDs reaching +deeper photometric depths. The residual Hess diagram +in the final panel of Figure 5 reveals this is a mixed +population of high-confidence Pegasus W member stars +and contamination. Specifically, there are an average of +11 blue plume sources in a region of equal area to Pega- +sus W compared with 30 found within 2 rh of the galaxy. +The BHeB star candidates and the blue plume sources +are both consistent with ages of ∼500 Myr, based on +BaSTI isochrones (Hidalgo et al. 2018). Examining the +spatial distribution shows that 2 of the 4 BHeB candi- +dates and 10 of the 30 blue plume sources are concen- +trated in the inner rh, with 4 blue plume sources within +0.5 rh. BSs in dwarf galaxies are expected to have a +flat radial distribution whereas younger stars are more +spatially clumped (e.g., Momany et al. 2007; Monelli +et al. 2012), lending support to the idea that some of +the sources may be young stars. +The specific fraction of blue stragglers to RGB stars +has been shown to be approximately constant in UFD +galaxies (Santana et al. 2013). Thus, to further investi- +gate the nature of the blue plume sources, we follow the +procedure in Santana et al. (2013) and calculate the frac- +tion of blue plume to RGB stars after subtracting the +number of sources in the field region scaled by the areal +coverage of the galaxy vs. field region. Note, however, +that the specific fraction we calculate is a lower limit on +how many blue stragglers are expected. Our data are +in different filters and do not reach the same photomet- +ric depths as the data from Santana et al. (2013). As +a result, the magnitude ranges used to select the popu- +lation of stars are not identical and, given the relative +faintness of blue plume to RGB stars, this has a greater +impact on the number of blue plume stars counted. Our +calculation yields a lower limit of the specific fraction of +0.21 − below the average value of 0.29±0.1 from San- +tana et al. (2013) − making our results inconclusive on +the nature of the blue plume sources using this metric. +Note, an Anderson-Darling statistical test performed on +the cumulative radial distributions of the blue plume +and galaxy population samples was also inconclusive. +The galaxy was not detected in GALEX near or far +UV imaging, although this is to be expected given the +shorter star formation timescale that UV emission traces +and the depth of the GALEX data. Pegasus W was also +not detected in the Hi ALFALFA survey (Haynes et al. +2018), but it is possible that the galaxy still harbors +some gas below the detection limit of the survey. + +Pegasus W +7 +0 +1 +2 +F606W-F814W (mag) +22 +23 +24 +25 +26 +27 +28 +F606W (mag) +Pegasus W +Star Density = 0.071 arcsec +2 +No. of stars=377 +0 +1 +2 +F606W-F814W (mag) +Field Region in ACS +Star Density = 0.013 arcsec +2 +0 +1 +2 +F606W-F814W (mag) +(Peg W + Field) Residual +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +No. of Stars +Figure 5. Left: CMD of the 377 sources located inside an ellipse centered on Pegasus W and extending to 2 rh. The CMD +is plotted to the ∼ 50% completeness limit; representative uncertainties per magnitude are shown. Middle: Hess diagram of +sources in the ACS field of view outside 3 rh that passed our quality cuts, scaled to the area encompassing the Pegasus W +region. Right: Residual Hess diagram made from subtracting the field scaled Hess diagram from the Pegasus W data. Red +indicates a higher number of sources in Pegasus W while blue indicates a higher amount of contamination. +While it is intriguing that Pegasus W may have hosted +late-time star formation, this is still speculative as it is +not possible to discern with confidence the true nature +of the BHeB candidates or blue plume sources without +deeper imaging or spectroscopic data. We return to this +when fitting for the SFH in Section 6. +5. THE DISTANCE TO PEGASUS W +The distance to the galaxy was measured to be 915+60 +−91 +kpc based on the luminosity of the HB stars. We chose +to use the HB stars for a standard candle distance de- +termination, instead of the tip of the red giant branch +stars, as the upper RGB is sparsely populated and the +brightest identified stars in the RGB sequence may be a +false tip (cf. McQuinn et al. 2013, 2015a). +The HB distance methodology arises from the nearly +constant V -band luminosity of helium burning stars in +the post-RGB phase of evolution. The absolute mag- +nitude of the HB has been calibrated in the Johnson +BVRI filter system. Thus, we convert the photometry +from the HST ACS filters to the Johnson filter system +using the transformations from Sirianni et al. (2005, see +their Equation 12 with coefficients from Table 22), and +iteratively solve for V and I band magnitudes using the +F606W−F814W colors as a starting point: +V = F606W + c0 + c1 ∗ (V − I) + c2 ∗ (V − I)2 +I = F814W + c0 + c1 ∗ (V − I) + c2 ∗ (V − I)2 +(7) +The coefficients c0, c1, c2 are given in Table 2. Before +transforming to the V, I magntiudes, the photometry +was corrected for Galactic foreground extinction using +the dust maps from Schlegel et al. (1998) and the re- +calibration by Schlafly & Finkbeiner (2011); values are +listed in Table 1. +The HB luminosity was measured using a maximum +likelihood approach to fitting a parametric luminosity +function to the stars in the region of the HB with the +following form: +P = eA(V −VHB)+B + e−0.5[(V −VHB)/C]2, +(8) +where V is the magnitude in the F606W filter, and A, +B, and C are free parameters. The parameter C has a +prior of 1 +C . An advantage of using a maximum likelihood +approach is that it takes into account the distribution +of photometric uncertainties and completeness measured +by the artificial star tests. +Figure 6 presents the extinction corrected CMD for +Pegasus W. Stars used for the fit lie in the range of + +8 +McQuinn et al. +Table 2. Photometry Transformation Coeffcients +Coeff. +V +I +V − I < 0.4 +V − I ≥ 0.4 +V − I < 0.1 +V − I < 0.1 +c0 +26.394±0.05 +26.331±0.008 +25.479±0.13 +25.496±0.010 +c1 +0.153±0.018 +0.340±0.008 +0.041±0.211 +−0.014 ± 0.013 +c2 +0.096±0.085 +0.038 ± 0.002 +−0.093 ± 0.803 +0.015±0.003 +Note—Coefficients used in Eq. 7 for transforming the ACS photometry to Johnson V and I magnitudes as a function of V − I +color. +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +F606W0-F814W0 (mag) +21 +22 +23 +24 +25 +26 +27 +F606W0 (mag) +D=915+60 +91 kpc +Pegasus W +BaSTI 12 Gyr +[M/H]=-2.2 +[M/H]=-1.9 +[M/H]=-1.7 +[M/H]=-1.4 +Figure 6. Extinction corrected CMD of Pegasus W with +12 Gyr BaSTI isochrones and varying [M/H] values overlaid. +The red box outlines the area of the CMD used to fit for the +HB luminosity. +the HB identified by eye in the CMD, as shown by the +red box in Figure 6, which avoids including stars on the +blue edge of the red clump. The best-fit HB luminosity +is V0 = 25.30+0.12 +−0.20 mag. +The HB luminosity was converted to a distance mod- +ulus using the calibration from Carretta et al. (2000): +MV = (0.13 ± 0.09) × [Fe/H] + 1.5 + (0.54 ± 0.07) (9) +We estimate [Fe/H] by overlaying isochrones on the +CMD. Shown in Figure 6 are a set of 12 Gyr BaSTI +isochrones with [M/H] ranging from −2.2 to −1.4. +Based on the CMD, we find that the [M/H] = −1.9 +and −2.2 are the best overall matches to the RGB. The +higher value of [M/H] = −1.9 is in agreement with the +best-fitting average stellar metallicity found from the +CMD-fitting technique using the BaSTI stellar library +(see Section 6). Thus, we adopt [M/H] = −1.9 as rep- +resentative of [Fe/H] in the distance calculation with +an uncertainty of 0.1 dex. This metallicity estimate is +somewhat higher than typically reported for UFD galax- +ies, but not unexpected given the extended SFH pre- +sented in the next section. +The final distance modulus to Pegasus W is 24.81+0.14 +−0.22 +mag, corresponding to a distance of 915+60 +−91 kpc. The +uncertainties are based on adding in quadrature the un- +certainties in the HB luminosity fit, the estimated un- +certainty on the adopted [Fe/H] value, and the uncer- +tainties from the calibration in Eq. 9. +5.1. Location within the Local Group +A secure distance enables the location of Pegasus W +within the Local Group to be firmly established. Fig- +ure 7 shows the distribution of sources within a physi- +cal 3-dimensional distance of 500 kpc of Pegasus W in +Supergalactic (SG) euclidian coordinates (SGX, SGY, +SGZ), which provide a way to visualize the physical spa- +tial distribution of galaxies and their linear separation. +The SG coordinates were determined using the spher- +ical coordinates and distance to each system following +the formalism described in McQuinn et al. (2014). For +reference, the MW sits at the SG origin. +Note that +the axis ranges on each plot are fixed at 600 kpc, but +the plot includes only galaxies found to be within 500 +kpc of Pegasus W in the updated compilation from Mc- +Connachie (2012) with revised distances from Savino +et al. (2022) were available. We also include M31 with +an adopted distance of 776.2+22 +−21 kpc determined from +RR Lyrae stars (Savino et al. 2022). +From Figure 7, Pegasus W (red star) lies on the far +side of the MW−M31 system at a distance of 348 kpc +from M31, which places it outside an assumed 300 kpc +virial radius (green square). +The galaxy is relatively +isolated; the two nearest neighbors are the M31 satellite, +AndVII (separation=92 kpc; blue square) and the low- +mass galaxy Peg Irr (separation=150 kpc; blue circle). +The clustering of black points are comprised mainly of +satellites of M31. + +Pegasus W +9 +400 +600 +800 +SGX (Mpc) +700 +600 +500 +400 +300 +200 +SGY (Mpc) +Rvir +M31 +PegW +AndVII +PegIrr +MW +400 +600 +800 +SGX (Mpc) +100 +0 +100 +200 +300 +400 +SGZ (Mpc) +600 +400 +200 +SGY (Mpc) +100 +0 +100 +200 +300 +400 +SGZ (Mpc) +Figure 7. Known systems in SG coordinates within 500 kpc of Pegasus W based on the updated compilation of galaxies from +McConnachie (2012) and revised distances for a subset from Savino et al. (2022). The location of Pegasus W is shown as a red +star; the uncertainties in the position of the galaxy based on distance uncertainties are smaller than the plot symbol. M31 is +shown as a green square; the majority of black points are satellites of M31. The nearest known neighbors to Pegasus W are +And VII (blue square) and the Pegasus dwarf galaxy (blue triangle). The MW is located at the SG origin on the far side of +M31 from Pegasus W; the black arrow in the first panel indicates the direction of the MW in the SGX−SGY plane. +6. THE STAR FORMATION HISTORY OF +Pegasus W +6.1. SFH Methodology +The SFH of the galaxy was reconstructed from the +stellar populations in the CMD using the CMD-fitting +software MATCH (Dolphin 2002). The basic approach of +a CMD-fitting technique is to find the best-fitting com- +bination of synthetic simple stellar populations (SSPs) +of different ages and metallicities from stellar evolution +libraries to an observed CMD. The SSPs are constructed +using an assumed initial mass function (IMF) and binary +fraction and then linearly combined until the best-fit is +found based on a Poisson likelihood statistic. +For the SFH fits for Pegasus W, we assumed a Kroupa +IMF (Kroupa 2001), a binary fraction of 35% with flat +secondary mass distribution, and used as primary inputs +the photometry and observational uncertainties and in- +completeness measured from the artificial star recovery +fractions. The SFHs were reconstructed using three stel- +lar evolution libraries, namely BaSTI, PARSEC (Bres- +san et al. 2012), and MIST (Choi et al. 2016), which +help to bracket the range of possible solutions and pro- +vide an estimate of the systematic uncertainties. Given +the photometric depth of the data, the CMD does not +fully constrain the metallicity evolution of Pegasus W. +Thus, we imposed a physically motivated constraint that +the metallicity is a continuous, non-decreasing function +over the lifetime of the galaxy. The SFH solutions were +fit with an age grid of log(t/yr) = 6.6–10.15 using a time +resolution δt = 0.1 dex for ages less than log(t/yr) = 9.0 +and δt = 0.05 dex for older ages, and a metallicity grid +[M/H]= −2.0 to −1.0 with a resolution of 0.15 dex. We +initially tested searching a metallicity grid encompassing +lower metallicity values, but as the best-fitting [M/H] so- +lutions were consistently above −2.0, we tightened the +search grid to reduce computational time. We fixed the +distance in the fits to the HB distance of 0.915 Mpc and +adopt a Galactic extinction value of AV = 0.315 mag +based on the dust maps from Schlegel et al. (1998) and +recalibration from Schlafly & Finkbeiner (2011). +Figure 8 shows an example of the quality of the fit to +the data using the BaSTI stellar library. The most im- +portant diagnostic is the residual significance plot (bot- +tom right panel; units in standard deviations); a checker- +board pattern of blue and red indicates no major resid- +uals. The overall fit is quite good and the different evo- +lutionary sequences are well reproduced by the model. +Random uncertainties due to the finite number of stars +in a CMD were estimated using a Markov Chain Monte +Carlo approach (Dolphin 2013). In addition, while the +SFH derived using the different stellar libraries provides +a measure of the range of possible solutions, systematic +uncertainties were also estimated using 50 Monte Carlo +simulations (Dolphin 2012). +6.2. Best-Fitting SFH +The left panel of Figure 9 presents the best-fitting SFH +using the BaSTI models; the uncertainties include both +random and systematic uncertainties. The right panel +shows the best-fitting SFH solutions with BaSTI, MIST, +and PARSEC models with random uncertainties only. +The SFH recovered by the three stellar libraries are in +overall good agreement. The range in solutions from the + +10 +McQuinn et al. +different libraries lie within the systematic uncertainties +estimated for the BaSTI library shown in the left panel. +For the downstream analysis, we adopt the SFH solution +based on the BaSTI library which produced a slightly +better fit overall than the PARSEC and MIST libraries. +The best-fitting SFH for Pegasus W shows a rising star +formation until ∼ 7 Gyr ago, with an indication of late- +time star formation activity over the past few Gyr. As +discussed above, there are a few BHeB star candidates +and sources in blue plume region of the CMD that may +be bona fide young stars. If these are true BHeB and +young main sequence stars, they can provide valuable +and unique constraints on late-time star formation in an +UFD. On the other hand, the BHeB candidates could be +contaminants in the CMD and the blue plume sources +could be BSs masquerading as young stars. To try to +account for the possibility that BHeB are contaminants, +we use the field catalog as representative of potential +background sources while fitting for the SFH. However, +BSs are currently not included in the stellar evolution +libraries used in the CMD-fitting and, thus, cannot be +explicitly taken into account in the SFH recovery. In +this case, including them in the fit will falsely result in +star formation activity ≲ 3 Gyr ago. +To explore the impact of the candidate BHeB stars +and blue plume on our results, we fit for the SFH with +and without these sources. Excluding these sources has +little impact on the total stellar mass recovered from +the data, reducing the stellar mass formed in the galaxy +by ∼ 1–2% depending on the stellar library. Removing +these sources eliminates star-formation activity < 500 +Myr ago, but not the low level of star formation 1–3 +Gyr ago. We include these sources in our final fit, but +note there is a larger uncertainty in the very recent SFH +(t < 500 Myr) given the ambiguity of the nature of these +sources. +6.3. Star-Formation Timescales +Of particular interest is the timescale by which the +star formation ceased, or the quenching timescale. We +adopt the lookback time at which the galaxy has formed +90% of its stellar mass as the quenching timescale (τ90). +Using τ90 reduces the impact blue plume stars have on +quenching timescale as it is not as dependent on whether +the galaxy has experienced a small level of recent star +formation activity. +We find τ90 = 7.4+2.2 +−2.6Gyr based on interpolating the +SFH derived using the BaSTI stellar library. The un- +certainties include both statistical and systematic un- +certainties shown in the left panel of Figure 9 which +were similarly determined by interpolating the uncer- +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +F606W - F814W +20.0 +22.0 +24.0 +26.0 +28.0 +F606W +Observed +10 +1 +100 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +F606W - F814W +20.0 +22.0 +24.0 +26.0 +28.0 +F606W +Model +10 +1 +100 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +F606W - F814W +20.0 +22.0 +24.0 +26.0 +28.0 +F606W +Residual (Obs - Model) +6 +4 +2 +0 +2 +4 +6 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +F606W - F814W +20.0 +22.0 +24.0 +26.0 +28.0 +F606W +Significance (Obs - Model) +4 +2 +0 +2 +4 +Figure 8. Example of the quality of fit to the observed CMD +using the BaSTI stellar library. Top left: observed CMD; +top right: modelled Hess diagram used in the fit; bottom +left: simple residual Hess diagram (data − model); bottom +right: residual significance Hess diagram where the pixels are +weighted by the variance. The checkerboard pattern in the +residual significance indicates there are no major residuals +and the modelled Hess diagram is a good fit to the data. +tainty envelopes. The value of τ90 with uncertainties is +overplotted on the SFH in the left panel of Figure 9. +6.4. Total Luminosity of Pegasus W +We calculate the present day MV of Pegasus W follow- +ing the approach outlined in Martin et al. (2008). To ac- +count for uncertainties in both distance (resulting from +our fit to the HB magnitude) and the structural param- +eters, we perform a Monte Carlo procedure. Specifically, +in each Monte Carlo iteration, we generate a synthetic +stellar population using the best-fit star formation his- +tory, a distance drawn from the HB-based distance de- +termination, and a value of N⋆ from the posterior distri- +bution in the previous section. Since N⋆ represents the +total number of stars in the system observed within our +chosen CMD limits of F606W < 26.6, F814W < 25.7 +and (F606W−F814W) < 1.5, we continue to draw stars +from the entire synthetic population until the number of +stars that would be observed within these CMD limits +is equal to N⋆. In each iteration, the F606W magni- +tudes of all stars drawn from the synthetic CMD are +corrected for extinction and distance and converted to +V -band magnitudes using the bolometric corrections of + +Pegasus W +11 +5 +10 +Lookback Time (Gyr) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative SFH +90 +5 +10 +Lookback Time (Gyr) +BaSTI +MIST +PARSEC +0 +0.25 +0.5 +1 +2 +3 +4 +10 +Redshift (z) +0 +0.25 +0.5 +1 +2 +3 +4 +10 +Redshift (z) +Figure 9. Left: Best-fitting SFH solution with the BaSTI +models. Uncertainties include both statistical and system- +atic uncertainties. The quenching timescale, τ90 is marked as +noted. Right: Best-fitting SFH solutions using the BaSTI, +MIST, and PARSEC models with statistical uncertainties +only. +The gray shaded vertical region shows the approxi- +mate timescale of reionization. +Chen et al. (2019) assuming [M/H] = −1.9, and their +sum then yields a total MV in each iteration. The value +and uncertainties we report correspond the median and +16th−84th percentiles obtained over all iterations, yield- +ing MV = −7.20+0.17 +−0.16 mag. +We also attempted to directly estimate the total lumi- +nosity of Pegasus W using publicly available images from +the DESI Legacy Imaging Surveys (Dey et al. 2019). +We obtained an estimate that is consistent with Monte +Carlo method; however, the uncertainty on the estimate +is much larger (≳ 0.5 mag) due to the image’s limited +depth. +6.5. Present-Day Stellar Mass +We estimate the present-day stellar mass, M∗, in Pe- +gasus W using two approaches. First, we obtain an es- +timate by using the results of the Monte Carlo simula- +tions used to determine the total luminosity the galaxy, +described above. +We sum the total mass of all stars +drawn in each Monte Carlo iteration, which yields M∗ += 6.9+1.1 +−1.0 × 104 M⊙. +Second, M∗ is estimated directly from the SFH. The +total stellar mass formed over the lifetime of the galaxy +is determined while fitting for the SFH solution. This +is akin to integrating the SFH over cosmic times, but is +a separate quantity fit by assuming one time bin (i.e., +log(t/yr) = 6.6–10.15). The stellar mass determined in +this way is equal to the value determined from integrat- +ing SFR(t), but the measurement avoids any co-variance +between time bins in the fit and results in lower total un- +certainties. +To calculate the present-day stellar mass of Pegasus W +we make two adjustments. First, as described in Telford +et al. (2020), match determines the stellar mass by in- +tegrating over a stellar mass range of [0, ∞]. This lower +IMF normalization has the effect of systematically in- +creasing the estimated stellar mass compared with a +Kroupa IMF with typical mass limits of 0.1–100 M⊙. +Thus, we subtract 0.12 from log(M∗/M⊙) to account +for the change in normalization, which lowers M∗ by +24%. +Second, we apply a recycling fraction, R, that +estimates the fraction of material returned to the ISM +from stars. We adopt a value of R = 0.41 from Vincenzo +et al. (2016), based on a Kroupa IMF and the metallicity +of Pegasus W, and reduce the stellar mass by a factor +of 1 − R. The final present-day stellar mass of Pega- +sus W is found to be 6.1+0.9 +−1.5 × 104 M⊙ and is in good +agreement with the present-day stellar mass recovered +from the SFH fits using the PARSEC and MIST stel- +lar libraries. It is also in very good agreement with our +first estimate based on the Monte Carlo simulations fol- +lowing the procedure in Martin et al. (2008). We adopt +an average of the two values and the largest uncertain- +ties from both methods as our final value, yielding M∗ += 6.5+1.1 +−1.5 × 104 M⊙. +7. DISCUSSION AND SUMMARY +Pegasus W is a very low-mass (M∗= 6.5+1.1 +−1.4 × 104 +M⊙), very faint (MV = −7.20+0.17 +−0.16 mag) dwarf galaxy +located on the far side of M31 at a distance of 915+60 +−91 +kpc measured from HB stars. The galaxy lies outside +the virial radius of M31 and is relatively isolated. Shown +in Figure 10, the size and luminosity of Pegasus W are +consistent with the properties of other local low-mass +galaxies, including UFDs, although Pegasus W is more +compact than galaxies with similar luminosities. Based +on the properties of Pegasus W reported here (distance, +luminosity, half-light radius), the galaxy is an UFD in +the Local Group. +The best-fitting SFH derived from the CMD shows +Pegasus W has formed 10% of its stellar mass within +the last several Gyr, and suggests star-formation activ- +ity as recently as the last Gyr. We also identify BHeB +candidate stars in the CMD that are consistent with age +estimates < 500 Myr. This late-time star formation is +unusual in an UFD galaxy and indicates that the galaxy +recently harbored gas, either through gas retention or +recent accretion. +In addition to the evidence of late-time star forma- +tion, the overall SFH of Pegasus W is extended: the +galaxy formed 50% of its stellar mass after z ∼ 6 and + +12 +McQuinn et al. +1 +10 +100 +1000 +rh (pc) +12.5 +10.0 +7.5 +5.0 +2.5 +0.0 +MV (mag) +PegW +Galaxies +GCCs +Figure 10. Distribution of galaxies and Galactic globular +clusters (GGCs) in the MV − rh plane based on the updated +compilation of galaxies from McConnachie (2012) and Baum- +gardt et al. (2020). Overlaid as a red star is the location of +Pegasus W using our measured parameters. The galaxy is +consistent with the properties of known local dwarfs with +comparable luminosity, although slightly more compact, and +is considered an UFD galaxy based on the definition from +Simon (2019). +quenched as recently as z ∼ 0.9 (i.e., 7.4 Gyr ago). Cos- +mological simulations predict the mass threshold where +reionization efficiently quenches galaxies to be M∗ ∼ 105 +M⊙, possibly with an assist by stellar feedback (e.g., +Bovill & Ricotti 2011; Ben´ıtez-Llambay et al. 2015; Ro- +driguez Wimberly et al. 2019; Garrison-Kimmel et al. +2019; Rey et al. 2020; Pereira Wilson et al. 2022). Above +this threshold mass, a galaxy’s potential should be deep +enough to continue to accrete gas and the gas within +a galaxy can self-shield sufficiently from the UV radia- +tion to continue to cool and form stars. However, even +though Pegasus W has a stellar mass below this thresh- +old (M∗ 6.5+1.1 +−1.4 × 104 M⊙), the extended SFH is in- +consistent with this expectation. Note that, while the +stellar mass is directly observable, using M∗ as a mea- +sure of an UFDs total potential has large uncertainties +given the spread in the stellar mass-halo mass relation +of UFDs (e.g., Munshi et al. 2021; Applebaum et al. +2021). It’s possible that despite having M∗ < 105 M⊙, +the halo mass of Pegasus W is large enough to be above +the reionization threshold, although not significantly. +Interestingly, there is evidence that some of the UFD +satellites of M31 in the mass range 104 M⊙ < M∗ < 105 +M⊙ also have somewhat extended SFHs and more recent +quenching timescales. Initial results derived from data +with a range of photometric depths (including And XVI +with imaging reaching below the oMSTO) found quench- +ing timescales similar to what we find for Pegasus W +(Skillman et al. 2017; Weisz et al. 2019). Deeper obser- +vations for a subset of these systems place tighter con- +straints on the SFHs and shift the quenching timescales +to slightly older lookback times (Alessandro Savino et al. +in preparation), but the timescales are still generally +more recent than the rapid quenching by reionization +scenario described above. +The more recent quenching timescale for Pegasus W +is in contrast to what has been derived for 6 UFDs of +the MW in a slightly lower but overlapping mass range +(6 × 103 M⊙ < M∗ < 5 × 104 M⊙). +These galaxies +formed 80% of their stars by z ∼ 6 (12.8 Gyr ago) and +100% of their stars by z ∼ 3 (11.6 Gyr ago; Brown et al. +2014). While the rapid quenching of the MW UFDs is +consistent with and often attributed to reionization, it +is possible that environment played a role in quench- +ing these systems. +Based on proper motions of MW +UFDs from Gaia and modeling the time-varying grav- +itational potential of the MW due to its dark matter +halo growth, there are indications that the SFHs of the +UFDs were impacted by early environment effects from +the MW halo (Miyoshi & Chiba 2020). If this is the case, +then the differences in quenching timescales for the MW +UFDs and Pegasus W could be explained by differences +in environment and orbital histories. +Indeed, rather than being quenched by reionization, +it is more likely that Pegasus W was quenched by en- +vironmental processing despite being outside the virial +radius of M31. Based on the dwarf galaxy population +around the MW and M31, there is a transition in the +structural and dynamical properties of dwarfs at a dis- +tance of ≳ 400 kpc from their host (corresponding to +∼ 1.33×Rvir), which is thought to be due to tidal inter- +actions (Higgs & McConnachie 2021). Simulations have +shown that tidal effects can extend well beyond the virial +radius (Rvir) of the main halo (Behroozi et al. 2014), +and that more massive dwarf galaxies (M∗ ∼ 105.5 −108 +M⊙) within 1–2.5 × Rvir of a more massive galaxy can +be quenched by environment (Fillingham et al. 2018). +It is also possible, even likely given Pegasus W’s current +location at 1.2 × Rvir, that the galaxy has previously +passed through the halo of M31, making it a ‘backsplash’ +galaxy. Recent, extremely high-resolution cosmological +simulations suggest that most galaxies within 1.5 × Rvir +fall in the backsplash category (Applebaum et al. 2021). +In this scenario, the galaxy would have experienced more +significant environmental processing including ram pres- +sure and tidal stripping (e.g., Teyssier et al. 2012; Buck +et al. 2019; Bla˜na et al. 2020). Such a fly-by interaction +with M31 could have quenched the majority of star for- +mation, but not completely stripped the Pegasus W of +gas. The remaining bound gas could have cooled over +Gyr timescales, reigniting star formation and providing + +Pegasus W +13 +an explanation for the possible late-time star formation +suggested in the CMD. +Note that the SFH of Pegasus W is based on a CMD +that is relatively deep, reaching ∼ 2 mag below the HB. +However, that depth still falls meaningfully short of the +oMSTO depth needed to break the age-metallicity de- +generacy in the CMD at old lookback times. Our overall +results would be improved should such data be acquired. +In addition, spectroscopic observations of the stars in +Pegasus W could provide important constraints on the +internal kinematics, dark matter content of the galaxy, +and its current radial trajectory within the Local Group. +Support for this work was provided by NASA through +grants no. HST-GO-16916 and support for Y.-Y.M. was +partly provided by NASA through the NASA Hubble +Fellowship grant no. HST-HF2-51441.001, both were +awarded by the Space Telescope Science Institute, which +is operated by the Association of Universities for Re- +search in Astronomy, Incorporated, under NASA con- +tract NAS5-26555. K.B.W.M. is also supported by NSF +grant AST-1940800. D.S. and M.R.B. are supported by +DOE grant DOE-SC0010008. This research has made +use of NASA Astrophysics Data System Bibliographic +Services, adstex2, and the arXiv preprint server. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Facilities: Hubble Space Telescope +Software: This research made use of DOLPHOT (Dol- +phin 2000, 2016), MATCH (Dolphin 2002, 2012, 2013), HST +drizzlepac (v3.0; Hack et al. 2013; Avila et al. 2015), +emcee (Foreman-Mackey et al. 2013), and Astropy,3 a +community-developed core Python package for Astron- +omy (The Astropy Collaboration 2018). +REFERENCES +Akins, H. B., Christensen, C. R., Brooks, A. M., et al. +2021, ApJ, 909, 139 +Applebaum, E., Brooks, A. M., Christensen, C. R., et al. +2021, ApJ, 906, 96 +Avila, R. J., Hack, W., Cara, M., et al. 2015, in +Astronomical Society of the Pacific Conference Series, +Vol. 495, Astronomical Data Analysis Software an +Systems XXIV (ADASS XXIV), ed. A. R. Taylor & +E. Rosolowsky, 281 +Baumgardt, H., Sollima, A., & Hilker, M. 2020, PASA, 37, +e046 +Bechtol, K., Drlica-Wagner, A., Balbinot, E., et al. 2015, +ApJ, 807, 50 +Behroozi, P. S., Wechsler, R. H., Lu, Y., et al. 2014, ApJ, +787, 156 +Ben´ıtez-Llambay, A., Navarro, J. F., Abadi, M. G., et al. +2015, MNRAS, 450, 4207 +Benson, A. J., Frenk, C. S., Lacey, C. G., Baugh, C. M., & +Cole, S. 2002, MNRAS, 333, 177 +Bla˜na, M., Burkert, A., Fellhauer, M., Schartmann, M., & +Alig, C. 2020, MNRAS, 497, 3601 +Bovill, M. S., & Ricotti, M. 2011, ApJ, 741, 18 +Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, +427, 127 +3 http://www.astropy.org +Brown, T. M., Tumlinson, J., Geha, M., et al. 2014, ApJ, +796, 91 +Buck, T., Macci`o, A. V., Dutton, A. A., Obreja, A., & +Frings, J. 2019, MNRAS, 483, 1314 +Bullock, J. S., & Boylan-Kolchin, M. 2017, ARA&A, 55, +343 +Bullock, J. S., Kravtsov, A. V., & Weinberg, D. H. 2000, +ApJ, 539, 517 +Carretta, E., Gratton, R. G., Clementini, G., & Fusi Pecci, +F. 2000, ApJ, 533, 215 +Cerny, W., Simon, J. D., Li, T. S., et al. 2022a, arXiv +e-prints, arXiv:2203.11788 +Cerny, W., Mart´ınez-V´azquez, C. E., Drlica-Wagner, A., +et al. 2022b, arXiv e-prints, arXiv:2209.12422 +Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, +arXiv e-prints, arXiv:1612.05560 +Chen, Y., Girardi, L., Fu, X., et al. 2019, A&A, 632, A105 +Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102 +Collins, M. L. M., Charles, E. J. E., Mart´ınez-Delgado, D., +et al. 2022, MNRAS, 515, L72 +Dey, A., Schlegel, D. J., Lang, D., et al. 2019, AJ, 157, 168 +Dolphin, A. 2016, DOLPHOT: Stellar photometry, +Astrophysics Source Code Library, record ascl:1608.013, +ascl:1608.013 +Dolphin, A. E. 2000, PASP, 112, 1383 +—. 2002, MNRAS, 332, 91 +—. 2012, ApJ, 751, 60 + +14 +McQuinn et al. +—. 2013, ApJ, 775, 76 +Drlica-Wagner, A., Bechtol, K., Rykoff, E. S., et al. 2015, +ApJ, 813, 109 +Drlica-Wagner, A., Bechtol, K., Mau, S., et al. 2020, ApJ, +893, 47 +Fillingham, S. P., Cooper, M. C., Boylan-Kolchin, M., et al. +2018, MNRAS, 477, 4491 +Ford, H. C., Bartko, F., Bely, P. Y., et al. 1998, in Society +of Photo-Optical Instrumentation Engineers (SPIE) +Conference Series, Vol. 3356, Space Telescopes and +Instruments V, ed. P. Y. Bely & J. B. Breckinridge, +234–248 +Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, +J. 2013, PASP, 125, 306 +Garrison-Kimmel, S., Wetzel, A., Hopkins, P. F., et al. +2019, MNRAS, 489, 4574 +Geha, M., Blanton, M. R., Yan, R., & Tinker, J. L. 2012, +ApJ, 757, 85 +Hack, W. J., Dencheva, N., & Fruchter, A. S. 2013, in +Astronomical Society of the Pacific Conference Series, +Vol. 475, Astronomical Data Analysis Software and +Systems XXII, ed. D. N. Friedel, 49 +Haynes, M. P., Giovanelli, R., Kent, B. R., et al. 2018, ApJ, +861, 49 +Hidalgo, S. L., Pietrinferni, A., Cassisi, S., et al. 2018, ApJ, +856, 125 +Higgs, C. R., & McConnachie, A. W. 2021, MNRAS, 506, +2766 +Kauffmann, G., White, S. D. M., & Guiderdoni, B. 1993, +MNRAS, 264, 201 +Koposov, S. E., Belokurov, V., Torrealba, G., & Evans, +N. W. 2015, ApJ, 805, 130 +Kroupa, P. 2001, MNRAS, 322, 231 +Mapelli, M., Ripamonti, E., Battaglia, G., et al. 2009, +MNRAS, 396, 1771 +Mapelli, M., Ripamonti, E., Tolstoy, E., et al. 2007, +MNRAS, 380, 1127 +Martin, N. F., de Jong, J. T. A., & Rix, H.-W. 2008, ApJ, +684, 1075 +Martin, N. F., Ibata, R. A., Lewis, G. F., et al. 2016, ApJ, +833, 167 +Mateo, M., Fischer, P., & Krzeminski, W. 1995, AJ, 110, +2166 +McConnachie, A. W. 2012, AJ, 144, 4 +McConnachie, A. W., Irwin, M. J., Ibata, R. A., et al. 2009, +Nature, 461, 66 +McQuinn, K. B. W., Skillman, E. D., Dalcanton, J. J., +et al. 2011, ApJ, 740, 48 +McQuinn, K. B. W., Skillman, E. D., Berg, D., et al. 2013, +AJ, 146, 145 +McQuinn, K. B. W., Cannon, J. M., Dolphin, A. E., et al. +2014, ApJ, 785, 3 +McQuinn, K. B. W., Skillman, E. D., Dolphin, A., et al. +2015a, ApJ, 812, 158 +—. 2015b, ApJL, 815, L17 +Miyoshi, T., & Chiba, M. 2020, ApJ, 905, 109 +Momany, Y., Held, E. V., Saviane, I., et al. 2007, A&A, +468, 973 +Monelli, M., Cassisi, S., Mapelli, M., et al. 2012, ApJ, 744, +157 +Moore, B., Ghigna, S., Governato, F., et al. 1999, ApJL, +524, L19 +Munshi, F., Brooks, A. M., Applebaum, E., et al. 2021, +ApJ, 923, 35 +Nadler, E. O., Drlica-Wagner, A., Bechtol, K., et al. 2021, +PhRvL, 126, 091101 +Okamoto, S., Arimoto, N., Yamada, Y., & Onodera, M. +2012, ApJ, 744, 96 +Pan, Y., Simpson, C. M., Kravtsov, A., et al. 2022, arXiv +e-prints, arXiv:2208.13805 +Pereira Wilson, M., Navarro, J., Santos Santos, I., & +Benitez Llambay, A. 2022, arXiv e-prints, +arXiv:2206.05338 +Rey, M. P., Pontzen, A., Agertz, O., et al. 2020, MNRAS, +497, 1508 +Rodriguez Wimberly, M. K., Cooper, M. C., Fillingham, +S. P., et al. 2019, MNRAS, 483, 4031 +Sand, D. J., Seth, A., Olszewski, E. W., et al. 2010, ApJ, +718, 530 +Santana, F. A., Mu˜noz, R. R., Geha, M., et al. 2013, ApJ, +774, 106 +Savino, A., Weisz, D. R., Skillman, E. D., et al. 2022, ApJ, +938, 101 +Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103 +Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, +500, 525 +Simon, J. D. 2019, ARA&A, 57, 375 +Sirianni, M., Jee, M. J., Ben´ıtez, N., et al. 2005, PASP, +117, 1049 +Skillman, E. D., Monelli, M., Weisz, D. R., et al. 2017, +ApJ, 837, 102 +Somerville, R. S. 2002, ApJL, 572, L23 +Telford, O. G., Dalcanton, J. J., Williams, B. F., et al. +2020, ApJ, 891, 32 +Teyssier, M., Johnston, K. V., & Kuhlen, M. 2012, +MNRAS, 426, 1808 +The Astropy Collaboration. 2018, astropy v3.1: a core +python package for astronomy, Zenodo, +doi:10.5281/zenodo.4080996 + +Pegasus W +15 +Vincenzo, F., Matteucci, F., Belfiore, F., & Maiolino, R. +2016, MNRAS, 455, 4183 +Weisz, D. R., Martin, N. F., Dolphin, A. E., et al. 2019, +ApJL, 885, L8 +Wetzel, A. R., Tollerud, E. J., & Weisz, D. R. 2015, ApJL, +808, L27 +Williams, B. F., Lang, D., Dalcanton, J. J., et al. 2014, +ApJS, 215, 9 +Williams, B. F., Durbin, M. J., Dalcanton, J. J., et al. +2021, ApJS, 253, 53 +York, D. G., Adelman, J., Anderson, John E., J., et al. +2000, AJ, 120, 1579 + diff --git a/t9E2T4oBgHgl3EQf2Ain/content/tmp_files/load_file.txt b/t9E2T4oBgHgl3EQf2Ain/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e5a22929fb210f2b8cc866efe6791a7abda407c --- /dev/null +++ b/t9E2T4oBgHgl3EQf2Ain/content/tmp_files/load_file.txt @@ -0,0 +1,1283 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf,len=1282 +page_content='Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX631 Pegasus W: An Ultra-Faint Dwarf Galaxy Outside the Halo of M31 Not Quenched by Reionization Kristen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' McQuinn,1 Yao-Yuan Mao,1, 2 Matthew R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Buckley,1 David Shih,1 Roger E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Cohen,1 and Andrew E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Dolphin3, 4 1Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Rd, Piscataway, NJ 08854, USA 2Department of Physics and Astronomy, University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA 3Raytheon Technologies, 1151 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Hermans Road, Tucson, AZ 85756, USA 4University of Arizona, Steward Observatory, 933 North Cherry Avenue, Tucson, AZ 85721, USA ABSTRACT We report the discovery of an ultrafaint dwarf (UFD) galaxy, Pegasus W, located on the far side of the Milky Way-M31 system and outside the virial radius of M31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The distance to the galaxy is 915+60 −91 kpc, measured using the luminosity of horizontal branch (HB) stars identified in Hubble Space Telescope optical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The galaxy has a half-light radius (rh) of 100+11 −13 pc, MV = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='16 mag, and a present-day stellar mass of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4×104 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We identify sources in the color-magnitude diagram (CMD) that may be younger than ∼ 500 Myr suggesting late-time star formation in the UFD galaxy, although further study is needed to confirm these are bona fide young stars in the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Based on fitting the CMD with stellar evolution libraries, Pegasus W shows an extended star formation history (SFH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Using the τ90 metric (defined as the timescale by which the galaxy formed 90% of its stellar mass), the galaxy was quenched only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6 Gyr ago, which is similar to the quenching timescale of a number of UFD satellites of M31 but significantly more recent than the UFD satellites of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Such late-time quenching is inconsistent with the more rapid timescale expected by reionization and suggests that, while not currently a satellite of M31, Pegasus W was nonetheless slowly quenched by environmental processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Keywords: stars: color-magnitude diagrams − galaxies: Local Group − galaxies: dwarf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' INTRODUCTION Very low-mass (M∗ ≲ 105 M⊙) galaxies are expected to be numerous in the present-day universe, based on a Λ Cold Dark Matter cosmology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Bullock & Boylan-Kolchin 2017, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Such low-mass systems have correspondingly low-luminosities (MV ≲ −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='7 mag) and are referred to as ultrafaint dwarfs (UFDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Si- mon 2019, for a recent definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The shallow potential wells of UFD galaxies make them extremely sensitive to both external perturbations (such as heating from meta- galactic UV background radiation and local environmen- tal conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bullock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Benson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Somerville 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Geha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Wetzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Rey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Akins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022) and internal perturbations (such kristen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='mcquinn@rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='edu as stellar feedback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Ap- plebaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, these small galaxies make excellent laboratories in which to test physical models and galaxy formation theories, and constrain the im- pact of reionization on the growth of low-mass halos in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' They are also critical components for tests of the hierarchical structure formation theories and dark matter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Nadler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' UFD galaxies went mostly undetected due to their low surface brightness, faint luminosities, and small physical sizes until ∼ 20 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' With the advent of wide-field, deeper optical surveys, such as the Sloan Digital Sky Survey (SDSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2000), the Panoramic Survey Telescope & Rapid Response System (Pan-STARRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2016), the Pan-Andromeda Archeolog- ical Survey (PAndAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' McConnachie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2009), the Dark Energy Survey (DES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Drlica-Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020), the DECam Local Volume Exploration (DELVE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Cerny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022a,b), and the DESI Legacy Imaging Surveys (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019), searches for very low-mass galaxies arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='04157v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='GA] 10 Jan 2023 2 McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' have been incredibly fruitful, evidenced by the growing number of UFD galaxies now known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Simon 2019, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The majority of the galaxies dis- covered lie in close proximity to the Milky Way (MW) and the Large Magellanic Cloud (LMC), as intrinsically faint systems are more readily detected at closer dis- tances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A smaller number have been identified as part of the M31 satellite system, found via the targeted PAn- dAS observations (McConnachie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Here, we report the discovery of an UFD galaxy in the Local Group (LG) named Pegasus W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The galaxy was identified in the DESI Legacy Imaging Surveys data as an over-density in the photometric stellar catalog, similar to previous discoveries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bechtol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Drlica-Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022), but the system is much farther than other known UFD galaxies in the LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Hubble Space Telescope (HST) optical imaging of Pegasus W enables a robust distance measurement which places the galaxy on the far side of M31 but outside M31’s virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, Pegasus W offers a unique opportunity to measure the properties and study the evolution of an UFD galaxy that is not a satellite galaxy but that is still in close enough proximity for detailed observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Section 2 describes the HST observations, photometry, and data process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Section 3 presents the structural parameters of Pe- gasus W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Section 4 explores the CMD of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Section 5 includes the distance measurement to the galaxy based on horizontal branch (HB) stars and in- vestigates the LG environment around Pegasus W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Sec- tion 6 presents the SFH based on fitting stellar evo- lution libraries to the CMD, calculates star formation timescales, and measures the integrated luminosity and stellar mass of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Section 7 summaries the over- all properties of Pegasus W in the context of other Local Group UFDs and discusses the implications of an UFD galaxy that was not quenched by reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' OBSERVATIONS AND DATA PROCESSING 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Observations HST observations of Pegasus W were obtained using the Wide Field Camera (WFC) of the Advanced Camera for Surveys (ACS) instrument (Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1998) on 2022 June 27 in the F606W and F814W filters as part of HST- GO-16916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' All the HST data used in this paper can be found in MAST: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='17909/x8qj-bn51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Two observations were made per filter during one orbit, with a 5×5 pixel dither (acs wfc dither line pattern #14) between exposures to help reject cosmic rays and to mitigate de- tector cosmetic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The ACS pointing was centered on J2000 RA = 23 : 53 : 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='229, Dec = +22 : 05 : 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='54, Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pegasus W Properties Property Value RA (J2000) 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='31248167◦±1′′ Dec (J2000) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='10197022◦±1′′ Position angle θ (◦ E of N) 92±3 ellipticity (ϵ = 1 − b a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='08 rh (′′) 23±2 rh (pc) 100+11 −13 MV (mag) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='16 M∗ (M⊙) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 × 104 HB mV,0 (mag) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='20 µ (mag) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='22 Σb (arcmin−2) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='8+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 Distance (kpc) 915+60 −91 [M/H] (dex) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 τ90 (Gyr) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6 AV (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='315 AF 606W (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='284 AF 814W (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='175 Note—The properties of Peg W were measured in this work, with the exception of the foreground extinction which is from Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (1998) with recalibration from Schlafly & Finkbeiner (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' which placed the galaxy slightly offset from the center of the ACS field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This selected pointing en- sured maximum coverage of the stellar disk while avoid- ing nearby bright stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The exposure time was evenly split between the two ACS/WFC filters with a final in- tegration time of 1140 s per filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A second field was simultaneously imaged in parallel using the Wide Field Camera 3 (WFC3) UVIS instru- ment in the same bandpasses, enabling an investigation of potential background and foreground contamination in a field near to the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The WFC3 pointing was centered on RA = 23 : 52 : 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='255, Dec = +22 : 07 : 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='48 with integration times of 1020 and 1045 s in F606W and F814W filters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Figure 1 presents 3-color images of the HST ACS ob- servations (left), a zoom-in on the region with Pega- sus W (center), and the parallel field from the WFC3 observations (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The 3-color images were made us- ing the F606W, F814W, and (F606W+F814W)/2 mo- saics created from the charge transfer efficiency cor- rected (flc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='fits) files with the HST drizzlepac v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 python package (Hack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Avila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Despite being low-luminosity, the galaxy is visible in the center image as an over-density of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pegasus W 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Three-color images of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Left: The ACS full field of view with an ellipse encircling the stellar component of Pegasus W out to 2 rh based on the best-fitting structural parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Middle: A zoom-in on the region of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Right: The WFC3 parallel field which can help quantify potential foreground and background contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The three-color images were created using F606W for red, and average of F606W and F814W images for green, and F814W for blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Photometry Photometry was performed on the flc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='fits images using the point spread function (PSF) fitting software DOLPHOT (Dolphin 2000, 2016), with includes specific ACS/WFC and WFC3/UVIS modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The DOLPHOT photometric parameters were set according to the val- ues recommended in Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2014, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The photometric output was filtered for well-recovered stars using a combination of quality metrics returned from DOLPHOT that help to characterize each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Specifically, we selected sources with an output error flag < 4, object type ≤ 2, signal-to-noise ratio ≥ 5 in both filters, sharp2 F 606W + sharp2 F 814W < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='075, and crowdF 606W + crowdF 814W < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The sharpness pa- rameter is a measure of how peaked or broad a source is relative to the PSF and helps to reject cosmic rays and background galaxies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We chose to apply strict sharpness cuts to limit contamination from faint, unresolved background galaxies in the stellar catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The crowding parameter measures how much brighter a source would have been if stars nearby on the sky had not been fit simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' While an important quality metric to consider, strict crowding cuts are not as criti- cal in creating a high-fidelity catalog given the spareness of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Artificial stars tests were performed on both the ACS and WFC3 data to measure the observational uncertain- ties and completeness of the images using the same pho- tometric software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Approximately 500k artificial stars were injected into each individual image following to the spatial distribution of all sources identified in the photometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', the pre-filtered DOLPHOT output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The sources were then recovered photometrically and the same quality cuts used for the photometry were applied to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The field was sufficiently uncrowded that we detected no significant trends of incompleteness with distance from the center of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' STRUCTURAL PARAMETERS We determined the structural parameters of Pega- sus W, including orientation on the sky (position angle, θ), shape (semi-major axis, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' ellipticity, ϵ = 1 − b a), and half-light radius (rh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' These parameters help char- acterize Pegasus W and provide a way to compare to the properties of Pegasus W to other UFD galaxies (see Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' They are also used to apply spatial cuts to the photometry to create our final stellar catalog for Pegasus W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The structural parameters were determined using an unbinned maximum likelihood approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Specifically, we perform a Markov Chain Monte Carlo (MCMC) fit of an exponential density profile (allowing for non-zero ellipticity) to the spatial distribution of observed sources over the full ACS/WFC field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' To avoid any impact of incompleteness while maximizing the contri- bution of galaxy members, we consider only sources with F606W < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6, F814W < 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='7 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 mag bright- ward of the 50% completeness limit in both filters, ascertained from the artificial star tests) and a color (F606W−F814W) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We verified that the resulting structural parameters presented below are quite robust to these cuts, such that either eliminating the color cut entirely and/or moving the magnitude cuts faintward yielded results that were consistent to within their 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Operationally, we fit for six free parameters: The tangent plane coordinates x0 and y0, which are the location of the center relative to an arbitrary posi- 22°08\' ACS/WFC Primary Field 07\' 06\' Dec 05\' 04\' 23h53m25s 20s 15s 10s 05s RA22°07\'00" Pegasus W 06\'30" Dec 00" 05\'30" 100 pc 23h53m18s 16s 14s 12s RAWFC3/UViS Parallel Field 22°09\' Dec 08\' 07\' 23h52m56s 52s 48s 44s RA4 McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' tional zeropoint1, the half-light radius rh, the ellipticity ϵ = (1 − b/a) where b/a is the ratio of minor to major axis lengths, the position angle θ in degrees east of north, and N⋆, which is the total number of stars in the galaxy (within the aforementioned CMD limits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Note that N⋆ is an extrapolation from the density profile rather than the number of stars we observe within the ACS/WFC field of view, which is Nobs = 314 after applying our CMD cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' To determine the best-fit values of the structural pa- rameters and their uncertainties, we search for the set of parameters for which the data are most likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In other words, for parameters (p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', p6) we maximize the likelihood function: L(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', p6) = � i ℓi(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', p6), (1) where ℓi(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', p6) is the probability of finding the ith datapoint given parameters (p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', p6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This proba- bility is calculated following Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2008, 2016), assuming the target galaxy has an exponential density profile ρgal(r) as a function of elliptical radius r: ρgal(r) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='682 2πr2 h(1 − ϵ)N⋆ exp(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='68r/rh), (2) where the elliptical radius r is related to the tangent plane coordinates x and y as r = �� 1 1 − ϵ((x − x0) cos θ − (y − y0) sin θ) �2 + ((x − x0) sin θ + (y − y0) cos θ)2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (3) We also allow for a spatially constant background den- sity Σb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The background density in each MCMC itera- tion is set by requiring that the number of background sources is equal to the difference between the model- predicted number of sources and the observed number of sources in the full field of view, with area A: Σb = � Nobs − � A ρgal dA � /A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (4) In practice, the integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2 over the field of view and the calculation of A are performed numerically by generating a spatial grid of pixels with an area of 1 arcsec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' With Σb in hand for each iteration, the full density profile for a given set of trial parameters is: ρmodel(r) = ρgal(r) + Σb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (5) 1 The positional zeropoint was set to a rough guess center of (RA,Dec) = (358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='312732◦,22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='102465◦) based on the distribution of sources recovered photometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' x0 ["] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='85+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 y0 ["] y0 ["] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='81+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='13 16 20 24 28 32 Rh ["] Rh ["] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='58+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='08 80 85 90 95 100 [ ] [ ] = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='67+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='63 6 3 0 3 6 x0 ["] 240 300 360 420 N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 y0 ["] 16 20 24 28 32 Rh ["] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 80 85 90 95 100 [ ] 240 300 360 420 N N = 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='34+31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='47 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='28 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Full posterior distributions of Pegasus W struc- tural parameters over 10,000 post-burn-in MCMC iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The black contour lines correspond to 1,2 and 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Best- fitting values are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1-5, the log likelihood is then: ln L = Nobs � i ρmodel(ri) − Nobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (6) We impose broad, flat, astrophysically motivated pri- ors on the free parameters: |x0|, |y0| < 100′′, essentially requiring the center of the galaxy to lie in our field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' rh > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 0 < ϵ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 0 < θ ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' N⋆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Σb ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We sample the posterior distributions of the param- eters using the emcee package (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013), running 50 affine-invariant walkers for 5000 burn- in iterations followed by 10000 production iterations (more than sufficient given autocorrelation lengths of <100 steps for all parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The best-fit parameter values we report are the medians over all post-burn-in Pegasus W 5 0 1 2 3 r [arcmin] 0 100 200 300 400 500 600 700 800 Density [arcmin 2] 10 1 100 r [arcmin] 101 102 103 Density [arcmin 2] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Binned radial density profile of Pegasus W, shown on linear (left) and logarithmic (right) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Black points are the binned observed data with errorbars indicating Pois- sonian uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The red line is the best-fit exponential profile from our maximum likelihood analysis, and the grey lines represent 100 random draws from the posterior distri- butions of the profile parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' iterations, with uncertainties corresponding to the 16th and 84th percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Figure 2 shows the full posterior distributions of our fits and their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' As an additional check, we calculate a binned radial density profile in Figure 3, where black points are the binned observed data and ver- tical errorbars represent Poissonian uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Over- plotted on the binned data is the maximum-likelihood solution in red, with 100 random individual draws from the posterior distributions shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The final structural parameters including center RA, Dec coordinates of Pegasus W, rh, ellipticity, and posi- tion angle, as well as the value of Σb, are listed in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We also overplot an ellipse based on these param- eters and encompassing 2 rh on the 3-color ACS images in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' COLOR-MAGNITUDE DIAGRAM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Spatial Selection of Sources Figure 4 presents the spatial distribution of all sources that pass our photometric quality cuts in X-Y coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Two ellipses are overplotted based on our struc- tural parameters with semi-major axes of 2 rh and 3 rh, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The overdensity of sources from Pegasus W is readily apparent in the smaller ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Outside of this region there are still a few hundred sources that pass our quality cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' While some of the sources may be bona fide members of Pegasus W, the majority are likely foreground stars or unresolved background galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, to create a high-fidelity photometric catalog of Pegasus W with minimal contamination, we used the structural parameters and applied spatial cuts to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Our final stellar catalog for Pegasus W includes all sources contained within the ellipse extending to 2 rh, shown as cyan stars in Figure 4, and includes 377 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' To quantify the potential contamination to the Pegasus W catalog, we use the sources outside the larger ellipse that extends to 3 rh and create a field catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' These sources are shown as orange circles and include a total of 465 sources from an area ∼ 7× larger than that used for Pegasus W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This field catalog is used to estimate contamination in the CMD in Section 4 and to model the contamination in the SFH fitting in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The sources in between 2 − 3rh are not used in either catalog and are shown as black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Note that the WFC3 UVIS parallel field was origi- nally intended to quantify the potential contamination of the final Pegasus W stellar catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We examined the WFC3 data and found a somewhat smaller number of sources (241) passed the photometric quality cuts from the full UVIS field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We opt to use the field cata- log from the ACS data as representative of the potential contamination as the field is closer spatially to Pega- sus W, avoids transforming from WFC3 UVIS to ACS magnitudes, and also avoids any systematics caused by slightly different spatial resolution and throughputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' CMD of Final Stellar Catalog Figure 5 shows the CMD from the final photometric catalog for Pegasus W in the left panel, a 2D histogram representation of the CMD (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', a Hess diagram) of the field catalog scaled to the same area as Pegasus W in the middle panel, and a residual Hess diagram of the field CMD subtracted from the Pegasus W CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The stellar density of the Pegasus W catalog is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='072 sources per arcsec2, compared with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='013 in the field catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The data are plotted to the 50% completeness limits and representative photometric uncertainties per magnitude bin are shown in the Pegasus W CMD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' no corrections for completeness have been applied to the stellar density values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The CMD for Pegasus W shows well-defined RGB, a blue and red horizontal branch (HB), and red clump (RC) features, which are largely absent in the field Hess diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The differences between the CMDs of Pega- sus W and the field region can be seen more quantita- tively in the residual Hess diagram in the right panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The red color represents higher star counts from Pegasus W over the field region and includes the RGB and HB features in the CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The blue color in- dicates that contamination dominates the source counts and is seen predominantly at fainter magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' There are also two groups of sources in the CMD that warrant additional attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' There are four sources blue-ward of the RGB and above the HB feature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', 6 McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 3500 4000 X 0 500 1000 1500 2000 2500 3000 3500 4000 Y 2rh 3rh Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Distribution of point sources passing the photo- metric quality cuts in the X–Y coordinates of the ACS field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pegasus W is clearly identifiable as an over den- sity of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The smaller ellipse is based on our measured structural parameters of Pegasus W extends to 2 rh, and encircles the point sources (cyan stars) used in our analysis of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The larger ellipse reaches 3 rh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' sources outside this area (orange circles) quantify the po- tential contamination of the final Pegasus W stellar catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The sources located between 2 < rh < 3 (black points) are not used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F606W−F814W < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F606W <24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='8) which have op- tical colors and magnitudes consistent with blue helium burning (BHeB) stars with ages of < 500 Myr (McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' There is one source in a similar region of CMD-space in the catalog from the off-galaxy field area that is ∼ 7× larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' There is also a grouping of thirty blue sources fainter than the HB with colors ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 mag that could be the bright end of what is referred to as the ‘blue plume’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The blue plume is often detected in the CMDs of globu- lar clusters and UFDs that reach the old main sequence turn-off (oMSTO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Momany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2007, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Sand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Okamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The nature of these sources is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' They may be young (≲ 3 Gyr) main sequence stars (possibly with unresolved binaries), indicating that a galaxy has hosted late-time star forma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' They could also be blue stragglers (BSs), which are older, hot, blue stellar sources with optical colors and magnitudes similar to younger main sequence stars (Mateo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' BSs are thought to be a product of stellar binary systems, with a collisional or mass-transfer origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In the case of Pegasus W, the population of faint, blue stars lie at the bright end of what is typically consid- ered the blue plume region identified in CMDs reaching deeper photometric depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The residual Hess diagram in the final panel of Figure 5 reveals this is a mixed population of high-confidence Pegasus W member stars and contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Specifically, there are an average of 11 blue plume sources in a region of equal area to Pega- sus W compared with 30 found within 2 rh of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The BHeB star candidates and the blue plume sources are both consistent with ages of ∼500 Myr, based on BaSTI isochrones (Hidalgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Examining the spatial distribution shows that 2 of the 4 BHeB candi- dates and 10 of the 30 blue plume sources are concen- trated in the inner rh, with 4 blue plume sources within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' BSs in dwarf galaxies are expected to have a flat radial distribution whereas younger stars are more spatially clumped (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Momany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012), lending support to the idea that some of the sources may be young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The specific fraction of blue stragglers to RGB stars has been shown to be approximately constant in UFD galaxies (Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, to further investi- gate the nature of the blue plume sources, we follow the procedure in Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2013) and calculate the frac- tion of blue plume to RGB stars after subtracting the number of sources in the field region scaled by the areal coverage of the galaxy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Note, however, that the specific fraction we calculate is a lower limit on how many blue stragglers are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Our data are in different filters and do not reach the same photomet- ric depths as the data from Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' As a result, the magnitude ranges used to select the popu- lation of stars are not identical and, given the relative faintness of blue plume to RGB stars, this has a greater impact on the number of blue plume stars counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Our calculation yields a lower limit of the specific fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='21 − below the average value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 from San- tana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2013) − making our results inconclusive on the nature of the blue plume sources using this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Note, an Anderson-Darling statistical test performed on the cumulative radial distributions of the blue plume and galaxy population samples was also inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The galaxy was not detected in GALEX near or far UV imaging, although this is to be expected given the shorter star formation timescale that UV emission traces and the depth of the GALEX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pegasus W was also not detected in the Hi ALFALFA survey (Haynes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2018), but it is possible that the galaxy still harbors some gas below the detection limit of the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pegasus W 7 0 1 2 F606W-F814W (mag) 22 23 24 25 26 27 28 F606W (mag) Pegasus W Star Density = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='071 arcsec 2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' of stars=377 0 1 2 F606W-F814W (mag) Field Region in ACS Star Density = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='013 arcsec 2 0 1 2 F606W-F814W (mag) (Peg W Field) Residual 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' of Stars Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Left: CMD of the 377 sources located inside an ellipse centered on Pegasus W and extending to 2 rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The CMD is plotted to the ∼ 50% completeness limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' representative uncertainties per magnitude are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Middle: Hess diagram of sources in the ACS field of view outside 3 rh that passed our quality cuts, scaled to the area encompassing the Pegasus W region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Right: Residual Hess diagram made from subtracting the field scaled Hess diagram from the Pegasus W data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Red indicates a higher number of sources in Pegasus W while blue indicates a higher amount of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' While it is intriguing that Pegasus W may have hosted late-time star formation, this is still speculative as it is not possible to discern with confidence the true nature of the BHeB candidates or blue plume sources without deeper imaging or spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We return to this when fitting for the SFH in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' THE DISTANCE TO PEGASUS W The distance to the galaxy was measured to be 915+60 −91 kpc based on the luminosity of the HB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We chose to use the HB stars for a standard candle distance de- termination, instead of the tip of the red giant branch stars, as the upper RGB is sparsely populated and the brightest identified stars in the RGB sequence may be a false tip (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013, 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The HB distance methodology arises from the nearly constant V -band luminosity of helium burning stars in the post-RGB phase of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The absolute mag- nitude of the HB has been calibrated in the Johnson BVRI filter system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, we convert the photometry from the HST ACS filters to the Johnson filter system using the transformations from Sirianni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2005, see their Equation 12 with coefficients from Table 22), and iteratively solve for V and I band magnitudes using the F606W−F814W colors as a starting point: V = F606W + c0 + c1 ∗ (V − I) + c2 ∗ (V − I)2 I = F814W + c0 + c1 ∗ (V − I) + c2 ∗ (V − I)2 (7) The coefficients c0, c1, c2 are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Before transforming to the V, I magntiudes, the photometry was corrected for Galactic foreground extinction using the dust maps from Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (1998) and the re- calibration by Schlafly & Finkbeiner (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' values are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The HB luminosity was measured using a maximum likelihood approach to fitting a parametric luminosity function to the stars in the region of the HB with the following form: P = eA(V −VHB)+B + e−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5[(V −VHB)/C]2, (8) where V is the magnitude in the F606W filter, and A, B, and C are free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The parameter C has a prior of 1 C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' An advantage of using a maximum likelihood approach is that it takes into account the distribution of photometric uncertainties and completeness measured by the artificial star tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Figure 6 presents the extinction corrected CMD for Pegasus W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Stars used for the fit lie in the range of 8 McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Photometry Transformation Coeffcients Coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' V I V − I < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 V − I ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 V − I < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 V − I < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 c0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='394±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='05 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='331±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='008 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='479±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='13 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='496±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='010 c1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='153±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='340±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='041±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='211 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='013 c2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='096±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='038 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='015±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='003 Note—Coefficients used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 7 for transforming the ACS photometry to Johnson V and I magnitudes as a function of V − I color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 F606W0-F814W0 (mag) 21 22 23 24 25 26 27 F606W0 (mag) D=915+60 91 kpc Pegasus W BaSTI 12 Gyr [M/H]=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 [M/H]=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9 [M/H]=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='7 [M/H]=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Extinction corrected CMD of Pegasus W with 12 Gyr BaSTI isochrones and varying [M/H] values overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The red box outlines the area of the CMD used to fit for the HB luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' the HB identified by eye in the CMD, as shown by the red box in Figure 6, which avoids including stars on the blue edge of the red clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The best-fit HB luminosity is V0 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='20 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The HB luminosity was converted to a distance mod- ulus using the calibration from Carretta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2000): MV = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='09) × [Fe/H] + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='07) (9) We estimate [Fe/H] by overlaying isochrones on the CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Shown in Figure 6 are a set of 12 Gyr BaSTI isochrones with [M/H] ranging from −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Based on the CMD, we find that the [M/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9 and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 are the best overall matches to the RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The higher value of [M/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9 is in agreement with the best-fitting average stellar metallicity found from the CMD-fitting technique using the BaSTI stellar library (see Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, we adopt [M/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9 as rep- resentative of [Fe/H] in the distance calculation with an uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This metallicity estimate is somewhat higher than typically reported for UFD galax- ies, but not unexpected given the extended SFH pre- sented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The final distance modulus to Pegasus W is 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='22 mag, corresponding to a distance of 915+60 −91 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The uncertainties are based on adding in quadrature the un- certainties in the HB luminosity fit, the estimated un- certainty on the adopted [Fe/H] value, and the uncer- tainties from the calibration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Location within the Local Group A secure distance enables the location of Pegasus W within the Local Group to be firmly established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Fig- ure 7 shows the distribution of sources within a physi- cal 3-dimensional distance of 500 kpc of Pegasus W in Supergalactic (SG) euclidian coordinates (SGX, SGY, SGZ), which provide a way to visualize the physical spa- tial distribution of galaxies and their linear separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The SG coordinates were determined using the spher- ical coordinates and distance to each system following the formalism described in McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' For reference, the MW sits at the SG origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Note that the axis ranges on each plot are fixed at 600 kpc, but the plot includes only galaxies found to be within 500 kpc of Pegasus W in the updated compilation from Mc- Connachie (2012) with revised distances from Savino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2022) were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We also include M31 with an adopted distance of 776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2+22 −21 kpc determined from RR Lyrae stars (Savino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' From Figure 7, Pegasus W (red star) lies on the far side of the MW−M31 system at a distance of 348 kpc from M31, which places it outside an assumed 300 kpc virial radius (green square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The galaxy is relatively isolated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' the two nearest neighbors are the M31 satellite, AndVII (separation=92 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' blue square) and the low- mass galaxy Peg Irr (separation=150 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' blue circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The clustering of black points are comprised mainly of satellites of M31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pegasus W 9 400 600 800 SGX (Mpc) 700 600 500 400 300 200 SGY (Mpc) Rvir M31 PegW AndVII PegIrr MW 400 600 800 SGX (Mpc) 100 0 100 200 300 400 SGZ (Mpc) 600 400 200 SGY (Mpc) 100 0 100 200 300 400 SGZ (Mpc) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Known systems in SG coordinates within 500 kpc of Pegasus W based on the updated compilation of galaxies from McConnachie (2012) and revised distances for a subset from Savino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The location of Pegasus W is shown as a red star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' the uncertainties in the position of the galaxy based on distance uncertainties are smaller than the plot symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M31 is shown as a green square;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' the majority of black points are satellites of M31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The nearest known neighbors to Pegasus W are And VII (blue square) and the Pegasus dwarf galaxy (blue triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The MW is located at the SG origin on the far side of M31 from Pegasus W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' the black arrow in the first panel indicates the direction of the MW in the SGX−SGY plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' THE STAR FORMATION HISTORY OF Pegasus W 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' SFH Methodology The SFH of the galaxy was reconstructed from the stellar populations in the CMD using the CMD-fitting software MATCH (Dolphin 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The basic approach of a CMD-fitting technique is to find the best-fitting com- bination of synthetic simple stellar populations (SSPs) of different ages and metallicities from stellar evolution libraries to an observed CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The SSPs are constructed using an assumed initial mass function (IMF) and binary fraction and then linearly combined until the best-fit is found based on a Poisson likelihood statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' For the SFH fits for Pegasus W, we assumed a Kroupa IMF (Kroupa 2001), a binary fraction of 35% with flat secondary mass distribution, and used as primary inputs the photometry and observational uncertainties and in- completeness measured from the artificial star recovery fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The SFHs were reconstructed using three stel- lar evolution libraries, namely BaSTI, PARSEC (Bres- san et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012), and MIST (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2016), which help to bracket the range of possible solutions and pro- vide an estimate of the systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Given the photometric depth of the data, the CMD does not fully constrain the metallicity evolution of Pegasus W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, we imposed a physically motivated constraint that the metallicity is a continuous, non-decreasing function over the lifetime of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The SFH solutions were fit with an age grid of log(t/yr) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='15 using a time resolution δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 dex for ages less than log(t/yr) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 and δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='05 dex for older ages, and a metallicity grid [M/H]= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 with a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='15 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We initially tested searching a metallicity grid encompassing lower metallicity values, but as the best-fitting [M/H] so- lutions were consistently above −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0, we tightened the search grid to reduce computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We fixed the distance in the fits to the HB distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='915 Mpc and adopt a Galactic extinction value of AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='315 mag based on the dust maps from Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (1998) and recalibration from Schlafly & Finkbeiner (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Figure 8 shows an example of the quality of the fit to the data using the BaSTI stellar library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The most im- portant diagnostic is the residual significance plot (bot- tom right panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' units in standard deviations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' a checker- board pattern of blue and red indicates no major resid- uals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The overall fit is quite good and the different evo- lutionary sequences are well reproduced by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Random uncertainties due to the finite number of stars in a CMD were estimated using a Markov Chain Monte Carlo approach (Dolphin 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In addition, while the SFH derived using the different stellar libraries provides a measure of the range of possible solutions, systematic uncertainties were also estimated using 50 Monte Carlo simulations (Dolphin 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Best-Fitting SFH The left panel of Figure 9 presents the best-fitting SFH using the BaSTI models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' the uncertainties include both random and systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The right panel shows the best-fitting SFH solutions with BaSTI, MIST, and PARSEC models with random uncertainties only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The SFH recovered by the three stellar libraries are in overall good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The range in solutions from the 10 McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' different libraries lie within the systematic uncertainties estimated for the BaSTI library shown in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' For the downstream analysis, we adopt the SFH solution based on the BaSTI library which produced a slightly better fit overall than the PARSEC and MIST libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The best-fitting SFH for Pegasus W shows a rising star formation until ∼ 7 Gyr ago, with an indication of late- time star formation activity over the past few Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' As discussed above, there are a few BHeB star candidates and sources in blue plume region of the CMD that may be bona fide young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' If these are true BHeB and young main sequence stars, they can provide valuable and unique constraints on late-time star formation in an UFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' On the other hand, the BHeB candidates could be contaminants in the CMD and the blue plume sources could be BSs masquerading as young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' To try to account for the possibility that BHeB are contaminants, we use the field catalog as representative of potential background sources while fitting for the SFH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' However, BSs are currently not included in the stellar evolution libraries used in the CMD-fitting and, thus, cannot be explicitly taken into account in the SFH recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In this case, including them in the fit will falsely result in star formation activity ≲ 3 Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' To explore the impact of the candidate BHeB stars and blue plume on our results, we fit for the SFH with and without these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Excluding these sources has little impact on the total stellar mass recovered from the data, reducing the stellar mass formed in the galaxy by ∼ 1–2% depending on the stellar library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Removing these sources eliminates star-formation activity < 500 Myr ago, but not the low level of star formation 1–3 Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We include these sources in our final fit, but note there is a larger uncertainty in the very recent SFH (t < 500 Myr) given the ambiguity of the nature of these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Star-Formation Timescales Of particular interest is the timescale by which the star formation ceased, or the quenching timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We adopt the lookback time at which the galaxy has formed 90% of its stellar mass as the quenching timescale (τ90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Using τ90 reduces the impact blue plume stars have on quenching timescale as it is not as dependent on whether the galaxy has experienced a small level of recent star formation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We find τ90 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6Gyr based on interpolating the SFH derived using the BaSTI stellar library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The un- certainties include both statistical and systematic un- certainties shown in the left panel of Figure 9 which were similarly determined by interpolating the uncer- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W - F814W 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W Observed 10 1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W - F814W 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W Model 10 1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W - F814W 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W Residual (Obs - Model) 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W - F814W 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 F606W Significance (Obs - Model) 4 2 0 2 4 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Example of the quality of fit to the observed CMD using the BaSTI stellar library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Top left: observed CMD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' top right: modelled Hess diagram used in the fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' bottom left: simple residual Hess diagram (data − model);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' bottom right: residual significance Hess diagram where the pixels are weighted by the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The checkerboard pattern in the residual significance indicates there are no major residuals and the modelled Hess diagram is a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' tainty envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The value of τ90 with uncertainties is overplotted on the SFH in the left panel of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Total Luminosity of Pegasus W We calculate the present day MV of Pegasus W follow- ing the approach outlined in Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' To ac- count for uncertainties in both distance (resulting from our fit to the HB magnitude) and the structural param- eters, we perform a Monte Carlo procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Specifically, in each Monte Carlo iteration, we generate a synthetic stellar population using the best-fit star formation his- tory, a distance drawn from the HB-based distance de- termination, and a value of N⋆ from the posterior distri- bution in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Since N⋆ represents the total number of stars in the system observed within our chosen CMD limits of F606W < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6, F814W < 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='7 and (F606W−F814W) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5, we continue to draw stars from the entire synthetic population until the number of stars that would be observed within these CMD limits is equal to N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In each iteration, the F606W magni- tudes of all stars drawn from the synthetic CMD are corrected for extinction and distance and converted to V -band magnitudes using the bolometric corrections of Pegasus W 11 5 10 Lookback Time (Gyr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 Cumulative SFH 90 5 10 Lookback Time (Gyr) BaSTI MIST PARSEC 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 1 2 3 4 10 Redshift (z) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 1 2 3 4 10 Redshift (z) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Left: Best-fitting SFH solution with the BaSTI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Uncertainties include both statistical and system- atic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The quenching timescale, τ90 is marked as noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Right: Best-fitting SFH solutions using the BaSTI, MIST, and PARSEC models with statistical uncertainties only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The gray shaded vertical region shows the approxi- mate timescale of reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2019) assuming [M/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9, and their sum then yields a total MV in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The value and uncertainties we report correspond the median and 16th−84th percentiles obtained over all iterations, yield- ing MV = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='16 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We also attempted to directly estimate the total lumi- nosity of Pegasus W using publicly available images from the DESI Legacy Imaging Surveys (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We obtained an estimate that is consistent with Monte Carlo method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' however, the uncertainty on the estimate is much larger (≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 mag) due to the image’s limited depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Present-Day Stellar Mass We estimate the present-day stellar mass, M∗, in Pe- gasus W using two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' First, we obtain an es- timate by using the results of the Monte Carlo simula- tions used to determine the total luminosity the galaxy, described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We sum the total mass of all stars drawn in each Monte Carlo iteration, which yields M∗ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 × 104 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Second, M∗ is estimated directly from the SFH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The total stellar mass formed over the lifetime of the galaxy is determined while fitting for the SFH solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This is akin to integrating the SFH over cosmic times, but is a separate quantity fit by assuming one time bin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', log(t/yr) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The stellar mass determined in this way is equal to the value determined from integrat- ing SFR(t), but the measurement avoids any co-variance between time bins in the fit and results in lower total un- certainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' To calculate the present-day stellar mass of Pegasus W we make two adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' First, as described in Telford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2020), match determines the stellar mass by in- tegrating over a stellar mass range of [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This lower IMF normalization has the effect of systematically in- creasing the estimated stellar mass compared with a Kroupa IMF with typical mass limits of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1–100 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Thus, we subtract 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='12 from log(M∗/M⊙) to account for the change in normalization, which lowers M∗ by 24%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Second, we apply a recycling fraction, R, that estimates the fraction of material returned to the ISM from stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We adopt a value of R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='41 from Vincenzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2016), based on a Kroupa IMF and the metallicity of Pegasus W, and reduce the stellar mass by a factor of 1 − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The final present-day stellar mass of Pega- sus W is found to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 × 104 M⊙ and is in good agreement with the present-day stellar mass recovered from the SFH fits using the PARSEC and MIST stel- lar libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' It is also in very good agreement with our first estimate based on the Monte Carlo simulations fol- lowing the procedure in Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We adopt an average of the two values and the largest uncertain- ties from both methods as our final value, yielding M∗ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 × 104 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' DISCUSSION AND SUMMARY Pegasus W is a very low-mass (M∗= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 × 104 M⊙), very faint (MV = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='16 mag) dwarf galaxy located on the far side of M31 at a distance of 915+60 −91 kpc measured from HB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The galaxy lies outside the virial radius of M31 and is relatively isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Shown in Figure 10, the size and luminosity of Pegasus W are consistent with the properties of other local low-mass galaxies, including UFDs, although Pegasus W is more compact than galaxies with similar luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Based on the properties of Pegasus W reported here (distance, luminosity, half-light radius), the galaxy is an UFD in the Local Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The best-fitting SFH derived from the CMD shows Pegasus W has formed 10% of its stellar mass within the last several Gyr, and suggests star-formation activ- ity as recently as the last Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' We also identify BHeB candidate stars in the CMD that are consistent with age estimates < 500 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This late-time star formation is unusual in an UFD galaxy and indicates that the galaxy recently harbored gas, either through gas retention or recent accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In addition to the evidence of late-time star forma- tion, the overall SFH of Pegasus W is extended: the galaxy formed 50% of its stellar mass after z ∼ 6 and 12 McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1 10 100 1000 rh (pc) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0 MV (mag) PegW Galaxies GCCs Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Distribution of galaxies and Galactic globular clusters (GGCs) in the MV − rh plane based on the updated compilation of galaxies from McConnachie (2012) and Baum- gardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Overlaid as a red star is the location of Pegasus W using our measured parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The galaxy is consistent with the properties of known local dwarfs with comparable luminosity, although slightly more compact, and is considered an UFD galaxy based on the definition from Simon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' quenched as recently as z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='9 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 Gyr ago).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Cos- mological simulations predict the mass threshold where reionization efficiently quenches galaxies to be M∗ ∼ 105 M⊙, possibly with an assist by stellar feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bovill & Ricotti 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Ben´ıtez-Llambay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Ro- driguez Wimberly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Garrison-Kimmel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Rey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Pereira Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Above this threshold mass, a galaxy’s potential should be deep enough to continue to accrete gas and the gas within a galaxy can self-shield sufficiently from the UV radia- tion to continue to cool and form stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' However, even though Pegasus W has a stellar mass below this thresh- old (M∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4 × 104 M⊙), the extended SFH is in- consistent with this expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Note that, while the stellar mass is directly observable, using M∗ as a mea- sure of an UFDs total potential has large uncertainties given the spread in the stellar mass-halo mass relation of UFDs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Munshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Applebaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' It’s possible that despite having M∗ < 105 M⊙, the halo mass of Pegasus W is large enough to be above the reionization threshold, although not significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Interestingly, there is evidence that some of the UFD satellites of M31 in the mass range 104 M⊙ < M∗ < 105 M⊙ also have somewhat extended SFHs and more recent quenching timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Initial results derived from data with a range of photometric depths (including And XVI with imaging reaching below the oMSTO) found quench- ing timescales similar to what we find for Pegasus W (Skillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Weisz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Deeper obser- vations for a subset of these systems place tighter con- straints on the SFHs and shift the quenching timescales to slightly older lookback times (Alessandro Savino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' in preparation), but the timescales are still generally more recent than the rapid quenching by reionization scenario described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The more recent quenching timescale for Pegasus W is in contrast to what has been derived for 6 UFDs of the MW in a slightly lower but overlapping mass range (6 × 103 M⊙ < M∗ < 5 × 104 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' These galaxies formed 80% of their stars by z ∼ 6 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='8 Gyr ago) and 100% of their stars by z ∼ 3 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='6 Gyr ago;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' While the rapid quenching of the MW UFDs is consistent with and often attributed to reionization, it is possible that environment played a role in quench- ing these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Based on proper motions of MW UFDs from Gaia and modeling the time-varying grav- itational potential of the MW due to its dark matter halo growth, there are indications that the SFHs of the UFDs were impacted by early environment effects from the MW halo (Miyoshi & Chiba 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' If this is the case, then the differences in quenching timescales for the MW UFDs and Pegasus W could be explained by differences in environment and orbital histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Indeed, rather than being quenched by reionization, it is more likely that Pegasus W was quenched by en- vironmental processing despite being outside the virial radius of M31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Based on the dwarf galaxy population around the MW and M31, there is a transition in the structural and dynamical properties of dwarfs at a dis- tance of ≳ 400 kpc from their host (corresponding to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='33×Rvir), which is thought to be due to tidal inter- actions (Higgs & McConnachie 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Simulations have shown that tidal effects can extend well beyond the virial radius (Rvir) of the main halo (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2014), and that more massive dwarf galaxies (M∗ ∼ 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 −108 M⊙) within 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 × Rvir of a more massive galaxy can be quenched by environment (Fillingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' It is also possible, even likely given Pegasus W’s current location at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='2 × Rvir, that the galaxy has previously passed through the halo of M31, making it a ‘backsplash’ galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Recent, extremely high-resolution cosmological simulations suggest that most galaxies within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5 × Rvir fall in the backsplash category (Applebaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In this scenario, the galaxy would have experienced more significant environmental processing including ram pres- sure and tidal stripping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Buck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Bla˜na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Such a fly-by interaction with M31 could have quenched the majority of star for- mation, but not completely stripped the Pegasus W of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' The remaining bound gas could have cooled over Gyr timescales, reigniting star formation and providing Pegasus W 13 an explanation for the possible late-time star formation suggested in the CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Note that the SFH of Pegasus W is based on a CMD that is relatively deep, reaching ∼ 2 mag below the HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' However, that depth still falls meaningfully short of the oMSTO depth needed to break the age-metallicity de- generacy in the CMD at old lookback times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Our overall results would be improved should such data be acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' In addition, spectroscopic observations of the stars in Pegasus W could provide important constraints on the internal kinematics, dark matter content of the galaxy, and its current radial trajectory within the Local Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Support for this work was provided by NASA through grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' HST-GO-16916 and support for Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' was partly provided by NASA through the NASA Hubble Fellowship grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' HST-HF2-51441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='001, both were awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Re- search in Astronomy, Incorporated, under NASA con- tract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' is also supported by NSF grant AST-1940800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' are supported by DOE grant DOE-SC0010008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' This research has made use of NASA Astrophysics Data System Bibliographic Services, adstex2, and the arXiv preprint server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 Facilities: Hubble Space Telescope Software: This research made use of DOLPHOT (Dol- phin 2000, 2016), MATCH (Dolphin 2002, 2012, 2013), HST drizzlepac (v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Hack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Avila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015), emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013), and Astropy,3 a community-developed core Python package for Astron- omy (The Astropy Collaboration 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' REFERENCES Akins, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Christensen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Brooks, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021, ApJ, 909, 139 Applebaum, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Brooks, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Christensen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021, ApJ, 906, 96 Avila, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Hack, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Cara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015, in Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 495, Astronomical Data Analysis Software an Systems XXIV (ADASS XXIV), ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Taylor & E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Rosolowsky, 281 Baumgardt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Sollima, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Hilker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020, PASA, 37, e046 Bechtol, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Drlica-Wagner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Balbinot, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015, ApJ, 807, 50 Behroozi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Wechsler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2014, ApJ, 787, 156 Ben´ıtez-Llambay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Navarro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Abadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015, MNRAS, 450, 4207 Benson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Frenk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Lacey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Baugh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Cole, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2002, MNRAS, 333, 177 Bla˜na, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Burkert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Fellhauer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Schartmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Alig, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020, MNRAS, 497, 3601 Bovill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Ricotti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2011, ApJ, 741, 18 Bressan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Marigo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Girardi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012, MNRAS, 427, 127 3 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='astropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='org Brown, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Tumlinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Geha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2014, ApJ, 796, 91 Buck, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Macci`o, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dutton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Obreja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Frings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019, MNRAS, 483, 1314 Bullock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Boylan-Kolchin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2017, ARA&A, 55, 343 Bullock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Kravtsov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Weinberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2000, ApJ, 539, 517 Carretta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Clementini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Fusi Pecci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2000, ApJ, 533, 215 Cerny, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Simon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022a, arXiv e-prints, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='11788 Cerny, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Mart´ınez-V´azquez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Drlica-Wagner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022b, arXiv e-prints, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='12422 Chambers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Magnier, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Metcalfe, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2016, arXiv e-prints, arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='05560 Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Girardi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Fu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019, A&A, 632, A105 Choi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dotter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Conroy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2016, ApJ, 823, 102 Collins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Charles, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Mart´ınez-Delgado, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022, MNRAS, 515, L72 Dey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Schlegel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Lang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019, AJ, 157, 168 Dolphin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2016, DOLPHOT: Stellar photometry, Astrophysics Source Code Library, record ascl:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='013, ascl:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='013 Dolphin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2000, PASP, 112, 1383 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2002, MNRAS, 332, 91 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012, ApJ, 751, 60 14 McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013, ApJ, 775, 76 Drlica-Wagner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bechtol, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Rykoff, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015, ApJ, 813, 109 Drlica-Wagner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bechtol, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Mau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020, ApJ, 893, 47 Fillingham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Cooper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Boylan-Kolchin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2018, MNRAS, 477, 4491 Ford, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bartko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bely, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1998, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 3356, Space Telescopes and Instruments V, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Bely & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Breckinridge, 234–248 Foreman-Mackey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Hogg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Lang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Goodman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013, PASP, 125, 306 Garrison-Kimmel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Wetzel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Hopkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019, MNRAS, 489, 4574 Geha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Blanton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Yan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Tinker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012, ApJ, 757, 85 Hack, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dencheva, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Fruchter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013, in Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 475, Astronomical Data Analysis Software and Systems XXII, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' Friedel, 49 Haynes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Giovanelli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Kent, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2018, ApJ, 861, 49 Hidalgo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Pietrinferni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Cassisi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2018, ApJ, 856, 125 Higgs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & McConnachie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021, MNRAS, 506, 2766 Kauffmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Guiderdoni, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1993, MNRAS, 264, 201 Koposov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Belokurov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Torrealba, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Evans, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015, ApJ, 805, 130 Kroupa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2001, MNRAS, 322, 231 Mapelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Ripamonti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Battaglia, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2009, MNRAS, 396, 1771 Mapelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Ripamonti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Tolstoy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2007, MNRAS, 380, 1127 Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', de Jong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Rix, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2008, ApJ, 684, 1075 Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Ibata, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Lewis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2016, ApJ, 833, 167 Mateo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Fischer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Krzeminski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1995, AJ, 110, 2166 McConnachie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012, AJ, 144, 4 McConnachie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Irwin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Ibata, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2009, Nature, 461, 66 McQuinn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Skillman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dalcanton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2011, ApJ, 740, 48 McQuinn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Skillman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Berg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013, AJ, 146, 145 McQuinn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Cannon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dolphin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2014, ApJ, 785, 3 McQuinn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Skillman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dolphin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015a, ApJ, 812, 158 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015b, ApJL, 815, L17 Miyoshi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Chiba, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020, ApJ, 905, 109 Momany, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Held, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Saviane, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2007, A&A, 468, 973 Monelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Cassisi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Mapelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012, ApJ, 744, 157 Moore, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Ghigna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Governato, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1999, ApJL, 524, L19 Munshi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Brooks, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Applebaum, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021, ApJ, 923, 35 Nadler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Drlica-Wagner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Bechtol, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021, PhRvL, 126, 091101 Okamoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Arimoto, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Yamada, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Onodera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012, ApJ, 744, 96 Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Simpson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Kravtsov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='13805 Pereira Wilson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Navarro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Santos Santos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Benitez Llambay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='05338 Rey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Pontzen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Agertz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020, MNRAS, 497, 1508 Rodriguez Wimberly, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Cooper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Fillingham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019, MNRAS, 483, 4031 Sand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Seth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Olszewski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2010, ApJ, 718, 530 Santana, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Mu˜noz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Geha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2013, ApJ, 774, 106 Savino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Weisz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Skillman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2022, ApJ, 938, 101 Schlafly, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Finkbeiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2011, ApJ, 737, 103 Schlegel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Finkbeiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Davis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 1998, ApJ, 500, 525 Simon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019, ARA&A, 57, 375 Sirianni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Jee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Ben´ıtez, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2005, PASP, 117, 1049 Skillman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Monelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Weisz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2017, ApJ, 837, 102 Somerville, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2002, ApJL, 572, L23 Telford, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dalcanton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Williams, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2020, ApJ, 891, 32 Teyssier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Johnston, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Kuhlen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2012, MNRAS, 426, 1808 The Astropy Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2018, astropy v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='1: a core python package for astronomy, Zenodo, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content='4080996 Pegasus W 15 Vincenzo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Matteucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Belfiore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Maiolino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2016, MNRAS, 455, 4183 Weisz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dolphin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2019, ApJL, 885, L8 Wetzel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', & Weisz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2015, ApJL, 808, L27 Williams, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Lang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dalcanton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2014, ApJS, 215, 9 Williams, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Durbin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Dalcanton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2021, ApJS, 253, 53 York, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Adelman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', Anderson, John E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} +page_content=' 2000, AJ, 120, 1579' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E2T4oBgHgl3EQf2Ain/content/2301.04157v1.pdf'} diff --git a/tNAzT4oBgHgl3EQfPft0/vector_store/index.faiss b/tNAzT4oBgHgl3EQfPft0/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6280135834ef3f1c92bef654c4c775393972f6c2 --- /dev/null +++ b/tNAzT4oBgHgl3EQfPft0/vector_store/index.faiss @@ -0,0 +1,3 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University of Science and Technology, Tehran, Iran +bLane Department of Computer Science and Electrical Engineering, West Virginia University, Morgantown, USA +A R T I C L E I N F O +Keywords: +Vision transformer +Robustness +Adversarial attacks +Traffic sign classification +A B S T R A C T +Vision transformers have been demonstrated to yield state-of-the-art results on a variety of computer +vision tasks using attention-based networks. However, research works in transformers mostly do not +investigate robustness/accuracy trade-off, and they still struggle to handle adversarial perturbations. +In this paper, we explore the robustness of vision transformers against adversarial perturbations and +try to enhance their robustness/accuracy trade-off in white box attack settings. To this end, we pro- +pose Locality iN Locality (LNL) transformer model. We prove that the locality introduction to LNL +contributes to the robustness performance since it aggregates local information such as lines, edges, +shapes, and even objects. In addition, to further improve the robustness performance, we encourage +LNL to extract training signal from the moments (a.k.a., mean and standard deviation) and the nor- +malized features. We validate the effectiveness and generality of LNL by achieving state-of-the-art +results in terms of accuracy and robustness metrics on German Traffic Sign Recognition Benchmark +(GTSRB) and Canadian Institute for Advanced Research (CIFAR-10). More specifically, for traffic +sign classification, the proposed LNL yields gains of 1.1% and 35% in terms of clean and robustness +accuracy compared to the state-of-the-art studies. +1. Introduction +Deep Neural Networks (DNNs) are widely deployed in +numerous computer vision applications, including object de- +tection [1–3], visual tracking [4, 5], object recognition [6], +and action recognition [7], yielding state-of-the-art perfor- +mance in a broad range of difficult tasks. Due to their widespread +success and ability to deploy in sensitive areas, these net- +works have now become the top choice for deployment in +real-world applications, including but not limited to autonomous +driving [8], recommender systems [9], health care [10], salient +object detection [11, 12], and defense-related applications [13, +14]. +Deep Neural Networks are susceptible to adversarial ex- +amples while these generated malicious perturbations are +hidden from human vision. Adversarial vulnerabilities have +raised concerns about security of computer vision systems, +which has led to a variety of studies on robustifying DNNs +and defense methods against such attacks. Defense meth- +ods try to strengthen the robustness of models in different +ways, e.g., carefully designed [15], stronger data augmenta- +tion [16–18], enhanced cost function [19], improved train- +ing strategy [20, 21], and better activation functions or pool- +ing [22], etc. Although these methods perform well on Con- +volutional Neural Networks, there is a lack of comparative +study to validate that they also keep the effectiveness on vi- +sion transformers. +Attention-based transformers have achieved great suc- +∗Corresponding author. +omid_nejaty@alumni.iust.ac.ir (O.N. Manzari); +hk00014@mix.wvu.edu (H. Kashiani); h_asgariandehkordi@elec.iust.ac.ir +(H.A. Dehkordi); bshokouhi@iust.ac.ir (S.B. Shokouhi) +ORCID(s): +cess in Natural Language Processing (NLP) and computer +vision tasks. Vision Transformer (ViT) is the first attention- +based image classification model proposed by Dosovitskiy +et al.[23]. On a variety of visual tasks [24–26], ViT models +have yielded state-of-the-art results by virtue of specific pre- +training phase. When trained with considerably large-scale +pre-train datasets like JFT-300M, the Vit models can out- +perform conventional Convolutional Neural Network (CNN) +based counterparts. The training is performed by processing +the image in patches. +Recently, different studies have demonstrated that vision +transformers could gain better robustness than state-of-the- +art CNNs with similar computational complexity [27, 28]. +However, vanilla vision transformers are vulnerable to ad- +versarial attacks, same as CNN. We aim to robustify ViT +models and maintain their state-of-art performance at the +same time. To this end, Locality iN Locality was proposed, +which is a robust transformer model. To be more specific, we +integrate locality to Feed-Forward Network (FFN) of Trans- +former iN Transformer (TNT) [29] by means of depth-wise +convolution [30–33] instate of multilayer perceptron. Since +locality could pertain a wide range of local structures such +as edge and shape of image feature, it would contribute to +higher robustness performance. In addition, an implicit data +augmentation method, called Moment Exchanger (MoEx), is +employed to make a better tradeoff between the robustness +and accuracy. Finally, the augmented LNL-MoEx model not +only improves the robustness accuracy, but also enhances the +clean accuracy in comparison with other vision transformer +studies. +To examine the adversarial robustness of the LNL model, +we gauge their performance in terms of the adversarial ro- +O.N Manzari et al. +Page 1 of 9 +arXiv:2301.11553v1 [cs.CV] 27 Jan 2023 + +aEngineering Science and Technology, an International Journal +bustness of vision transformers on traffic sign classification +task in white box attack setting shown in Figure 1. We be- +gin with an exhaustive set of experiments to compare the +performance of ViT model variants under different perturba- +tions. Various adversarial attack methods have been adopted +utilize to generate robust and imperceptible adversarial ex- +amples. In this work, two methods have been used, includ- +ing Fast Gradient Sign Method (FGSM) [34] and Projected +Gradient Descent (PGD) [35]. The proposed LNL is trained +from scratch on the German Traffic Sign Recognition Bench- +mark (GTSRB) [36] and Canadian Institute for Advanced +Research (CIFAR-10) [37] without ImageNet [38] pre-training +requirement. Our contributions are summarized as follows: +∙ The depth-wise convolutional filters are integrated in +the conventional FFN modules in the transformers (ViT) +to account locality principle and gauge its impact on +the accuracy/robustness trade-off. +∙ To further improve the robustness of our proposed method, +an implicit data augmentation method called MoEx is +introduce to encourage the model to utilize moment +information. +∙ The robustness of ViT models are investigated on GT- +SRB are measured in addition, and the experimental +evaluation demonstrates the LNL-MoEx performed bet- +ter than counterparts. +2. Related Works +Transformers have made significant contributions to the +area of NLP. Thanks to the self-attention module, Trans- +former can now properly capture the non-local interactions +between all different parts of the input sequence, resulting in +state-of-the-art performance on a wide range of NLP tasks +[39–44]. The Vision Transformer recently showed that trans- +formers could achieve state-of-the-art performance by pre- +training the model on massive data. To this end, the Trans- +formers sequence the input images into patches. ViT re- +quires a computationally expensive pre-training phase on a +larger dataset (such as ImageNet-21k [38]) because of the +lower amounts of inductive biases, as described in [23], to +achieve decent state-of-the-art performance. +Multiple Transformers models have been developed to +demonstrate that comparable performance may be achieved +without extra data. DeiT [26] created a transformer-specific +teacher-student technique and trained a transformer architec- +ture only on the ImageNet-1K dataset to relax the require- +ment of large-scale training dataset in the conventional trans- +formers. Simultaneously, T2T-ViT [45], Transformer iN Trans- +former (TNT) [29], and CvT [46] models have been devel- +oped to improve low level feature extractions and further re- +ducing their need on large-scale datasets. These models are +known as hybrid-ViTs. Further research is being done to +improve the efficiency and performance of transformer ar- +chitectures by enhancing the ViT architecture [47, 48]. Our +study achieves a high level of performance without a large- +scale training by using just GTSRB [36]. +0 +1 +2 +3 +4 +5 +6 +FLOPS(G) +90 +92 +94 +96 +98 +100 +Clean Accuracy(%) + PVT + TNT + T2T + RVT + LNL +0 +1 +2 +3 +4 +5 +6 +FLOPS(G) +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Robust Accuracy(%) + PVT + TNT + T2T + RVT + LNL +Figure 1: Comparison between LNL and ViT model variants. The +robust accuracy is tested on FGSM attack. +Concurrent to our work, several recent works analyze the +robustness of ViTs from different aspects. Early works focus +on the adversarial robustness of ViTs. They demonstrated +that ViTs are more robust to adversarial attacks than CNNs. +[49], and the adversarial transferability between CNNs and +ViTs is significantly low [50]. Follow-up studies [51–53] ex- +pand the robustness of ViT models to much common image +corruption and distribution shift and prove that ViTs are ro- +bust learners. Although several findings are consistent with +the above works, in this paper, we do not make a simple com- +parison of robustness between ViTs and CNNs. We exten- +sively compare the ViT variants architecture from an adver- +sarial robustness standpoint with white-box attack methods. +We design a robust vision transformer and introduce LNL- +MoEx to further reduce the fragility of ViT models against +adversarial attack. +3. METHODOLOGY +In this paper, we propose Locality iN Locality (LNL) +transformer architecture for visual recognition as illustrated +in Figure 2. Firstly, we give a brief overview of the primary +components of TNT in subsection 3.1. We then explain our +proposed Locally FeedForward and build our LNL by adding +locality mechanism into the FFN component of TNT. Fi- +nally, the Moment Exchanger (MoEx) implicit augmentation +is integrated into the proposed LNL model in subsection 3.2 +so that we can encourage the LNL-MoEx to utilize the mo- +ment information for enhancing robustness/accuracy trade- +off. +3.1. Transformer iN Transformer (TNT) +TNT splits a 2D image into 푛 patches 푋 = [푥1, 푥2, ..., 푥푛] ∈ +ℝ푛×푝×푝×3 uniformly, where (푝, 푝) is the resolution of each +image patch. In TNT, the patches are viewed as visual sen- +tences for representing the input images. Then each patch is +divided into 푚 sub-patches. That is to say, a visual sentence +is composed of a sequence of visual words. This operation +can be formulated as follows: +푋푖 → [푥푖,1, 푥푖,2, ..., 푥푖,푚], +(1) +where 푥(푖,푗) ∈ ℝ푠×푠×3 is the 푗 − 푡ℎ visual word of the 푖 − 푡ℎ +visual sentence, (푠, 푠) is the spatial size of each sub-patches, +O.N Manzari et al. +Page 2 of 9 + +Engineering Science and Technology, an International Journal +D +Linear Projection of Sentences and Words +1 +1 +4 +7 +2 +8 +3 +9 +5 +6 +2 +1 +4 +7 +2 +8 +3 +9 +5 +6 +9 +1 +4 +7 +2 +8 +3 +9 +5 +6 +0 + +* +Outer Transformer Block +Inner +Transformer +Block ++ +Inner +Transformer +Block ++ +Inner +Transformer +Block ++ +output +Patch Moment +Exchanger +Patch Moment +Exchanger +Patch Moment +Exchanger +… +…… +… +MSA +LFF +LayerNorm +LFF +3x3 DW Conv +Conv 1 X 1 +Seq2Img +Conv 1 X 1 +Img2Seq +MSA +Q +K +V +Avg_pool +Multi-Head +Self-Attention +Figure 2: Overall architecture of the proposed Locality iN Locality (LNL). Outer Transformer Block has the same structure as Inner +Transformer Block, which is composed of Locally-FeedForward (LFF), Multi-head Self-Attention (MSA), and LayerNorm. Pach Moment +Exchanger is our implicit data augmentation method that appears only in the training stage. +and 푗 = 1, 2, ..., 푚. With a linear projection, the visual words +푥푖,푗 are transformed into a sequence of word embedding as +follows: +푦푖,푗 = 퐹퐶(푉 퐸퐶(푥푖,푗)), +(2) +푌 푖 = [푦푖,1, 푦푖,2, ..., 푦푖,푚], +(3) +where 푦(푖,푗) ∈ ℝ푐 is the 푗 − 푡ℎ word embedding of the 푖 − +푡ℎ visual sentence, 푐 is the dimension of word embedding, +and 푉 퐸퐶(.) denotes the vectorization function. The TNT +method has two data flows across and inside the visual sen- +tences. More specifically, one data flow is employed for vi- +sual sentences and the other one is utilized for the visual +words inside each sentence. A transformer block is used +to investigate the relationship among visual words and their +embedding as follows: +푌 +′푖 +푙 = 푌 푖 +푙−1 + 푀푆퐴(퐿푁(푌 푖 +푙−1)), +(4) +푌 푖 +푙 = 푌 +′푖 +푙 + 푀퐿푃(퐿푁(푌 +′푖 +푙 )), +(5) +where the standard transformer blocks consist of a Multi- +head Self-Attention module (MSA), a Multiple Layer Per- +ceptron (MLP) and Layer Normalization (LN). In inner trans- +former blocks of TNT, 푙 = (1, 2, ..., 퐿) indicate the index +of the 푙 − 푡ℎ transformer block, and 퐿 indicates the overall +quantity of stacked blocks. After transformation, the word +embedding would be converted as 푌푙 = [푌 1 +푙 , 푌 2 +푙 , ..., 푌 푛 +푙 ] , +which is transformer block 푇푖푛. This can be viewed as an +inner transformer block, denoted as 푇푖푛 process. This op- +eration could account the connections among visual words. +The series of sentence are kept in the embedding memory at +the sentence level 푍0 = [푍푐푙푎푠푠, 푍1 +0, 푍2 +0, ..., 푍푛 +0] ∈ ℝ(푛+1)×푑 +In each layer, the word embedding are linear projected onto +the sentence embedding domain in each layer as follows: +푍푖 +푙−1 = 푍푖 +푙−1 + 퐹퐶(푉 퐸퐶(푌 푖 +푙 )), +(6) +where 푍푖 +푙−1 ∈ ℝ푑 and the FC layer ensures that the di- +mension is addable. With the above addition operation, Eq. +6 augments the representation of sentence embedding with +word-level features. The sentence embeddings are converted +by vanilla transformer block as follows: +푍 +′ +푙 = 푍푙−1 + 푀푆퐴(퐿푁(푍푙−1)), +(7) +푍푙 = 푍 +′ +푙 + 푀퐿푃 (퐿푁(푍 +′ +푙)). +(8) +In short, the visual word embeddings and sentence em- +beddings be expressed can be as follows: +푌푙, 푍푙 = 푇 푁푇 (푌푙−1, 푍푙−1). +(9) +In TNT, the inner transformer block represents word re- +lationships for local feature extraction, and the outside trans- +former block encodes intrinsic sentence-level context. The +TNT is built by stacking the TNT components 퐿 times. +3.2. LNL-MoEx +Locally-FeedForward Previous researches [46, 54] have +demonstrated that the FFN in the classical vision transformer +struggles to effectively model local dependencies. However, +neighboring pixels are valuable references to provide vital +information of local details in input images. To capture local +dependency, inspired by previews works [30, 31], a depth- +wise convolution block is integrated into FFN module in the +LNL model. A (푘 × 푘) convolution kernel is utilized in each +channel in the depth-wise convolution to properly incorpo- +rate locality into the LNL model. For features commutation, +O.N Manzari et al. +Page 3 of 9 + +Engineering Science and Technology, an International Journal +3x3 DW Conv +Conv 1 X 1 +Seq2Img +Conv 1 X 1 +Img2Seq +Conv 1 X 1 +Conv 1 X 1 +(b) +(a) +Figure 3: Block modifications of Feed Forward Network. (a) +simplified version of Feed Forward Network in TNT. For simplic- +ity of notation, we replace fully-connected layers by 1 × 1 convo- +lutions. (b) Locally-FeedForward Network, which is composed of +depth-wise convolution (DW) and 1 × 1 convolutions to inject local- +ity mechanism into FFN. +the feature inside the (푘 × 푘) kernel is integrated to compute +a new feature. To enhance the feature dimension for each +token, we first apply a linear projection layer. The tokens +are then reshaped into 2D feature maps; as a result, the lo- +cal information would captured by means of the depth-wise +convolution. The features are then flattened to reshape to- +kens, and the channels are shrunk down via another linear +layer to match the channel dimensions of input data. The +computation could be represented as: +퐿푂퐶(푍) = 퐼2푆((푆2퐼(푍) ⊙ 푊푑)). +(10) +Where 푊푑 ∈ ℝ훾푑×1×푘×푘 is the kernel of the depth-wise +convolution, ⊙ is a convolutional operation, 퐼2푆 is the func- +tion that converts an image into a sequence of tokens, and +푆2퐼 performs the reverse process. Figure 3 depicts the en- +tire proposed architecture. To introduce locality bias into the +TNT, the MLP layer (Eq. 8) is replaced by: +푍푙 = 푍 +′ +푙 + 퐿푂퐶(푍 +′ +푙). +(11) +MoEx After feeding the word-level features 푍 +′ +푙 to the +depth-wise convolution, the feature representation 푍푙 would +be a 2D tensor. MoEx integrates the feature normalization +into the data augmentation, and fuses features with labels +across two training samples. The normalized features of one +data sample are unified with the feature moments of another. +This nonsymmetric combination in feature space aims to en- +code various directions for the decision boundary. To nor- +malize features in the LNL model, a normalization function +푁 is defined. This function takes the word level features +푍푖 +푙 of the 푖 − 푡ℎ input 푥푖 at block 푙 of LNL model. Finally +function 푁 generates three outputs which include: the nor- +malized word-level features ̂푍푖 +푙, the first-moment 휇푖, and the +second moment 휎푖 as follows: +푁(푍푖 +푙) = ( ̂푍푖 +푙, 휇푖, 휎푖). +(12) +푁 calculates the value of the first and second momen- +tum after feeding the word level feature 푍 +′ +푙 through the local +module, i.e., depth-wise convolution. This operation rela- +tively resembles PONO function [55] in the realm of CNNs. +Along patch dimension, the normalized features pose zero- +mean and standard deviation one. To employ the LNL model, +the input images are taken and fused by means of calculated +moments. First image (푥퐴 ∶ 푍퐴 = 푁−1( ̂푍퐴, 휇퐴, 휎퐴)) takes +the moments of second image (푥퐵 ∶ 휇퐵, 휎퐵) as follows: +푍(퐵) +퐴 += 휎퐵 +푍퐴 − 휇퐴 +휎퐴 ++ 휇퐵, +(13) +where 푍퐴, 휇퐴, 휎퐴 are the word-level features, the first-moment, +and the second moment of image 퐴. In addition, 휇퐵, 휎퐵 are +the first-moment, and the second moment of image 퐵. We +now proceed with these features 푍(퐵) +퐴 . To emphasize on in- +jected features of 퐵, the loss function would be adjusted as +follows: +휆 ⋅ 퓁(푍(퐵) +퐴 , 푦퐴) + (1 − 휆) ⋅ 퓁(푍(퐵) +퐴 , 푦퐵), +(14) +where 휆 is a fixed variable for setting the combination +of the features and the moments 휆 ∈ (0, 1), and (푦퐴, 푦퐵) are +labels of images. Finally, the overall loss function would be +the straight-forward combination of the two losses. +4. EXPERIMENTAL RESULTS +4.1. Experimental Settings +Our experimental evaluations are carried out on the NVIDIA +Tesla P100 GPUs. The LNL is assessed under tiny and small +model sizes, which are coined LNL-Ti and LNL-S, respec- +tively. Performance of the proposed method is evaluated on +German Traffic Sign Recognition Benchmark (GTSRB) [36] +for focusing on object recognition tasks in autonomous vehi- +cles, and Canadian Institute for Advanced Research (CIFAR- +10) [37] dataset, as a well-known dataset in deep learning. +We also implement LNL-MoEx by adding implicit Moment +Changer augmentation into the our transformer blocks. +The models are all trained on the GTSRB dataset, which +contains 39,209 labeled traffic sign images in different 43 +classes. The images in the GTSRB dataset are divided into +35,209 training images, 4,000 validation images, and 12,630 +test images. +CIFAR-10 is a subset of the Tiny Images dataset and con- +sists of 60000 (32 x 32) color images. The images are di- +vided into 10 different categories including bird, automobile, +airplane, frog, deer, dog, cat, horse, truck, and ship. Train +and Test sets of this dataset contain 50000, and 10000 im- +ages, respectively. To be more specific, each class in CIFAR- +10 provides 5000 training images and 1000 test images. When +constructing the dataset, the test set was constructed by se- +lecting 1000 images randomly, and the rest of the images +were dedicated to the train set. +All ViT models are trained with the same hyperparame- +ters. The number of training epochs is set to 100 and 150 for +GTSRB and CIFAR-10, respectively. The training hyperpa- +rameters for the training phase are reported in Table 1. +O.N Manzari et al. +Page 4 of 9 + +Engineering Science and Technology, an International Journal +Table 1 +Training model parameters for GTSRB and CIFAR-10 +datasets. +Parameter +GTSRB +CIFAR-10 +Batch Size +50 +128 +Epoch +100 +150 +Learning Rate +0.007 +0.001 +Optimizer +SGD +SGD +4.2. LNLs are usable for small dataset +Table 2 demonstrates the results of the proposed LNL +compared to the state-of-the-art studies with respect to the +Top-1 and Top-5 robustness accuracy and computational com- +plexity. As reported in Table 2, the proposed LNL ranks first +in terms of clean accuracy and efficiency, and adversarial +robustness compared with the state-of-the-art works. To be +more specific, the proposed LNL-S yields gains of 0.3% and +0.5% on Top-5 metric compared to the second [53] and third +[56] best methods. While the obtained gains in comparison +with the second-best approach [45] is not pronounced, our +superiority in terms of Top-1 metric is noticeable with a ac- +curacy of 98.2%. This is due to the fact that the results with +respect to the Top-5 metric consider higher probability out- +puts. +Table 3 reports the comparison results on CIFAR-10. +The results show that our LNL model significantly outper- +forms the state-of-the-art methods in terms of clean accu- +racy. Specifically, we achieve accuracies of 97.6% and 98.9% +on the validation set using LNL-Ti and LNL-S, respectively. +Comparing clean accuracies of GTSRB and CIFAR-10 datasets +in Table 2 and Table 3 show that traffic sign images are more +difficult to predict the correct label. However, it can be seen +LNL achieves the best performance on both datasets using +fewer or comparable model complexity. These results demon- +strate the strong classification ability of the proposed model. +It is worth highlighting that all the counterpart model vi- +sion transformers are pre-trained in the ImageNet [38] dataset, +while our proposed LNL models are only trained on CIFAR- +10 and GTSBR datasets from scratch without ImageNet pre- +training requirement. +4.3. Adversarial Robustness +To evaluate robustness against adversarial attacks, we +adopt single-step attack algorithm FGSM [34] and multi- +step attack algorithm PGD [35] with step number 푡 = 5, +and step size 훼 = 0 ∶ 5. The input image is perturbed +by both attackers with a maximum magnitude of 1. Results +in Table 2 demonstrate that the adversarial robustness dra- +matically impacts on the LNL architecture with similar com- +putational complexity, the proposed LNL models represent +high adversarial robustness under adversarial attacks. This +is ascribed to the proposed modifications on LNLs, which +aims to strengthen the adversarial robustness. More specifi- +cally, the depth-wise convolution introduces locality induc- +tive Bias into FFN to model local dependencies and intro- +duce inductive bias in favor of better adversarial robustness. +Moreover, the MoEx module utilizes implicit data augmen- +tation, which is helpful for adversarial robustness. +Table 2 and Table 3 show that the proposed LNL model +achieves superior performance on both admired FGSM and +PGD attacks in compared state-of-the-art models. In detail, +LNL-Ti and LNL-S considerably outperforms the state-of- +the-art models with a gain of 32.4% on FGSM attack and a +gain of 34.7% on PGD attack compared to its counterpart vi- +sion transformers on GTSRB dataset. Our model similarly +outperforms prior arts by a large margin on both adversar- +ial benchmarks for the CIFAR-10 dataset. Nonetheless, our +LNL model generally yields the best accuracy/robustness +trade-off. +This advance is further expanded by our MoEx augmen- +tation. Table 2 additionally shows that LNL-MoEx improves +both the accuracy and robustness. Compared to other aug- +mentation methods, when LNL-Ti is trained using MoEx +achieves the top-1 accuracy of 98.6% and robust accuracy of +71.3%, 55.4% on FGSM and PGD benchmarks. Notably, the +highest robust accuracy is obtained by LNL-MoEx-S with +77.8% on the FGSM. Our enhanced model with MoEx aug- +mentation outperforms other methods by significant improve- +ments on multiple standard benchmarks. +4.4. Visualizing Attention Maps +To present qualitative result comparisons to TNT as base- +line for our improved model, we compare the attention maps +generated by the TNT [29] and the LNL using CAM [58]. +The attention maps were extracted using a visualization pro- +cedure inspired by Caron et al. [59]. In Table 4, the first +column shows the input images of traffic signs from GTSRB +[36]. The second and third columns represent the attention +maps of the clean images. In contrast, the fourth and fifth +columns represent the attention maps of the adversarial im- +ages generated by FGSM [34] under the same settings in +subsection 4.3. Furthermore, the predicted labels are shown +with confidence scores for each sample. +From the attention map, we can see TNT just gives at- +tention to widespread features, such as in all cases, it focuses +on the center of traffic signs, whereas LNL diverts the atten- +tion to more diverse and significant features. For example, +in the case of ‘no passing’, ‘road work’ and ‘Roundabout +mandatory’ LNL focuses on the frame, color, and shape of +traffic signs, respectively, which are important classification +features. We can also see from Table 4 that the LNL has +strong background separability, and it does not concentrate +on trivial background features. It is clear that LNL has the +capability to track the local features of the input image while +TNT tracks global and centered features. +We can see from the adversarial part that the perturba- +tions highly target the main object of images, the ‘Round- +about mandatory’ for both TNT and LNL. Moreover, the +perturbation effect can be noticed on the attention maps. For +instance, the FGSM fools the TNT to mislabel ‘Road work’ +instate of ‘No passing’ with high confidence. In contrast, +the LNL predicts labels with high confidence in almost 3/4 +adversarial traffic signs. +O.N Manzari et al. +Page 5 of 9 + +Engineering Science and Technology, an International Journal +Table 2 +The performance (%) of LNL and Transformers on GTSRB and two robustness benchmarks. In Transformers part, models +are trained without any extra modifications. In Augmentations part, we trained LNL-Ti model with our proposed MoEx and +some augmentation methods such as CutMix. +Group +Model +Model Complexity +GTSRB +Robustness Benchmarks +FLOPs (G) +Params (M) +Top-1 +Top-5 +FGSM +PGD +Transformers +PVT-Tiny [56] +1.3 +12.7 +96.2 +99.0 +10.8 +2.0 +TNT-T [29] +1.2 +5.9 +91.9 +98.4 +10.9 +3.9 +T2T-ViT-t-10 [45] +1.3 +5.6 +97.8 +99.7 +10.2 +0.4 +RVT-Ti [53] +1.2 +8.6 +96.1 +99.1 +31.5 +13.9 +LNL-Ti +1.2 +6.1 +97.9 +99.7 +57.7 +37.9 +Swin-T [47] +4.1 +28.5 +96.8 +99.1 +20.2 +7.8 +PVT-Small [56] +3.6 +24.0 +96.8 +99.3 +21.4 +2.4 +TNT-S [29] +4.8 +23.4 +97.1 +98.9 +21.2 +6.8 +T2T-ViT-t-14 [45] +4.3 +21.1 +98.0 +99.2 +21.7 +4.9 +RVT-S [53] +4.3 +21.9 +97.2 +99.5 +46.1 +25.3 +LNL-S +4.1 +23.8 +98.2 +99.8 +64.5 +45.7 +Augmentations +DeepAugment [16] +1.2 +6.1 +97.6 +99.5 +65.1 +42.2 +CutMix [18] +1.2 +6.1 +98.5 +99.5 +62.6 +40.1 +AugMix [17] +1.2 +6.1 +98.3 +99.4 +61.7 +39.8 +Puzzle-Mix [57] +1.2 +6.1 +98.5 +99.8 +67.9 +44.2 +RVT-Ti* [53] +1.2 +10.6 +96.9 +99.4 +38.9 +20.7 +LNL-MoEx-Ti +1.2 +6.1 +98.6 +99.7 +71.3 +55.4 +RVT-S* [53] +4.3 +23.0 +97.7 +99.6 +51.2 +32.9 +LNL-MoEx-S +4.1 +23.8 +99.1 +99.9 +77.8 +59.4 +Table 3 +The performance (%) of LNL and Transformers on CIFAR-10 +and two robustness benchmarks. +Model +Top-1. Acc +Robustness Benchmarks +FGSM +PGD +PVT-Tiny [56] +94.2 +16.6 +3.1 +TNT-T [29] +94.9 +13.2 +4.2 +T2T-ViT-t-10 [45] +95.3 +10.9 +1.4 +RVT-Ti [53] +95.1 +34.3 +14.2 +LNL-Ti +97.6 +60.8 +38.2 +Swin-T [47] +96.6 +29.2 +11.3 +PVT-Small [56] +96.8 +23.1 +7.3 +TNT-S [29] +98.7 +29.3 +11.3 +T2T-ViT-t-14 [45] +97.5 +26.7 +17.5 +RVT-S [53] +97.4 +46.3 +25.9 +LNL-S +98.9 +69.0 +46.9 +4.5. Ablation Study +To understand our LNL architecture better, we ablate +each critical design by evaluating its performance on GT- +SRB classification and adversarial benchmarks. Firstly, a +study is conducted to demonstrate how the Locally FeedFor- +ward could influence the performance of other vision trans- +formers. We also evaluate the performance of our implicit +Moment Exchanger augmentation in this section. +Impact of Locally FeedForward. To show the effec- +tiveness of our proposed Locally FeedForward, we intro- +duced local information into the FeedForward network of +some transformer architectures. T2T-Ti, PVT-Ti, and Swin- +Ti are chosen as the base model. Table 5 illustrates the ex- +perimental results of enhanced models by replacing the base +FFN module with the proposed Locally FeedForward, all the +enhanced models yield significant improvements. Specifi- +cally, all the enhanced models achieve more than 2.2% and +1.4% promotion on robust and standard accuracy on average. +The greatest improvement is for Local-T2T, where the Lo- +cally FeedForward leads to a gain of 23% in robust accuracy. +Compared with the base model, there is only a negligible in- +crease in the amount of computation and a marginal increase +in the number of parameters. +Impact of Moment Exchanger augmentation. We fur- +ther study the impact of MoEx augmentation on multiple +transformer blocks. As shown in Table 6, we can see our +augmentation improves standard and robust accuracy of all +models. Among them, PVT∗ and T2T∗ achieve significant +improvements of 8.0% and 1.1% on robust and standard ac- +curacy compared to the baselines. +5. CONCLUSION +We propose the LNL model and study the robustness of +vision transformers on traffic sign classification task. The +proposed model relaxes the requirement of large-scale pre- +training phase in the conventional vision transformer and at +the same time outperforms the state-of-the-art transformer- +based studies in relation to the clean and adversarial robust- +ness. Furthermore, we integrate the MoEx data augmen- +tation into the vanilla vision transformers to improve ad- +O.N Manzari et al. +Page 6 of 9 + +Engineering Science and Technology, an International Journal +Table 4 +Comparison of the attention maps generated by the LNL model and the TNT.(first column) input images; (second and +third columns) attention maps generated by models with clean images; and (fourth and fifth columns) attention map +generated by models with adversarial images. +Input Image +No Attack +FGSM Attack +TNT [29] +LNL +TNT [29] +LNL +Label: ‘over 3.5 tons prohibited’ +‘over 3.5 tons prohibited’ 99% +‘over 3.5 tons prohibited’ 99% +‘No passing’ 77% +‘over 3.5 tons prohibited’ 79% +Label: ‘Roundabout mandatory’ +‘Roundabout mandatory’ 99% +‘Roundabout mandatory’ 99% +‘Keep right’ 99% +‘Roundabout mandatory’ 25% +Label: ‘No passing’ +‘No passing’ 99% +‘No passing’ 99% +‘Road work’ 100% +‘No passing’ 98% +Label: ‘Road work’ +‘Road work’ 99% +‘Road work’ 99% +‘Priority road’ 99% +‘Road work’ 84% +versarial robustness. Instead of disregarding the moments +extracted by the normalization layer, the MoEx data aug- +mentation forces the neural network to pay attention towards +robust feature. Experimental evaluations approves that the +proposed LNL-MoEx consistently achieves outstanding per- +formance on the GTSRB dataset in terms of adversarial ro- +bustness and performance clean accuracy. In short, concern- +ing the trade-off between FLOPs, clean and robustness ac- +curacy, extensive experiments validate the superiority of our +LNL-MoEx-Ti and LNL-MoEx-S. +CRediT authorship contribution statement +Omid Nejati Manzari: Conceptualization, Software, +Writing- Original draft preparation, Validation, Resources. +Hossein Kashiani: Methodology, Writing- Reviewing and +Editing. Hojat Asgarian Dehkordi: Modification for the +final layout. Shahriar Baradaran Shokouhi: Supervision, +Review & Editing. +Declaration of competing interest +The authors declare that they have no known competing +financial interests or personal relationships that could have +appeared to influence the work reported in this paper. +O.N Manzari et al. +Page 7 of 9 + +3Engineering Science and Technology, an International Journal +Table 5 +Effect of Locally FeedForward on other ViT architectures. Per- +formance (%) of clean accuracy and adversarial robustness under +FGSM attack on GTSRB. +Network +Params +FLOPs +Acc +Rob. Acc +(M) +(G) +T2T-Ti +5.9 +1.2 +97.8 +10.2 +Local-T2T +5.9 +1.2 +99.2 (1.4↑) +33.3 (23.1↑) +PVT-Ti +12.7 +1.3 +96.2 +10.8 +Local-PVT +13.0 +1.4 +99.1 (2.9↑) +13.0 (2.2↑) +Swin-Ti +28.5 +4.1 +96.8 +20.2 +Local-Swin +28.8 +4.1 +98.5 (1.7↑) +26.8 (6.6↑) +Table 6 +Effect of our Patch Moment Exchanger augmentation on other +ViT architectures. Performance (%) of clean accuracy and adver- +sarial robustness under FGSM attack on GTSRB. Best Results are +in bold face. +Vanilla +Acc +Rob. Acc +Improved +Acc +Rob. Acc +models +models +T2T-Ti +97.8 +10.2 +T2T-Ti* +98.1 +16.5 +PVT-Ti +96.2 +10.8 +PVT-Ti* +96.7 +18.8 +Swin-Ti +96.8 +20.2 +Swin-Ti* +97.9 +25.7 +Acknowledgment +we would like to thank the editors and anonymous reviewers +for providing insightful suggestions and comments to improve +the quality of research paper. +References +[1] J. Zhang, Z. Xie, J. Sun, X. Zou, and J. Wang, “A cascaded r-cnn +with multiscale attention and imbalanced samples for traffic sign +detection,” IEEE access, vol. 8, pp. 29 742–29 754, 2020. +[2] J. Zhang, X. Zou, L.-D. Kuang, J. Wang, R. S. Sherratt, and +X. Yu, “Cctsdb 2021: a more comprehensive traffic sign detection +benchmark,” Human-centric Computing and Information Sciences, +vol. 12, 2022. +[3] O. N. Manzari, A. Boudesh, and S. B. Shokouhi, “Pyramid +transformer +for +traffic +sign +detection,” +arXiv +preprint +arXiv:2207.06067, 2022. +[4] J. Zhang, J. Sun, J. Wang, Z. Li, and X. Chen, “An object tracking +framework with recapture based on correlation filters and siamese +networks,” Computers & Electrical Engineering, vol. 98, p. 107730, +2022. +[5] J. Zhang, W. Feng, T. Yuan, J. Wang, and A. K. Sangaiah, “Scstcf: +spatial-channel selection and temporal regularized correlation filters +for visual tracking,” Applied Soft Computing, vol. 118, p. 108485, +2022. +[6] A. Tourani, A. Shahbahrami, S. Soroori, S. Khazaee, and C. Y. +Suen, “A robust deep learning approach for automatic iranian vehicle +license plate detection and recognition for surveillance systems,” +IEEE Access, vol. 8, pp. 201 317–201 330, 2020. +[7] H. A. Dehkordi, A. S. Nezhad, S. S. Ashrafi, and S. B. Shokouhi, +“Still image action recognition using ensemble learning,” in 2021 7th +International Conference on Web Research (ICWR). +IEEE, 2021, +pp. 125–129. +[8] H. Asgarian, A. Amirkhani, and S. B. Shokouhi, “Fast drivable area +detection for autonomous driving with deep learning,” in 2021 5th +International Conference on Pattern Recognition and Image Analysis +(IPRIA). +IEEE, 2021, pp. 1–6. +[9] J. Wang, Y. Gao, X. Yin, F. Li, and H.-J. Kim, “An enhanced pegasis +algorithm with mobile sink support for wireless sensor networks,” +Wireless Communications and Mobile Computing, vol. 2018, 2018. +[10] H. A. Dehkordi, H. Kashiani, A. A. H. Imani, and S. B. Shokouhi, +“Lightweight local transformer for covid-19 detection using chest +ct scans,” in 2021 11th International Conference on Computer +Engineering and Knowledge (ICCKE). +IEEE, 2021, pp. 328–333. +[11] S. Mohammadi, M. Noori, A. Bahri, S. G. Majelan, and M. Havaei, +“Cagnet: +Content-aware guidance for salient object detection,” +Pattern Recognition, vol. 103, p. 107303, 2020. +[12] M. Noori, S. Mohammadi, S. G. Majelan, A. Bahri, and M. Havaei, +“Dfnet: Discriminative feature extraction and integration network +for salient object detection,” Engineering Applications of Artificial +Intelligence, vol. 89, p. 103419, 2020. +[13] O. N. Manzari and S. B. Shokouhi, “A robust network for embedded +traffic sign recognition,” in 2021 11th International Conference on +Computer Engineering and Knowledge (ICCKE). +IEEE, 2021, pp. +447–451. +[14] A. Tourani, +S. Soroori, +A. Shahbahrami, +S. Khazaee, +and +A. Akoushideh, “A robust vehicle detection approach based on faster +r-cnn algorithm,” in 2019 4th International Conference on Pattern +Recognition and Image Analysis (IPRIA). IEEE, 2019, pp. 119–123. +[15] B. Wu, J. Chen, D. Cai, X. He, and Q. Gu, “Do wider neural networks +really help adversarial robustness?” Advances in Neural Information +Processing Systems, vol. 34, pp. 7054–7067, 2021. +[16] D. Hendrycks, S. Basart, N. Mu, S. Kadavath, F. Wang, E. Dorundo, +R. Desai, T. Zhu, S. Parajuli, M. Guo et al., “The many faces of +robustness: A critical analysis of out-of-distribution generalization,” +in Proceedings of the IEEE/CVF International Conference on +Computer Vision, 2021, pp. 8340–8349. +[17] D. Hendrycks, +N. Mu, +E. D. Cubuk, +B. Zoph, +J. Gilmer, +and B. Lakshminarayanan, “Augmix: +A simple data processing +method to improve robustness and uncertainty,” arXiv preprint +arXiv:1912.02781, 2019. +[18] S. Yun, D. Han, S. J. Oh, S. Chun, J. Choe, and Y. Yoo, “Cutmix: +Regularization strategy to train strong classifiers with localizable +features,” in Proceedings of the IEEE/CVF international conference +on computer vision, 2019, pp. 6023–6032. +[19] A. S. Hashemi, S. Mozaffari, and S. Alirezaee, “Improving adversarial +robustness of traffic sign image recognition networks,” Displays, +vol. 74, p. 102277, 2022. +[20] A. +S. +Hashemi +and +S. +Mozaffari, +“Cnn +adversarial +attack +mitigation using perturbed samples training,” Multimedia Tools +and Applications, vol. 80, no. 14, pp. 22 077–22 095, 2021. +[21] ——, +“Secure deep neural networks using adversarial image +generation and training with noise-gan,” Computers & Security, +vol. 86, pp. 372–387, 2019. +[22] R. Zhang, “Making convolutional networks shift-invariant again,” in +International conference on machine learning. +PMLR, 2019, pp. +7324–7334. +[23] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, +T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly et al., +“An image is worth 16x16 words: Transformers for image recognition +at scale,” arXiv preprint arXiv:2010.11929, 2020. +[24] M. Zhu, K. Han, Y. Tang, and Y. Wang, “Visual transformer pruning,” +arXiv preprint arXiv:2104.08500, 2021. +[25] Y. Jiang, S. Chang, and Z. Wang, “Transgan: Two transformers can +make one strong gan,” arXiv preprint arXiv:2102.07074, vol. 1, no. 3, +2021. +[26] H. Touvron, M. Cord, M. Douze, F. Massa, A. Sablayrolles, and +H. Jégou, “Training data-efficient image transformers & distillation +through attention,” in International Conference on Machine Learning. +PMLR, 2021, pp. 10 347–10 357. +[27] P. Benz, S. Ham, C. Zhang, A. Karjauv, and I. S. Kweon, “Adversarial +robustness comparison of vision transformer and mlp-mixer to cnns,” +O.N Manzari et al. +Page 8 of 9 + +Engineering Science and Technology, an International Journal +arXiv preprint arXiv:2110.02797, 2021. +[28] Y. Bai, J. Mei, A. L. Yuille, and C. Xie, “Are transformers more robust +than cnns?” Advances in Neural Information Processing Systems, +vol. 34, 2021. +[29] K. Han, A. Xiao, E. Wu, J. Guo, C. Xu, and Y. Wang, “Transformer +in transformer,” arXiv preprint arXiv:2103.00112, 2021. +[30] Y. Li, +K. Zhang, +J. Cao, +R. Timofte, +and L. Van Gool, +“Localvit: Bringing locality to vision transformers,” arXiv preprint +arXiv:2104.05707, 2021. +[31] Z. Wang, X. Cun, J. Bao, and J. Liu, “Uformer: A general u-shaped +transformer for image restoration,” arXiv preprint arXiv:2106.03106, +2021. +[32] Q. Zhang and Y. Yang, “Rest: An efficient transformer for visual +recognition,” arXiv preprint arXiv:2105.13677, 2021. +[33] Q. Yu, Y. Xia, Y. Bai, Y. Lu, A. Yuille, and W. Shen, “Glance-and- +gaze vision transformer,” arXiv preprint arXiv:2106.02277, 2021. +[34] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and +harnessing adversarial examples,” arXiv preprint arXiv:1412.6572, +2014. +[35] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, +“Towards deep learning models resistant to adversarial attacks,” arXiv +preprint arXiv:1706.06083, 2017. +[36] J. Stallkamp, M. Schlipsing, J. Salmen, and C. Igel, “The german +traffic sign recognition benchmark: +a multi-class classification +competition,” in The 2011 international joint conference on neural +networks. +IEEE, 2011, pp. 1453–1460. +[37] A. Krizhevsky, G. Hinton et al., “Learning multiple layers of features +from tiny images,” 2009. +[38] A. Krizhevsky, +I. Sutskever, +and G. E. Hinton, +“Imagenet +classification with deep convolutional neural networks,” Advances in +neural information processing systems, vol. 25, pp. 1097–1105, 2012. +[39] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, +A. Neelakantan, P. Shyam, G. Sastry, A. Askell et al., “Language +models are few-shot learners,” arXiv preprint arXiv:2005.14165, +2020. +[40] Z. Dai, Z. Yang, Y. Yang, J. Carbonell, Q. V. Le, and R. Salakhutdinov, +“Transformer-xl: Attentive language models beyond a fixed-length +context,” arXiv preprint arXiv:1901.02860, 2019. +[41] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: +Pre-training +of +deep +bidirectional +transformers +for +language +understanding,” arXiv preprint arXiv:1810.04805, 2018. +[42] A. Radford, +K. Narasimhan, +T. Salimans, +and I. Sutskever, +“Improving language understanding by generative pre-training,” +2018. +[43] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever et al., +“Language models are unsupervised multitask learners,” OpenAI +blog, vol. 1, no. 8, p. 9, 2019. +[44] Z. Yang, Z. Dai, Y. Yang, J. Carbonell, R. R. Salakhutdinov, and +Q. V. Le, “Xlnet: Generalized autoregressive pretraining for language +understanding,” Advances in neural information processing systems, +vol. 32, 2019. +[45] L. Yuan, Y. Chen, T. Wang, W. Yu, Y. Shi, Z. Jiang, F. E. Tay, J. Feng, +and S. Yan, “Tokens-to-token vit: Training vision transformers from +scratch on imagenet,” arXiv preprint arXiv:2101.11986, 2021. +[46] H. Wu, B. Xiao, N. Codella, M. Liu, X. Dai, L. Yuan, and +L. Zhang, “Cvt: Introducing convolutions to vision transformers,” +arXiv preprint arXiv:2103.15808, 2021. +[47] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and B. Guo, +“Swin transformer: Hierarchical vision transformer using shifted +windows,” arXiv preprint arXiv:2103.14030, 2021. +[48] X. Chu, Z. Tian, B. Zhang, X. Wang, X. Wei, H. Xia, and C. Shen, +“Conditional positional encodings for vision transformers,” arXiv +preprint arXiv:2102.10882, 2021. +[49] R. Shao, Z. Shi, J. Yi, P.-Y. Chen, and C.-J. Hsieh, “On the adversarial +robustness of visual transformers,” arXiv preprint arXiv:2103.15670, +2021. +[50] K. Mahmood, R. Mahmood, and M. Van Dijk, “On the robustness +of vision transformers to adversarial examples,” arXiv preprint +arXiv:2104.02610, 2021. +[51] S. Bhojanapalli, A. Chakrabarti, D. Glasner, D. Li, T. Unterthiner, +and A. Veit, “Understanding robustness of transformers for image +classification,” arXiv preprint arXiv:2103.14586, 2021. +[52] S. Paul and P.-Y. Chen, “Vision transformers are robust learners,” +arXiv preprint arXiv:2105.07581, 2021. +[53] X. Mao, G. Qi, Y. Chen, X. Li, R. Duan, S. Ye, Y. He, +and H. Xue, “Towards robust vision transformer,” arXiv preprint +arXiv:2105.07926, 2021. +[54] K. Yuan, +S. Guo, +Z. Liu, +A. Zhou, +F. Yu, +and W. Wu, +“Incorporating convolution designs into visual transformers,” arXiv +preprint arXiv:2103.11816, 2021. +[55] B. Li, F. Wu, S.-N. Lim, S. Belongie, and K. Q. Weinberger, “On +feature normalization and data augmentation,” in Proceedings of the +IEEE/CVF Conference on Computer Vision and Pattern Recognition, +2021, pp. 12 383–12 392. +[56] W. Wang, E. Xie, X. Li, D.-P. Fan, K. Song, D. Liang, T. Lu, +P. Luo, and L. Shao, “Pyramid vision transformer: +A versatile +backbone for dense prediction without convolutions,” arXiv preprint +arXiv:2102.12122, 2021. +[57] J.-H. Kim, W. Choo, and H. O. Song, “Puzzle mix: Exploiting +saliency and local statistics for optimal mixup,” in International +Conference on Machine Learning. +PMLR, 2020, pp. 5275–5285. +[58] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba, +“Learning +deep +features +for +discriminative +localization,” +in +Proceedings of the IEEE conference on computer vision and +pattern recognition, 2016, pp. 2921–2929. +[59] M. Caron, H. Touvron, I. Misra, H. Jégou, J. Mairal, P. Bojanowski, +and A. Joulin, “Emerging properties in self-supervised vision +transformers,” arXiv preprint arXiv:2104.14294, 2021. +O.N Manzari et al. +Page 9 of 9 + diff --git a/u9FJT4oBgHgl3EQfeiys/content/tmp_files/load_file.txt b/u9FJT4oBgHgl3EQfeiys/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbb9126dc54e49e3e3da422a2b8e2029d146f652 --- /dev/null +++ b/u9FJT4oBgHgl3EQfeiys/content/tmp_files/load_file.txt @@ -0,0 +1,1009 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf,len=1008 +page_content='Robust Transformer with Locality Inductive Bias and Feature Normalization Omid Nejati Manzaria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hossein Kashianib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hojat Asgarian Dehkordia and Shahriar Baradaran Shokouhia aSchool of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Iran University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Iran bLane Department of Computer Science and Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' West Virginia University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Morgantown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' USA A R T I C L E I N F O Keywords: Vision transformer Robustness Adversarial attacks Traffic sign classification A B S T R A C T Vision transformers have been demonstrated to yield state-of-the-art results on a variety of computer vision tasks using attention-based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' However, research works in transformers mostly do not investigate robustness/accuracy trade-off, and they still struggle to handle adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In this paper, we explore the robustness of vision transformers against adversarial perturbations and try to enhance their robustness/accuracy trade-off in white box attack settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To this end, we pro- pose Locality iN Locality (LNL) transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We prove that the locality introduction to LNL contributes to the robustness performance since it aggregates local information such as lines, edges, shapes, and even objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In addition, to further improve the robustness performance, we encourage LNL to extract training signal from the moments (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', mean and standard deviation) and the nor- malized features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We validate the effectiveness and generality of LNL by achieving state-of-the-art results in terms of accuracy and robustness metrics on German Traffic Sign Recognition Benchmark (GTSRB) and Canadian Institute for Advanced Research (CIFAR-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' More specifically, for traffic sign classification, the proposed LNL yields gains of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1% and 35% in terms of clean and robustness accuracy compared to the state-of-the-art studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Introduction Deep Neural Networks (DNNs) are widely deployed in numerous computer vision applications, including object de- tection [1–3], visual tracking [4, 5], object recognition [6], and action recognition [7], yielding state-of-the-art perfor- mance in a broad range of difficult tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Due to their widespread success and ability to deploy in sensitive areas, these net- works have now become the top choice for deployment in real-world applications, including but not limited to autonomous driving [8], recommender systems [9], health care [10], salient object detection [11, 12], and defense-related applications [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Deep Neural Networks are susceptible to adversarial ex- amples while these generated malicious perturbations are hidden from human vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Adversarial vulnerabilities have raised concerns about security of computer vision systems, which has led to a variety of studies on robustifying DNNs and defense methods against such attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Defense meth- ods try to strengthen the robustness of models in different ways, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', carefully designed [15], stronger data augmenta- tion [16–18], enhanced cost function [19], improved train- ing strategy [20, 21], and better activation functions or pool- ing [22], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Although these methods perform well on Con- volutional Neural Networks, there is a lack of comparative study to validate that they also keep the effectiveness on vi- sion transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Attention-based transformers have achieved great suc- ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' omid_nejaty@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='iust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='ir (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Manzari);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' hk00014@mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='wvu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='edu (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kashiani);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' h_asgariandehkordi@elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='iust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='ir (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dehkordi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' bshokouhi@iust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='ir (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shokouhi) ORCID(s): cess in Natural Language Processing (NLP) and computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Vision Transformer (ViT) is the first attention- based image classification model proposed by Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' On a variety of visual tasks [24–26], ViT models have yielded state-of-the-art results by virtue of specific pre- training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' When trained with considerably large-scale pre-train datasets like JFT-300M, the Vit models can out- perform conventional Convolutional Neural Network (CNN) based counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The training is performed by processing the image in patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Recently, different studies have demonstrated that vision transformers could gain better robustness than state-of-the- art CNNs with similar computational complexity [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' However, vanilla vision transformers are vulnerable to ad- versarial attacks, same as CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We aim to robustify ViT models and maintain their state-of-art performance at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To this end, Locality iN Locality was proposed, which is a robust transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To be more specific, we integrate locality to Feed-Forward Network (FFN) of Trans- former iN Transformer (TNT) [29] by means of depth-wise convolution [30–33] instate of multilayer perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Since locality could pertain a wide range of local structures such as edge and shape of image feature, it would contribute to higher robustness performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In addition, an implicit data augmentation method, called Moment Exchanger (MoEx), is employed to make a better tradeoff between the robustness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Finally, the augmented LNL-MoEx model not only improves the robustness accuracy, but also enhances the clean accuracy in comparison with other vision transformer studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To examine the adversarial robustness of the LNL model, we gauge their performance in terms of the adversarial ro- O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 1 of 9 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='11553v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='CV] 27 Jan 2023 aEngineering Science and Technology, an International Journal bustness of vision transformers on traffic sign classification task in white box attack setting shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We be- gin with an exhaustive set of experiments to compare the performance of ViT model variants under different perturba- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Various adversarial attack methods have been adopted utilize to generate robust and imperceptible adversarial ex- amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In this work, two methods have been used, includ- ing Fast Gradient Sign Method (FGSM) [34] and Projected Gradient Descent (PGD) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The proposed LNL is trained from scratch on the German Traffic Sign Recognition Bench- mark (GTSRB) [36] and Canadian Institute for Advanced Research (CIFAR-10) [37] without ImageNet [38] pre-training requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Our contributions are summarized as follows: The depth-wise convolutional filters are integrated in the conventional FFN modules in the transformers (ViT) to account locality principle and gauge its impact on the accuracy/robustness trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To further improve the robustness of our proposed method, an implicit data augmentation method called MoEx is introduce to encourage the model to utilize moment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The robustness of ViT models are investigated on GT- SRB are measured in addition, and the experimental evaluation demonstrates the LNL-MoEx performed bet- ter than counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Related Works Transformers have made significant contributions to the area of NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Thanks to the self-attention module, Trans- former can now properly capture the non-local interactions between all different parts of the input sequence, resulting in state-of-the-art performance on a wide range of NLP tasks [39–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The Vision Transformer recently showed that trans- formers could achieve state-of-the-art performance by pre- training the model on massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To this end, the Trans- formers sequence the input images into patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' ViT re- quires a computationally expensive pre-training phase on a larger dataset (such as ImageNet-21k [38]) because of the lower amounts of inductive biases, as described in [23], to achieve decent state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Multiple Transformers models have been developed to demonstrate that comparable performance may be achieved without extra data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' DeiT [26] created a transformer-specific teacher-student technique and trained a transformer architec- ture only on the ImageNet-1K dataset to relax the require- ment of large-scale training dataset in the conventional trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Simultaneously, T2T-ViT [45], Transformer iN Trans- former (TNT) [29], and CvT [46] models have been devel- oped to improve low level feature extractions and further re- ducing their need on large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' These models are known as hybrid-ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Further research is being done to improve the efficiency and performance of transformer ar- chitectures by enhancing the ViT architecture [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Our study achieves a high level of performance without a large- scale training by using just GTSRB [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 0 1 2 3 4 5 6 FLOPS(G) 90 92 94 96 98 100 Clean Accuracy(%) PVT TNT T2T RVT LNL 0 1 2 3 4 5 6 FLOPS(G) 0 10 20 30 40 50 60 70 80 90 Robust Accuracy(%) PVT TNT T2T RVT LNL Figure 1: Comparison between LNL and ViT model variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The robust accuracy is tested on FGSM attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Concurrent to our work, several recent works analyze the robustness of ViTs from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Early works focus on the adversarial robustness of ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' They demonstrated that ViTs are more robust to adversarial attacks than CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [49], and the adversarial transferability between CNNs and ViTs is significantly low [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Follow-up studies [51–53] ex- pand the robustness of ViT models to much common image corruption and distribution shift and prove that ViTs are ro- bust learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Although several findings are consistent with the above works, in this paper, we do not make a simple com- parison of robustness between ViTs and CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We exten- sively compare the ViT variants architecture from an adver- sarial robustness standpoint with white-box attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We design a robust vision transformer and introduce LNL- MoEx to further reduce the fragility of ViT models against adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' METHODOLOGY In this paper, we propose Locality iN Locality (LNL) transformer architecture for visual recognition as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Firstly, we give a brief overview of the primary components of TNT in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We then explain our proposed Locally FeedForward and build our LNL by adding locality mechanism into the FFN component of TNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Fi- nally, the Moment Exchanger (MoEx) implicit augmentation is integrated into the proposed LNL model in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 so that we can encourage the LNL-MoEx to utilize the mo- ment information for enhancing robustness/accuracy trade- off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Transformer iN Transformer (TNT) TNT splits a 2D image into 푛 patches 푋 = [푥1, 푥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', 푥푛] ∈ ℝ푛×푝×푝×3 uniformly, where (푝, 푝) is the resolution of each image patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In TNT, the patches are viewed as visual sen- tences for representing the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Then each patch is divided into 푚 sub-patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' That is to say, a visual sentence is composed of a sequence of visual words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This operation can be formulated as follows: 푋푖 → [푥푖,1, 푥푖,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', 푥푖,푚], (1) where 푥(푖,푗) ∈ ℝ푠×푠×3 is the 푗 − 푡ℎ visual word of the 푖 − 푡ℎ visual sentence, (푠, 푠) is the spatial size of each sub-patches, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 2 of 9 Engineering Science and Technology, an International Journal D Linear Projection of Sentences and Words 1 1 4 7 2 8 3 9 5 6 2 1 4 7 2 8 3 9 5 6 9 1 4 7 2 8 3 9 5 6 0 Outer Transformer Block Inner Transformer Block + Inner Transformer Block + Inner Transformer Block + output Patch Moment Exchanger Patch Moment Exchanger Patch Moment Exchanger … ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' … MSA LFF LayerNorm LFF 3x3 DW Conv Conv 1 X 1 Seq2Img Conv 1 X 1 Img2Seq MSA Q K V Avg_pool Multi-Head Self-Attention Figure 2: Overall architecture of the proposed Locality iN Locality (LNL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Outer Transformer Block has the same structure as Inner Transformer Block, which is composed of Locally-FeedForward (LFF), Multi-head Self-Attention (MSA), and LayerNorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Pach Moment Exchanger is our implicit data augmentation method that appears only in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' and 푗 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' With a linear projection, the visual words 푥푖,푗 are transformed into a sequence of word embedding as follows: 푦푖,푗 = 퐹퐶(푉 퐸퐶(푥푖,푗)), (2) 푌 푖 = [푦푖,1, 푦푖,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', 푦푖,푚], (3) where 푦(푖,푗) ∈ ℝ푐 is the 푗 − 푡ℎ word embedding of the 푖 − 푡ℎ visual sentence, 푐 is the dimension of word embedding, and 푉 퐸퐶(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=') denotes the vectorization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The TNT method has two data flows across and inside the visual sen- tences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' More specifically, one data flow is employed for vi- sual sentences and the other one is utilized for the visual words inside each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' A transformer block is used to investigate the relationship among visual words and their embedding as follows: 푌 ′푖 푙 = 푌 푖 푙−1 + 푀푆퐴(퐿푁(푌 푖 푙−1)), (4) 푌 푖 푙 = 푌 ′푖 푙 + 푀퐿푃(퐿푁(푌 ′푖 푙 )), (5) where the standard transformer blocks consist of a Multi- head Self-Attention module (MSA), a Multiple Layer Per- ceptron (MLP) and Layer Normalization (LN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In inner trans- former blocks of TNT, 푙 = (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', 퐿) indicate the index of the 푙 − 푡ℎ transformer block, and 퐿 indicates the overall quantity of stacked blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' After transformation, the word embedding would be converted as 푌푙 = [푌 1 푙 , 푌 2 푙 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', 푌 푛 푙 ] , which is transformer block 푇푖푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This can be viewed as an inner transformer block, denoted as 푇푖푛 process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This op- eration could account the connections among visual words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The series of sentence are kept in the embedding memory at the sentence level 푍0 = [푍푐푙푎푠푠, 푍1 0, 푍2 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', 푍푛 0] ∈ ℝ(푛+1)×푑 In each layer, the word embedding are linear projected onto the sentence embedding domain in each layer as follows: 푍푖 푙−1 = 푍푖 푙−1 + 퐹퐶(푉 퐸퐶(푌 푖 푙 )), (6) where 푍푖 푙−1 ∈ ℝ푑 and the FC layer ensures that the di- mension is addable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' With the above addition operation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 6 augments the representation of sentence embedding with word-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The sentence embeddings are converted by vanilla transformer block as follows: 푍 ′ 푙 = 푍푙−1 + 푀푆퐴(퐿푁(푍푙−1)), (7) 푍푙 = 푍 ′ 푙 + 푀퐿푃 (퐿푁(푍 ′ 푙)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (8) In short, the visual word embeddings and sentence em- beddings be expressed can be as follows: 푌푙, 푍푙 = 푇 푁푇 (푌푙−1, 푍푙−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (9) In TNT, the inner transformer block represents word re- lationships for local feature extraction, and the outside trans- former block encodes intrinsic sentence-level context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The TNT is built by stacking the TNT components 퐿 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' LNL-MoEx Locally-FeedForward Previous researches [46, 54] have demonstrated that the FFN in the classical vision transformer struggles to effectively model local dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' However, neighboring pixels are valuable references to provide vital information of local details in input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To capture local dependency, inspired by previews works [30, 31], a depth- wise convolution block is integrated into FFN module in the LNL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' A (푘 × 푘) convolution kernel is utilized in each channel in the depth-wise convolution to properly incorpo- rate locality into the LNL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' For features commutation, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 3 of 9 Engineering Science and Technology, an International Journal 3x3 DW Conv Conv 1 X 1 Seq2Img Conv 1 X 1 Img2Seq Conv 1 X 1 Conv 1 X 1 (b) (a) Figure 3: Block modifications of Feed Forward Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (a) simplified version of Feed Forward Network in TNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' For simplic- ity of notation, we replace fully-connected layers by 1 × 1 convo- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (b) Locally-FeedForward Network, which is composed of depth-wise convolution (DW) and 1 × 1 convolutions to inject local- ity mechanism into FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' the feature inside the (푘 × 푘) kernel is integrated to compute a new feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To enhance the feature dimension for each token, we first apply a linear projection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The tokens are then reshaped into 2D feature maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' as a result, the lo- cal information would captured by means of the depth-wise convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The features are then flattened to reshape to- kens, and the channels are shrunk down via another linear layer to match the channel dimensions of input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The computation could be represented as: 퐿푂퐶(푍) = 퐼2푆((푆2퐼(푍) ⊙ 푊푑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (10) Where 푊푑 ∈ ℝ훾푑×1×푘×푘 is the kernel of the depth-wise convolution, ⊙ is a convolutional operation, 퐼2푆 is the func- tion that converts an image into a sequence of tokens, and 푆2퐼 performs the reverse process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Figure 3 depicts the en- tire proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To introduce locality bias into the TNT, the MLP layer (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 8) is replaced by: 푍푙 = 푍 ′ 푙 + 퐿푂퐶(푍 ′ 푙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (11) MoEx After feeding the word-level features 푍 ′ 푙 to the depth-wise convolution, the feature representation 푍푙 would be a 2D tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' MoEx integrates the feature normalization into the data augmentation, and fuses features with labels across two training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The normalized features of one data sample are unified with the feature moments of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This nonsymmetric combination in feature space aims to en- code various directions for the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To nor- malize features in the LNL model, a normalization function 푁 is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This function takes the word level features 푍푖 푙 of the 푖 − 푡ℎ input 푥푖 at block 푙 of LNL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Finally function 푁 generates three outputs which include: the nor- malized word-level features ̂푍푖 푙, the first-moment 휇푖, and the second moment 휎푖 as follows: 푁(푍푖 푙) = ( ̂푍푖 푙, 휇푖, 휎푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (12) 푁 calculates the value of the first and second momen- tum after feeding the word level feature 푍 ′ 푙 through the local module, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', depth-wise convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This operation rela- tively resembles PONO function [55] in the realm of CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Along patch dimension, the normalized features pose zero- mean and standard deviation one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To employ the LNL model, the input images are taken and fused by means of calculated moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' First image (푥퐴 ∶ 푍퐴 = 푁−1( ̂푍퐴, 휇퐴, 휎퐴)) takes the moments of second image (푥퐵 ∶ 휇퐵, 휎퐵) as follows: 푍(퐵) 퐴 = 휎퐵 푍퐴 − 휇퐴 휎퐴 + 휇퐵, (13) where 푍퐴, 휇퐴, 휎퐴 are the word-level features, the first-moment, and the second moment of image 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In addition, 휇퐵, 휎퐵 are the first-moment, and the second moment of image 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We now proceed with these features 푍(퐵) 퐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To emphasize on in- jected features of 퐵, the loss function would be adjusted as follows: 휆 ⋅ 퓁(푍(퐵) 퐴 , 푦퐴) + (1 − 휆) ⋅ 퓁(푍(퐵) 퐴 , 푦퐵), (14) where 휆 is a fixed variable for setting the combination of the features and the moments 휆 ∈ (0, 1), and (푦퐴, 푦퐵) are labels of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Finally, the overall loss function would be the straight-forward combination of the two losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' EXPERIMENTAL RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Experimental Settings Our experimental evaluations are carried out on the NVIDIA Tesla P100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The LNL is assessed under tiny and small model sizes, which are coined LNL-Ti and LNL-S, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Performance of the proposed method is evaluated on German Traffic Sign Recognition Benchmark (GTSRB) [36] for focusing on object recognition tasks in autonomous vehi- cles, and Canadian Institute for Advanced Research (CIFAR- 10) [37] dataset, as a well-known dataset in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We also implement LNL-MoEx by adding implicit Moment Changer augmentation into the our transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The models are all trained on the GTSRB dataset, which contains 39,209 labeled traffic sign images in different 43 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The images in the GTSRB dataset are divided into 35,209 training images, 4,000 validation images, and 12,630 test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' CIFAR-10 is a subset of the Tiny Images dataset and con- sists of 60000 (32 x 32) color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The images are di- vided into 10 different categories including bird, automobile, airplane, frog, deer, dog, cat, horse, truck, and ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Train and Test sets of this dataset contain 50000, and 10000 im- ages, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To be more specific, each class in CIFAR- 10 provides 5000 training images and 1000 test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' When constructing the dataset, the test set was constructed by se- lecting 1000 images randomly, and the rest of the images were dedicated to the train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' All ViT models are trained with the same hyperparame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The number of training epochs is set to 100 and 150 for GTSRB and CIFAR-10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The training hyperpa- rameters for the training phase are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 4 of 9 Engineering Science and Technology, an International Journal Table 1 Training model parameters for GTSRB and CIFAR-10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Parameter GTSRB CIFAR-10 Batch Size 50 128 Epoch 100 150 Learning Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='001 Optimizer SGD SGD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' LNLs are usable for small dataset Table 2 demonstrates the results of the proposed LNL compared to the state-of-the-art studies with respect to the Top-1 and Top-5 robustness accuracy and computational com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' As reported in Table 2, the proposed LNL ranks first in terms of clean accuracy and efficiency, and adversarial robustness compared with the state-of-the-art works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To be more specific, the proposed LNL-S yields gains of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5% on Top-5 metric compared to the second [53] and third [56] best methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' While the obtained gains in comparison with the second-best approach [45] is not pronounced, our superiority in terms of Top-1 metric is noticeable with a ac- curacy of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This is due to the fact that the results with respect to the Top-5 metric consider higher probability out- puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Table 3 reports the comparison results on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The results show that our LNL model significantly outper- forms the state-of-the-art methods in terms of clean accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Specifically, we achieve accuracies of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6% and 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9% on the validation set using LNL-Ti and LNL-S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Comparing clean accuracies of GTSRB and CIFAR-10 datasets in Table 2 and Table 3 show that traffic sign images are more difficult to predict the correct label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' However, it can be seen LNL achieves the best performance on both datasets using fewer or comparable model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' These results demon- strate the strong classification ability of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' It is worth highlighting that all the counterpart model vi- sion transformers are pre-trained in the ImageNet [38] dataset, while our proposed LNL models are only trained on CIFAR- 10 and GTSBR datasets from scratch without ImageNet pre- training requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Adversarial Robustness To evaluate robustness against adversarial attacks, we adopt single-step attack algorithm FGSM [34] and multi- step attack algorithm PGD [35] with step number 푡 = 5, and step size 훼 = 0 ∶ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The input image is perturbed by both attackers with a maximum magnitude of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Results in Table 2 demonstrate that the adversarial robustness dra- matically impacts on the LNL architecture with similar com- putational complexity, the proposed LNL models represent high adversarial robustness under adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This is ascribed to the proposed modifications on LNLs, which aims to strengthen the adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' More specifi- cally, the depth-wise convolution introduces locality induc- tive Bias into FFN to model local dependencies and intro- duce inductive bias in favor of better adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Moreover, the MoEx module utilizes implicit data augmen- tation, which is helpful for adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Table 2 and Table 3 show that the proposed LNL model achieves superior performance on both admired FGSM and PGD attacks in compared state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In detail, LNL-Ti and LNL-S considerably outperforms the state-of- the-art models with a gain of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4% on FGSM attack and a gain of 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7% on PGD attack compared to its counterpart vi- sion transformers on GTSRB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Our model similarly outperforms prior arts by a large margin on both adversar- ial benchmarks for the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Nonetheless, our LNL model generally yields the best accuracy/robustness trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' This advance is further expanded by our MoEx augmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Table 2 additionally shows that LNL-MoEx improves both the accuracy and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Compared to other aug- mentation methods, when LNL-Ti is trained using MoEx achieves the top-1 accuracy of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6% and robust accuracy of 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3%, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4% on FGSM and PGD benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Notably, the highest robust accuracy is obtained by LNL-MoEx-S with 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8% on the FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Our enhanced model with MoEx aug- mentation outperforms other methods by significant improve- ments on multiple standard benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Visualizing Attention Maps To present qualitative result comparisons to TNT as base- line for our improved model, we compare the attention maps generated by the TNT [29] and the LNL using CAM [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The attention maps were extracted using a visualization pro- cedure inspired by Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In Table 4, the first column shows the input images of traffic signs from GTSRB [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The second and third columns represent the attention maps of the clean images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In contrast, the fourth and fifth columns represent the attention maps of the adversarial im- ages generated by FGSM [34] under the same settings in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Furthermore, the predicted labels are shown with confidence scores for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' From the attention map, we can see TNT just gives at- tention to widespread features, such as in all cases, it focuses on the center of traffic signs, whereas LNL diverts the atten- tion to more diverse and significant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' For example, in the case of ‘no passing’, ‘road work’ and ‘Roundabout mandatory’ LNL focuses on the frame, color, and shape of traffic signs, respectively, which are important classification features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We can also see from Table 4 that the LNL has strong background separability, and it does not concentrate on trivial background features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' It is clear that LNL has the capability to track the local features of the input image while TNT tracks global and centered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We can see from the adversarial part that the perturba- tions highly target the main object of images, the ‘Round- about mandatory’ for both TNT and LNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Moreover, the perturbation effect can be noticed on the attention maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' For instance, the FGSM fools the TNT to mislabel ‘Road work’ instate of ‘No passing’ with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In contrast, the LNL predicts labels with high confidence in almost 3/4 adversarial traffic signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 5 of 9 Engineering Science and Technology, an International Journal Table 2 The performance (%) of LNL and Transformers on GTSRB and two robustness benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In Transformers part, models are trained without any extra modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In Augmentations part, we trained LNL-Ti model with our proposed MoEx and some augmentation methods such as CutMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Group Model Model Complexity GTSRB Robustness Benchmarks FLOPs (G) Params (M) Top-1 Top-5 FGSM PGD Transformers PVT-Tiny [56] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 TNT-T [29] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 T2T-ViT-t-10 [45] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 RVT-Ti [53] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 LNL-Ti 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 Swin-T [47] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 PVT-Small [56] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 TNT-S [29] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 T2T-ViT-t-14 [45] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 RVT-S [53] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 LNL-S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 Augmentations DeepAugment [16] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 CutMix [18] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 AugMix [17] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 Puzzle-Mix [57] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 RVT-Ti* [53] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 LNL-MoEx-Ti 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 RVT-S* [53] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 LNL-MoEx-S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 Table 3 The performance (%) of LNL and Transformers on CIFAR-10 and two robustness benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Model Top-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Acc Robustness Benchmarks FGSM PGD PVT-Tiny [56] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 TNT-T [29] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 T2T-ViT-t-10 [45] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 RVT-Ti [53] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 LNL-Ti 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 Swin-T [47] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 PVT-Small [56] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 TNT-S [29] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 T2T-ViT-t-14 [45] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 RVT-S [53] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 LNL-S 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Ablation Study To understand our LNL architecture better, we ablate each critical design by evaluating its performance on GT- SRB classification and adversarial benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Firstly, a study is conducted to demonstrate how the Locally FeedFor- ward could influence the performance of other vision trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We also evaluate the performance of our implicit Moment Exchanger augmentation in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Impact of Locally FeedForward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' To show the effec- tiveness of our proposed Locally FeedForward, we intro- duced local information into the FeedForward network of some transformer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' T2T-Ti, PVT-Ti, and Swin- Ti are chosen as the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Table 5 illustrates the ex- perimental results of enhanced models by replacing the base FFN module with the proposed Locally FeedForward, all the enhanced models yield significant improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Specifi- cally, all the enhanced models achieve more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4% promotion on robust and standard accuracy on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The greatest improvement is for Local-T2T, where the Lo- cally FeedForward leads to a gain of 23% in robust accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Compared with the base model, there is only a negligible in- crease in the amount of computation and a marginal increase in the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Impact of Moment Exchanger augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' We fur- ther study the impact of MoEx augmentation on multiple transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' As shown in Table 6, we can see our augmentation improves standard and robust accuracy of all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Among them, PVT∗ and T2T∗ achieve significant improvements of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1% on robust and standard ac- curacy compared to the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' CONCLUSION We propose the LNL model and study the robustness of vision transformers on traffic sign classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' The proposed model relaxes the requirement of large-scale pre- training phase in the conventional vision transformer and at the same time outperforms the state-of-the-art transformer- based studies in relation to the clean and adversarial robust- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Furthermore, we integrate the MoEx data augmen- tation into the vanilla vision transformers to improve ad- O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 6 of 9 Engineering Science and Technology, an International Journal Table 4 Comparison of the attention maps generated by the LNL model and the TNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (first column) input images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' (second and third columns) attention maps generated by models with clean images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' and (fourth and fifth columns) attention map generated by models with adversarial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Input Image No Attack FGSM Attack TNT [29] LNL TNT [29] LNL Label: ‘over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 tons prohibited’ ‘over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 tons prohibited’ 99% ‘over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 tons prohibited’ 99% ‘No passing’ 77% ‘over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 tons prohibited’ 79% Label: ‘Roundabout mandatory’ ‘Roundabout mandatory’ 99% ‘Roundabout mandatory’ 99% ‘Keep right’ 99% ‘Roundabout mandatory’ 25% Label: ‘No passing’ ‘No passing’ 99% ‘No passing’ 99% ‘Road work’ 100% ‘No passing’ 98% Label: ‘Road work’ ‘Road work’ 99% ‘Road work’ 99% ‘Priority road’ 99% ‘Road work’ 84% versarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Instead of disregarding the moments extracted by the normalization layer, the MoEx data aug- mentation forces the neural network to pay attention towards robust feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Experimental evaluations approves that the proposed LNL-MoEx consistently achieves outstanding per- formance on the GTSRB dataset in terms of adversarial ro- bustness and performance clean accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' In short, concern- ing the trade-off between FLOPs, clean and robustness ac- curacy, extensive experiments validate the superiority of our LNL-MoEx-Ti and LNL-MoEx-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' CRediT authorship contribution statement Omid Nejati Manzari: Conceptualization, Software, Writing- Original draft preparation, Validation, Resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hossein Kashiani: Methodology, Writing- Reviewing and Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hojat Asgarian Dehkordi: Modification for the final layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shahriar Baradaran Shokouhi: Supervision, Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 7 of 9 3Engineering Science and Technology, an International Journal Table 5 Effect of Locally FeedForward on other ViT architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Per- formance (%) of clean accuracy and adversarial robustness under FGSM attack on GTSRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Network Params FLOPs Acc Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Acc (M) (G) T2T-Ti 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 Local-T2T 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4↑) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1↑) PVT-Ti 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='3 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 Local-PVT 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9↑) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2↑) Swin-Ti 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 Local-Swin 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7↑) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6↑) Table 6 Effect of our Patch Moment Exchanger augmentation on other ViT architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Performance (%) of clean accuracy and adver- sarial robustness under FGSM attack on GTSRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Best Results are in bold face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Vanilla Acc Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Acc Improved Acc Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Acc models models T2T-Ti 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 T2T-Ti* 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='5 PVT-Ti 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 PVT-Ti* 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 Swin-Ti 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='2 Swin-Ti* 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='7 Acknowledgment we would like to thank the editors and anonymous reviewers for providing insightful suggestions and comments to improve the quality of research paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zou, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, “A cascaded r-cnn with multiscale attention and imbalanced samples for traffic sign detection,” IEEE access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 29 742–29 754, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kuang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sherratt, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yu, “Cctsdb 2021: a more comprehensive traffic sign detection benchmark,” Human-centric Computing and Information Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 12, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [3] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Manzari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Boudesh, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shokouhi, “Pyramid transformer for traffic sign detection,” arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='06067, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chen, “An object tracking framework with recapture based on correlation filters and siamese networks,” Computers & Electrical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 98, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 107730, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Feng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sangaiah, “Scstcf: spatial-channel selection and temporal regularized correlation filters for visual tracking,” Applied Soft Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 118, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 108485, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Tourani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shahbahrami, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Soroori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Khazaee, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Suen, “A robust deep learning approach for automatic iranian vehicle license plate detection and recognition for surveillance systems,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 201 317–201 330, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dehkordi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Nezhad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Ashrafi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shokouhi, “Still image action recognition using ensemble learning,” in 2021 7th International Conference on Web Research (ICWR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 125–129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Asgarian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Amirkhani, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shokouhi, “Fast drivable area detection for autonomous driving with deep learning,” in 2021 5th International Conference on Pattern Recognition and Image Analysis (IPRIA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Li, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kim, “An enhanced pegasis algorithm with mobile sink support for wireless sensor networks,” Wireless Communications and Mobile Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 2018, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dehkordi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kashiani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Imani, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shokouhi, “Lightweight local transformer for covid-19 detection using chest ct scans,” in 2021 11th International Conference on Computer Engineering and Knowledge (ICCKE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 328–333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mohammadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Noori, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Bahri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Majelan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Havaei, “Cagnet: Content-aware guidance for salient object detection,” Pattern Recognition, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 103, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 107303, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Noori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mohammadi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Majelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Bahri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Havaei, “Dfnet: Discriminative feature extraction and integration network for salient object detection,” Engineering Applications of Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 89, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 103419, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [13] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Manzari and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shokouhi, “A robust network for embedded traffic sign recognition,” in 2021 11th International Conference on Computer Engineering and Knowledge (ICCKE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 447–451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Tourani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Soroori, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shahbahrami, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Khazaee, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Akoushideh, “A robust vehicle detection approach based on faster r-cnn algorithm,” in 2019 4th International Conference on Pattern Recognition and Image Analysis (IPRIA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 119–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Cai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' He, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Gu, “Do wider neural networks really help adversarial robustness?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 7054–7067, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hendrycks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Basart, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kadavath, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dorundo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Desai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Parajuli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', “The many faces of robustness: A critical analysis of out-of-distribution generalization,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 8340–8349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hendrycks, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Cubuk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zoph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Gilmer, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lakshminarayanan, “Augmix: A simple data processing method to improve robustness and uncertainty,” arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='02781, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Han, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Choe, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yoo, “Cutmix: Regularization strategy to train strong classifiers with localizable features,” in Proceedings of the IEEE/CVF international conference on computer vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 6023–6032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hashemi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mozaffari, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Alirezaee, “Improving adversarial robustness of traffic sign image recognition networks,” Displays, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 74, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 102277, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hashemi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mozaffari, “Cnn adversarial attack mitigation using perturbed samples training,” Multimedia Tools and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 80, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 22 077–22 095, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [21] ——, “Secure deep neural networks using adversarial image generation and training with noise-gan,” Computers & Security, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 86, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 372–387, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, “Making convolutional networks shift-invariant again,” in International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' PMLR, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 7324–7334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dosovitskiy, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Beyer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kolesnikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Weissenborn, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Unterthiner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dehghani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Minderer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Heigold, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Gelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', “An image is worth 16x16 words: Transformers for image recognition at scale,” arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='11929, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Tang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, “Visual transformer pruning,” arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='08500, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, “Transgan: Two transformers can make one strong gan,” arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='07074, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 3, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Touvron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Cord, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Douze, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Massa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sablayrolles, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Jégou, “Training data-efficient image transformers & distillation through attention,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' PMLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 10 347–10 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Benz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Ham, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Karjauv, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kweon, “Adversarial robustness comparison of vision transformer and mlp-mixer to cnns,” O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 8 of 9 Engineering Science and Technology, an International Journal arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='02797, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yuille, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xie, “Are transformers more robust than cnns?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Han, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xiao, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, “Transformer in transformer,” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='00112, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Cao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Timofte, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Van Gool, “Localvit: Bringing locality to vision transformers,” arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='05707, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [31] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Cun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Bao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Liu, “Uformer: A general u-shaped transformer for image restoration,” arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='03106, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [32] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yang, “Rest: An efficient transformer for visual recognition,” arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='13677, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [33] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Bai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yuille, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shen, “Glance-and- gaze vision transformer,” arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='02277, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [34] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shlens, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Szegedy, “Explaining and harnessing adversarial examples,” arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='6572, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Madry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Makelov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Schmidt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Tsipras, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Vladu, “Towards deep learning models resistant to adversarial attacks,” arXiv preprint arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='06083, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Stallkamp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Schlipsing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Salmen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Igel, “The german traffic sign recognition benchmark: a multi-class classification competition,” in The 2011 international joint conference on neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' IEEE, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 1453–1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Krizhevsky, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', “Learning multiple layers of features from tiny images,” 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Krizhevsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sutskever, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hinton, “Imagenet classification with deep convolutional neural networks,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 25, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 1097–1105, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [39] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Brown, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Ryder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Subbiah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kaplan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dhariwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Neelakantan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shyam, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sastry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Askell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', “Language models are few-shot learners,” arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='14165, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [40] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Carbonell, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Le, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Salakhutdinov, “Transformer-xl: Attentive language models beyond a fixed-length context,” arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='02860, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Devlin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lee, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='04805, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Radford, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Narasimhan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Salimans, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sutskever, “Improving language understanding by generative pre-training,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Radford, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Child, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Luan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Amodei, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Sutskever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=', “Language models are unsupervised multitask learners,” OpenAI blog, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [44] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Carbonell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Salakhutdinov, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Le, “Xlnet: Generalized autoregressive pretraining for language understanding,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [45] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Jiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Tay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Feng, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yan, “Tokens-to-token vit: Training vision transformers from scratch on imagenet,” arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='11986, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [46] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xiao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Codella, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Dai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yuan, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, “Cvt: Introducing convolutions to vision transformers,” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='15808, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [47] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Cao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Guo, “Swin transformer: Hierarchical vision transformer using shifted windows,” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='14030, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [48] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Tian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xia, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shen, “Conditional positional encodings for vision transformers,” arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='10882, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [49] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Hsieh, “On the adversarial robustness of visual transformers,” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='15670, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [50] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mahmood, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mahmood, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Van Dijk, “On the robustness of vision transformers to adversarial examples,” arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='02610, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [51] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Bhojanapalli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chakrabarti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Glasner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Unterthiner, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Veit, “Understanding robustness of transformers for image classification,” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='14586, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [52] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Paul and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chen, “Vision transformers are robust learners,” arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='07581, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [53] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Duan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Ye, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' He, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xue, “Towards robust vision transformer,” arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='07926, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [54] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yuan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Yu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wu, “Incorporating convolution designs into visual transformers,” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='11816, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [55] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Belongie, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Weinberger, “On feature normalization and data augmentation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 12 383–12 392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [56] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Xie, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Fan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Song, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Liang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Luo, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Shao, “Pyramid vision transformer: A versatile backbone for dense prediction without convolutions,” arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='12122, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [57] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Kim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Choo, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Song, “Puzzle mix: Exploiting saliency and local statistics for optimal mixup,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 5275–5285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [58] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Khosla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Lapedriza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Oliva, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Torralba, “Learning deep features for discriminative localization,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' 2921–2929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' [59] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Caron, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Touvron, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Misra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Jégou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Mairal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Bojanowski, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Joulin, “Emerging properties in self-supervised vision transformers,” arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='14294, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content='N Manzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} +page_content=' Page 9 of 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FJT4oBgHgl3EQfeiys/content/2301.11553v1.pdf'} diff --git a/utE5T4oBgHgl3EQfLg6S/content/tmp_files/2301.05474v1.pdf.txt b/utE5T4oBgHgl3EQfLg6S/content/tmp_files/2301.05474v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..793893ff91e9fce72f4a5fc85d95bbd26d3dfcb6 --- /dev/null +++ b/utE5T4oBgHgl3EQfLg6S/content/tmp_files/2301.05474v1.pdf.txt @@ -0,0 +1,1254 @@ +LOCATING TOPOLOGICAL STRUCTURES IN DIGITAL IMAGES VIA +LOCAL HOMOLOGY +A PREPRINT +Chuan-Shen Hu +School of Physical and Mathematical Sciences +Nanyang Technological University +50 Nanyang Avenue 639798, Singapore +chuanshen.hu@ntu.edu.sg +peterbill26@hotmail.com +January 16, 2023 +ABSTRACT +Topological data analysis (TDA) is a rising branch in modern applied mathematics. It extracts +topological structures as features of a given space and uses these features to analyze digital data. +Persistent homology, one of the central tools in TDA, defines persistence barcodes to measure the +changes in local topologies among deformations of topological spaces. Although local spatial changes +characterize barcodes, it is hard to detect the locations of corresponding structures of barcodes due to +computational limitations. The paper provides an efficient and concise way to divide the underlying +space and applies the local homology of the divided system to approximate the locations of local +holes in the based space. We also demonstrate this local homology framework on digital images. +Keywords Topological data analysis · Persistent homology · Local hole structures · Persistence barcodes · Local +systems and patches · Short filtrations · Cellular sheaves · Global sections · Merging and outer-merging numbers +1 +Introduction +Homology is an algebraic description of topological spaces and has become one of the foundations of modern geometry +and topology. It uses algebra to detect genera in topological spaces, such as loops and high-dimensional voids, and to +classify the topological types and shapes of manifolds. In addition to its importance in pure mathematics, over the past +two decades or so, data scientists have noticed the benefits and potential of homology in numerical data and raised a +new field called topological data analysis (TDA) [59, 7, 22, 8, 21]. +Persistent homology plays a central role in TDA, which transforms a sequence of topological spaces linked by continuous +functions into a homology chain. By checking the birth and death of elements in the chain, one can understand which +homological generator can have a longer lifespan and shows its importance in the continuous process [59]. Persistent +homology and related techniques have been applied in many data science tasks, such as bioinformatics [44, 42, 31], +molecular analysis [56, 24, 3, 53], image processing [11, 10, 16, 13, 43], and material science [10]. +Persistent barcodes (Section 2.2) record the lifespans of connected components, loops, and voids. Many applications +use persistence barcodes and related statistical features as machine learning features [6, 1, 12]. Although persistent +homology and persistence barcode has shown their potential in many real applications, it still has some limitations. +One is it can only capture the global information of how connected components and holes behave during geometric +deformation, while the local merging relations are usually omitted. This information is theoretically present in the +definition of persistent homology and persistent barcodes. However, for computational efficiency, hole representations +(e.g., q-circular representation of q-holes) or positions are often buried in the Gaussian elimination of the matrices in +the computation of persistence barcodes. +arXiv:2301.05474v1 [math.AT] 13 Jan 2023 + +A PREPRINT - JANUARY 16, 2023 +Recently, some scholars noticed the importance of local information on persistent homology and proposed some +interesting works on the local behavior of persistent homology [50, 46]. For example, Vandaele et al. [50] investigated +the local Vietoris-Rips complexes of the point cloud and applied the local Betti pairs to form a global descriptor of the +point cloud. This descriptor can be viewed as a heatmap of the whole space. Regions with higher heat values usually +mean they have more significant topological/geometric information, such as higher local branch numbers or loops. +Also, Stolz described in her doctoral dissertation [46] how to apply the Mayer–Vietoris sequence to compute the local +Vietoris–Rips complex linked from a data point. +On the other hand, some of the research also aims to detect the locations of loop or hole structures in the topological +space. For example, Akai et al. [2] generate persistence barcodes of the Vietoris–Rips complex as inputs of a neural +network model and apply them for the ego-vehicle localization application. Similarly, Keros et al. [38] train on a +Hodge Laplacian-based graph neural network to detect the nearest optimal homology as a location representation of +homologies. Furthermore, Xu et al. [57] apply the distance measurement (DTM) function [9] to enhance the robustness +of Vietoris–Rips complex construction, and apply persistence and distance information to detect holes and voids in +point cloud data. +However, while image structures are more regular than point clouds, making it easier to compare local-global attributes, +most current methods are designed for point cloud data. Theoretical assurance methods for localized hole detection +are still limited. The paper provides a theoretically guaranteed framework for hole position detection in arbitrary +topological spaces, demonstrated on digital images. +This paper is an extension of our previous work presented as a workshop paper at CVPR 2021 (2021 Conference on +Computer Vision and Pattern Recognition) [30]. The work [46] introduces the concept of cellular sheaves and connects +them to persistent homology. In [46], we define the local merging number can consider its geometric meaning in +0-dimensional objects. This paper extends the framework in [30] and focuses on 1-dimensional merging relations. In +addition to theoretical promotion, we have a preliminary demonstration of images. It shows that the 1-dimensional +merging relations can estimate the position of holes in the space, which provides a way to analyze the local topological +characteristics. +1.1 +Organization +The organization of the paper is as follows. Section 2 quickly recaps the homology, Betti numbers, persistent homology, +and barcodes. We present the main results in Section 3 and separate the section into two parts. Section 3.1 introduces +how we divide the ambient space by a local region and apply the divided system to compute its persistent homology. +We also interpret the geometric meaning of the computed barcodes and explain how they detect the cycle locations. We +also compare the proposed framework with previous methods in Section 3.3. Section 4 shows how to adapt the theory +developed in Section 3 on digital images and demonstrates the proposed locating method. Finally, we discuss future +directions and summarize the paper in Section 5. +2 +Persistent Homology and Barcodes +We briefly introduce the standard notions and terminologies of singular homology, including its functoriality, Betti +numbers, and geometric meanings in Section 2.1. Section 2.2 focuses on persistent homology and barcodes. We will +also show in this section typical ways for building filtrations, especially the construction relying on the thresholding +technique, which is the foundation of the paper. +2.1 +Homology +This section briefly recalls the singular homology and related properties of topological spaces. One can find these +materials in several classic textbooks on algebraic topology [27, 51, 41, 25]. We start the section with the following +definitions. +Definition 1. For any non-negative integer q, we define the geometric q-simplex, denoted by ∆q, as the convex hull of +the standard basis {e0, e1, ..., eq} for the (q + 1)-dimensional Euclidean space Rq+1. That is, +∆q = conv(e0, e1, ..., eq) = +� +t0e0 + t1e1 + · · · + tqeq : ti ∈ [0, 1] and +q +� +i=0 +ti = 1 +� +. +For any (q + 1) points x0, ..., xq in Rn we can define the affine map [x0, ..., xq] : ∆q → Rn by +(t0, t1, ..., tq) �−→ t0x0 + · · · + tqxq. +(1) +2 + +A PREPRINT - JANUARY 16, 2023 +Then [x0, ..., xq] is a continuous map. A continuous function from ∆q to a topological space X is called a singular +q-simplex in X. In particular, any affine map [x0, ..., xq] : ∆q → Rn is a singular q-simplex in Rn. For q ∈ Z≥0 +and i ∈ {0, 1, ..., q + 1} we define f i +q+1 = [e0, ..., �ei, ..., eq+1]. In other words, f i +q+1 is a singular (q − 1)-simplex in +Rq+1. One can see that the image of f i +q+1 is actually the convex hull of the set {e0, ..., �ei, ..., eq+1}, which is the i-th +(q − 1)-face of the geometric simplex ∆q [25]. +Definition 2. Let X be a topological space, R a commutative ring with identity, and q a non-negative integer. We +define Sq(X; R) as the free R-module generated by all continuous maps σ : ∆q → X. For convenience, we usually +define Sq(X; R) = 0 for q < 0. +The singular simplexes give us a way to express geometric simplexes in arbitrary topological spaces. In Euclidean +spaces, one can explore the faces as boundaries of geometric simplexes by using convex analysis, while it is not +applicable in general spaces. In algebraic topology, we use the following boundary maps to read the boundary data of +singular simplexes. +Definition 3. Let X be a topological space, R a commutative ring with identity, and q ∈ N a positive integer. The +q-boundary map is the function ∂q : Sq(X; R) → Sq−1(X; R) that extends by the mapping +σ �−→ +q +� +i=0 +(−1)i · σ ◦ f i +q +for all continuous σ : ∆q → X. Note that ∂q is well-defined since each σ ◦ f i +q is a singular (q − 1)-simplex in X. +Because Sq(X; R) is defined as the zero space for q < 0, we also define ∂q as the zero maps for q ≤ 0. The following +proposition is the foundation of homology theory. +Proposition 1 ([25], (9.2)). Let X, R, q and ∂q be defined as above. Then ∂q−1 ◦ ∂q = 0. +The equation ∂q−1 ◦ ∂q = 0 shows that im(∂q) ⊆ ker(∂q−1) for every q ∈ Z, and hence we can define the q-th singular +homology of X as the R-module +Hq(X; R) = ker(∂q) +im(∂q+1). +Notation ([25, 51, 41, 18]). To simply the notations, for a topological space X and q ≥ 0, we use Zq(X; R) and +Bq(X; R) to denote the modules ker(∂q) and im(∂q+1). That is, +Zq(X; R) = ker(∂q), Bq(X; R) = im(∂q+1), and Hq(X; R) = Zq(X; R) +Bq(X; R). +(2) +Chains in Zq(X; R) and Bq(X; R) are called the q-cycles and q-boundaries of X. +Except for sending each topological space X to an R-module Hq(X; R), for every continuous map f : X → Y and +q ∈ Z≥0 we can define an R-module homomorphism Sq(f; R) : Sq(X; R) → Sq(Y ; R) that extends the mapping +σ �−→ f ◦ σ +for all singular q-simplexes σ : ∆q → X. Note that the mapping is well-defined since f ◦ σ is also a continuous map +from ∆q to Y . This observation leads to the following proposition. +Proposition 2 ([25]). Let Top and ModR be the categories of topological spaces and R-modules. For each q ∈ Z≥0, +the assignments X ∈ Ob(Top) �→ Sq(X; R) and f ∈ HomTop(X, Y ) �→ Sq(f; R) form a functor from Top to ModR. +In fact, for a continuous map f : X → Y , one can prove that the rectangles in the ladder +· · · +� Sq+1(X; R) +∂q+1(X) � +Sq+1(f;R) +� +Sq(X; R) +∂q(X) � +Sq(f;R) +� +Sq−1(X; R) +� +Sq−1(f;R) +� +· · · +· · · +� Sq+1(Y ; R) +∂q+1(Y ) � Sq(Y ; R) +∂q(Y ) � Sq−1(Y ; R) +� · · · +of R-modules and R-module homomorphisms are commutative. Therefore, for every q, this ladder induces an R-module +homomorphism +Hq(f; R) : Hq(X; R) −→ Hq(Y ; R) +that sends each equivalence class [c] in Hq(X; R) to the class [Sq(f; R)(c)] in Hq(Y ; R). Furthermore, we can see +that the assignment Hq(·; R) of topological spaces and continuous maps also forms a functor from Top to ModR: +3 + +A PREPRINT - JANUARY 16, 2023 +(a) f = f0 +(b) f1 +(c) f2 +(d) f3 +(e) f4 +(f) g = g0 +(g) g1 +(h) g2 +(i) g3 +(j) g4 +Figure 1: Two filtrations of 2-dimensional black pixels; that is, f −1 +0 (0) ⊆ f −1 +1 (0) ⊆ f −1 +2 (0) ⊆ f −1 +3 (0) ⊆ f −1 +4 (0) and +g−1 +0 (0) ⊆ g−1 +1 (0) ⊆ g−1 +2 (0) ⊆ g−1 +3 (0) ⊆ g−1 +4 (0). Although images f and g share the same 1-dimensional homology +space Z2, the persistent homologies of these two images depict different lifespans. Indeed, the 1-dimensional hole in +(a)-(e) has the barcode (0, 2) while the hole in (f)-(j) has (0, 4). +Proposition 3 ([25]). Let Top and ModR be the categories of topological spaces and R-modules. For each q ∈ Z≥0, +the assignments X ∈ Ob(Top) �→ Hq(X; R) and f ∈ HomTop(X, Y ) �→ Hq(f; R) form a functor from Top to +ModR. +An important purpose of developing singular homology is to detect holes in a topological space in any dimension. This +property of singular homology is sometimes called the Poincaré lemma of singular homology. We state this lemma as +follows. +Proposition 4 (Corollary (15.5), [25]). Let n ≥ 1 be a positive integer, and let +Sn = {(x1, x2, ..., xn+1) ∈ Rn+1 : x2 +1 + · · · + x2 +n+1 = 1} +be the n-sphere in Rn+1. Then, for every commutative ring R with identity and a non-negative integer q ≥ 0, we have +Hq(Sn; R) ≃ +�R +if q = n or q = 0, +0 +otherwise. +(3) +In particular, for every topological space X, we have H0(X; R) ≃ Rm, where m is the number of path-connected +components of X, and each path-connected component of X can be represented by a constant function from [0, 1] to X. +The Poincaré lemma provides us with a reliable measurement to detect the number of q-dimensional holes in a +topological space. This number is called the q-th Betti number. +Definition 4 ([25]). Let R be a PID. For any topological space X and integer q ≥ 0, we define the q-th Betti +number βq = βq(X) of X to be the rank of the R-module Hq(X; R). In particular, when R = F is a field, we have +βq = dimF Hq(X; R). +In applications, we often set R as the binary field Z2 = Z/2Z and simplify the notation Hq(X; Z2) to Hq(X). In the +paper, we will focus on homology over Z2 and the singular homology of (binary) images (see Section 3). +2.2 +Prescient Homology +Homology detects the hole structure in a given topological space, while it may omit some geometry of the based space. +For example, two geometric objects with a single 1-dimensional hole in different sizes share the same first homology +group (Figure 1). As a generalization of homology, persistent homology (PH) concerns sequences of topological spaces +and their homologies. It was motivated by the works related to the Morse theory of Patrizio Frosini [19] and Vanessa +Robins [45] in the 1990s. In Morse theory, a height function f : M → R on a smooth manifold M can form a sublevel +4 + +A PREPRINT - JANUARY 16, 2023 +set filtration of subspaces of M [18]. The topological changes of such sublevel sets (e.g., the changes of Betti numbers) +track the shape of M along the direction of the height function and hence a descriptor (or fingerprint) of M. Persistent +homology of height functions is now a well-known and fundamental tool in Morse theory and has many applications in +theory [40, 5, 49] and data science [10, 15, 26, 37]. +More generally, besides the smooth structures, suppose we have a sequence X1 +f1 +−→ X2 +f2 +−→ · · · +fn−1 +−−−→ Xn of topological +spaces and continuous maps, then the functoriality of singular homology shown in Proposition 3 induces a sequence of +homologies as follows: +Hq(X1; R) +Hq(f1) +−−−−→ Hq(X2; R) +Hq(f2) +−−−−→ · · · +Hq(fn−1) +−−−−−−→ Hq(Xn; R), +where q is an arbitrary non-negative integer, and Hq(Xi), Hq(fi) are vector spaces and linear transformations over +Z2. Because continuous maps can deform the geometry of spaces (e.g., sizes, lengths, and connectivity), the changes +in homological cycles and Betti numbers depict how the hole structures in the spaces changed among the continuous +deformation. +Computing homologies connected by continuous maps is challenging in real applications, so one usually considers a +chain of filtered topological spaces with subspace relations. A tower of such topological spaces is called a filtration. We +list the formal definition of filtration as follows. +Definition 5 ([18]). A filtration of topological spaces is a sequence ∅ = X0, X1, X2, ..., Xn of topological spaces +such that Xi is a subspace of Xi+1 for each i ∈ {0, 1, ..., n − 1}. We usually use the chain +F : ∅ = X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn +of topological spaces to denote a filtration of topological spaces. +Because Hq(·; R) : Top → ModR is a functor, a filtration of topological spaces ∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn and a +non-negative integer q ≥ 0 induce a sequence of R-modules and R-module homomorphisms: +PHq : 0 = Hq(∅; R) +ρ0,1 +−−→ Hq(X1; R) +ρ1,2 +−−→ Hq(X2; R) → · · · → Hq(Xn; R) +(4) +where the R-module homomorphism ρi,j : Hq(Xi; R) → Hq(Xj; R) for i ≤ j is induced by the inclusion Xi �→ Xj. +Based on the functoriality of singular homology on the sequence (4), we define ρi,j = ρj−1,j ◦ρj−2,j−1 ◦· · ·◦ρi,i+1 for +every i ≤ j in {0, 1, ..., n}, then the ρi,j is also the R-module homomorphism induced by the inclusion map Xi �→ Xj. +Definition 6 ([18]). Suppose F : ∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn is a filtration of topological spaces. Then, for every ring +R and q ∈ Z≥0, we call the sequence defined in (4) is the q-th persistent homology of the filtration F. +One of the primary purposes of persistent homology is to track the lifespans of local holes, i.e., the births/deaths of +connected components, loops, and higher dimensional voids. To tackle this problem, H. Edelsbrunner and J. Harer +proposed the persistence barcode of persistent homology to detect such topological changes [17, 18]. We refer to the +definition of persistence barcodes as follows. +Definition 7 ([17, 18]). Suppose ∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn is a filtration of topological spaces and 0 → +Hq(X1; F) → · · · → Hq(Xn; F) is the induced qth persistent homology over a field F. Let si be an element in +Hq(Xi; F) (i ≥ 1). Then we have the following definitions: +(a) si is said to be born at i if si /∈ im(ρi−1,i), i is called the birth of si; +(b) si is said to die at j if ρi,j−1(si) /∈ im(ρi−1,j−1) and ρi,j(si) ∈ im(ρi−1,j), j is called the death of si. +If si is still alive at n, we define the death of si to be +∞ (up to this filtration). The tuple (i, j) of si is called +the persistence barcode of the element si ∈ Hq(Xi; F). The multiset of all persistence barcodes of non-repeated +representative generators in all Hq(Xi; F) is called the persistence diagram of the filtration. +For example, by considering the geometry of 2D black objects, rows in Figure 1 define two filtrations of subspaces in +R2, and the induced first persistent homologies (over Z2) are +Z2 +idZ2 � Z2 +� 0 +� 0 +� 0 +and +Z2 +idZ2 � Z2 +idZ2 � Z2 +idZ2 � Z2 +� 0 . +(5) +By definition, the 1-dimensional hole in Figure 1(a)-(e) has the barcode (0, 2). On the other hand, the hole in Figure +1(f)-(j) has barcode (0, 4). +There are many different ways to construct filtrations and compute their persistent homology. A typical one is the +Vietoris–Rips complexes for the point-cloud data. For a (finite) set X in the n-dimensional Euclidean space Rn and +5 + +A PREPRINT - JANUARY 16, 2023 +a fixed positive real number ϵ > 0, one explores the intersections of n-dimensional balls centered at points x in X +with radius ϵ. Regarding points in X as the vertices of a simplicial complex, higher repeated regions lead to higher +dimensional simplexes in Rn. The strategy of the Vietoris–Rips complex is to enlarge the radius to construct a filtration +of simplicial complexes [39, 23, 14]. +As shown in Figure 1 and equation (5), except for the point-cloud data, one can also construct filtrations of digital images +and compute their persistent homology. We referred to an m-dimensional digital image as a function f : P −→ R≥0 +from a non-empty set P of Zm to the set of all non-negative real numbers (cf. [11]). An image f is called binary if its +range is contained in the binary set {0, 1} and called grayscale for otherwise. For a binary image, the primage of zero +f −1(0) referred to the set of all black pixels of f, and f −1(1) denotes the set of all white pixels of f. Viewing each +black pixel as a closed cube in Rm, we regard f −1(0) as a subspace of Rm and consider its topological properties. The +first row in Figure 2 provides examples of 2-dimensional grayscale and binary digital images. +As in Figure 1, one can construct filtrations of images by using image processing techniques on a given binary one. +Another typical method of building filtrations is operating the sub-level sets of a grayscale image. For a image +f : P −→ R≥0 and a threshold t ∈ R, we define a binary image ft : P −→ {0, 1} by setting ft(x) = 0 if f(x) ≤ t and +ft(x) = 1 for otherwise. Then f −1 +t1 (0) ⊆ f −1 +t2 (0) ⊆ · · · ⊆ f −1 +tn (0) for t1 ≤ t2 ≤ · · · ≤ tn. The second the third row +in Figure 2 illustrate how sub-level sets of a grayscale image form a filtration of black pixels. In particular, the 0-th +and 1-th persistence diagrams of the filtrations are {(0, +∞)} and {(0, 3), (2, 3)}. For readers who are interested in +persistent homology on digital images, see [36] for more information. +In this paper, we focus on 2-dimensional binary images and their local homology. We combine image segmentation +techniques, local homology, and persistence barcodes to illustrate how to estimate and detect the positions of holes +in 2D binary images. The combination of this detection method with more image processing techniques (such as +mathematical morphology and sub-level set filtration) will be our future work. +3 +Our Approaches +The section is separated into three parts. First, we quote the definitions of local systems and short persistent homology +in our previous work [30]. Local systems and short persistent homology induce a cellular sheaf structure of topological +spaces and can depict the spatial merging relations via their global/local sections [30]. We discuss the relationship +between hole positions, global/local sections, and persistence barcodes on local systems (Section 3.1). Second, we +introduce how we adapt the theory to digital images and implement the method (Section 3.2). Finally, we discuss some +properties of the proposed framework, such as the relationship between local systems, the location of holes, and image +noises (Section 3.3). +3.1 +Persistent Homology of Local Systems +For a topological space X and a concerned local region A of X, the relative homology Hq(X, A) considers the +equivalence classes of cycles in X that do not meet the subspace A. One can formulate the relative homology of X and +A by Hq(X, A) = Zq(X, A)/Bq(X, A), where Zq(X, A) = {c ∈ Sq(X) : ∂q(c) ∈ Sq−1(A)} = ∂−1 +q (Sq(A)) is the +set of all chains in Sq(X) with boundaries in Sq−1(A), and Bq(X, A) = Bq(X) + Sq(A) is the submodule generated +by all q-boundaries of X and q-chains in A. Elements in Zq(X, A) and Bq(X, A) are called relative q-cycles and +relative q-boundaries of X, respectively [25]. +Roughly speaking, relative homology detects holes in X except for holes that are totally contained in A. More precisely, +one can apply the snake lemma on the short exact sequence 0 −→ S•(A) +ι• +−→ S•(X) +π• +−→ S•(X)/S•(A) −→ 0 with +canonical inclusion and projection to obtain the long exact sequence +· · · −→ Hq(A) +ιq +−→ Hq(X) +πq +−→ Hq(X, A) +δq +−→ Hq−1(A) +ιq−1 +−−−→ Hq−1(X) +πq−1 +−−−→ Hq−1(X, A) −→ · · · . +(6) +One can use barcode representation to detect hole structures in the spaces H•(A), H•(X), and H•(X, A). For example, +a non-zero element in Hq(X) \ im(ιq) represents a hole in X that is not totally emerged in the region A. On the other +hand, c ∈ Hq(X) dies at Hq(X, A) if the cycle c does not represent a hole in A. +Remark. Because the long exact sequence in (6) is a chain complex, every lifespan b − d of a barcode (b, d) is 1. +Relative homology can capture holes contributed by A, X \ A, or both. However, it is difficult and expensive to +implement and compute due to the complicated data representation. This paper proposes a relatively efficient method to +detect hole relations and positions via persistent homology. To achieve this goal, we introduce here two main ideas +proposed in our previous work, called local system and short filtration [30]. +6 + +A PREPRINT - JANUARY 16, 2023 +(a) Image domain P ⊆ Z2 +0 +0 +0 +0 +0 +0 +0 +1 +3 +3 +3 +0 +0 +2 +1 +2 +3 +0 +0 +3 +2 +1 +2 +0 +0 +3 +3 +3 +2 +0 +0 +0 +0 +0 +0 +0 +(b) Grayscale image g +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +0 +0 +0 +0 +0 +1 +0 +0 +1 +0 +0 +0 +0 +0 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +(c) Binary image f +(d) Binary image f +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +0 +0 +1 +1 +1 +1 +0 +0 +1 +1 +1 +1 +0 +0 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +(e) Binary image g0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +0 +0 +1 +0 +1 +1 +0 +0 +1 +1 +0 +1 +0 +0 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +(f) Binary image g1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +0 +0 +0 +0 +0 +1 +0 +0 +1 +0 +0 +0 +0 +0 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +(g) Binary image g2 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +(h) Binary image g3 +(i) Binary image g0 +(j) Binary image g1 +(k) Binary image g2 +(l) Binary image g3 +Figure 2: First row: a 6 × 6 image domain P in Z2, a grayscale image g : P → {0, 1, 2, 3}, and a binary image +f : P → {0, 1}. Figures (c) and (d) are two different representations for the image f. In a binary image f as in (d), +pixels with a value of 0 represent the black pixels of the image. Second row: a filtration of binary images made by +image g and thresholds 0, 1, 2, and 3. Third row: the white-black pixel representations of images in the second row. +Definition 8 ([30]). Let X be a topological space and X1, X2 be subspaces of X. The triad (X, X1, X2) is called a +local system (or an admissible triad) if clX(X1) ∩ clX(X2) = ∅. +For any topological space X and its subspaces X1 and X2, we have the following definition. +Definition 9 ([30]). Let (X, X1, X2) be a triad of topological spaces with X1 ⊆ X and X2 ⊆ X. This triad leads +to two filtrations ∅ ⊆ X1 ⊆ X1 ∪ X2 ⊆ X and ∅ ⊆ X2 ⊆ X1 ∪ X2 ⊆ X. We call them short filtrations of the triad +(X, X1, X2). +Focusing on the first one in Definition 9, the birth information at H•(X1 ∪ X2; F) depicts whether X2 contains a +homological generator that cannot be represented via generators in H•(X1; F). When clX(X1) ∩ clX(X2) = ∅, the +homology H•(X1 ∪ X2; F) is canonically isomorphic to the space H•(X1; F) ⊕ H•(X2; F) since X1 and X2 are two +path-connected components of X1 ∪ X2. In this case, every generator s2 in H•(X2; F) is born at H•(X1 ∪ X2; F) +of the persistent homology 0 −→ H•(X1; F) −→ H•(X1 ∪ X2; F) −→ H•(X; F) and dies at H•(X; F) if there is an +s1 ∈ H•(X1; F) such that s1 and s2 represent the same homological generator in H•(X; F). This property will benefit +computing the homological changes of holes in X1, X2, and X. Furthermore, we will show in Section 3.2 that the +condition clX(X1) ∩ clX(X2) = ∅ can be easily established in image data through elementary image processing +techniques. +7 + +A PREPRINT - JANUARY 16, 2023 +Figure 3: Two local systems (X, X1, X2) of topological spaces. Let Γ0 denote the global section space of the sheaf +structure H0(X1) −→ H0(X) ←− H0(X2). Then we have the following information. The first row: H0(X1) ≃ Z2, +H0(X2) ≃ Z5 +2, H0(X1 ∪ X2) ≃ Z6 +2, H0(X) ≃ Z2 +2, and Γ0 ≃ Z4 +2. The second row: H0(X1) ≃ Z4 +2, H0(X2) ≃ Z2 +2, +H0(X1 ∪ X2) ≃ Z6 +2, H0(X) ≃ Z2, and Γ0 ≃ Z5 +2. +In [30], we applied the two filtrations of a local system (X, X1, X2) to construct the following cellular sheaf structure: +Hq(X1; F) +ρ1 +� Hq(X; F) +Hq(X2; F) +ρ2 +� +where q is any non-negative integer, F is a fixed field, and ρ1, ρ2 are the F-linear transformations induced by the +inclusions X1 �→ X and X2 �→ X. We often call the maps ρ1, ρ2 restriction maps. A pair (s1, s2) ∈ Hq(X1; F) ⊕ +Hq(X2; F) is called a global section of the sheaf if ρ1(s1) = ρ2(s2). We use Γ to denote the subspace of all global +sections in Hq(X1; F) ⊕ Hq(X2; F), and it can be sculptured by the following theorem. +Theorem 1. For the following sheaf structure of F-vector spaces and F-linear maps: +V +f +� P +, +W +g +� +we define φ : V ⊕W → P by (v, w) �−→ f(v)−g(w). Then, φ is also an F-linear linear map and (V ⊕W)/Γ ≃ im(φ), +where Γ = {(v, w) : f(v) = g(w)} is the space of global sections. +In particular, dim(Γ) = dim(V ) + dim(W) − dim(im(φ)) if the spaces V, W, P are finite-dimensional. In addition, +dim(Γ) = dim(V ) + dim(W) − dim(P) if φ is onto. +Proof. It is evident that φ is F-linear. Because (v, w) ∈ ker(φ) if and only if f(v) = f(w). By the first isomorphism +theorem of modules, the theorem follows. +8 + +(a) Xi +(b) X2 +(c) Xi U X2 +(d) X +二亚Ⅲ +三 +(e) Xi +(f) X2 +(g) Xi U X2 +(h) XA PREPRINT - JANUARY 16, 2023 +Let (X, X1, X2) be a local system of topological spaces and Γ the global section space of the sheaf structure +Hq(X1; F) −→ Hq(X; F) ←− Hq(X2; F). Examples shown in Figure 3 depict that the vector spaces Hq(X1; F), +Hq(X2; F), Hq(X1 ∪ X2; F), Hq(X; F), and Γ can be totally different. In other words, the global section space +provides an additional than the homology of X1, X2, X1 ∪ X2, and X. Actually, suppose we have a sequence +(Xi, Xi1, Xi2) of local systems that satisfy Xi1 ⊆ X(i+1)1, Xi2 ⊆ X(i+1)2, and Xi ⊆ Xi+1, then we have the +following commutative diagram: +Hq(X11; F) +res. +� +φ11 +� Hq(X21; F) +res. +� +φ21 +� Hq(X31; F) +� +res. +� +· · · +Hq(X1; F) +φ1 +� Hq(X2; F) +φ2 +� Hq(X3; F) +� · · · +Hq(X12; F) +res. +� +φ12 +� Hq(X22; F) +res. +� +φ22 +� Hq(X32; F) +� +res. +� +· · · +where φij and φi are the F-linear maps induced by the inclusions. One can check the following sequence is also valid: +Γ1 +φ11⊕φ12|Γ1 +� Γ2 +φ21⊕φ22|Γ2 +� Γ3 +� · · · . +In other words, except for computing single global section spaces, one can also consider the persistent homology of +global section spaces induced by any filtered local systems of topological spaces. +Theorem 1 presents a way to compute global section spaces. However, on many occasions, computing the image of φ +in the theorem may be infeasible. To tackle this, we previously proposed an approximation method using persistent +homology [30]. We quote the method as the following theorem. +Theorem 2 (Theorem 2.3.1 [30]). Let R be a commutative ring with identity. Let (X, X1, X2) be a local system +of topological spaces and q a non-negative integer. Let G1 be the short filtration ∅ ⊆ X1 ⊆ X1 ∪ X2 ⊆ X and +s2 ∈ Hq(X2; R) a non-zero element. Then the followings are equivalent: +(a) There is an s1 ∈ Hq(X1; R) such that (s1, s2) ∈ Hq(X1; R) ⊕ Hq(X2; R) is global section; +(b) �s2 := ω2(s2) has barcode (2, 3) in the PH Pq(G1) : 0 −→ Hq(X1; R) −→ Hq(X1 ∪ X2; R) −→ Hq(X; R). +For a local system (X, X1, X2), numbers of barcode (2, 3) in Pq(G1) records how many homological non-zero +generators in Hq(X2; R) that merge to a generator in Hq(X1; R). In [30], we defined it as the q-th local merging +number. +Definition 10 ([30]). Let (X, X1, X2) be a local system of topological spaces and q ≥ 0. We define the q-th local +merging number of X1 and X2 as the numbers of barcodes (2, 3) in Pq(G1) and denote it by mq(X1; X2). +We use the two pairs in Figure 3 to explain the local merging numbers. For the first row, we have m0(X1; X2) = 5 +since there are 5 connected components that merge to X1. On the other hand, m0(X2; X1) = 5. Similarly, the +m0(X1; X2) and m0(X2; X1) of the second row are 2 and 4, respectively. In particular, these two examples show +the local merging numbers m0(X1; X2) and m0(X2; X1) are not equal in general. Actually, one can prove that +max{m0(X1; X2), m0(X2; X1)} ≤ dim(Γ) ≤ m0(X1; X2) + m0(X2; X1) [29]. +When q = 0, the local merging numbers m0(X1; X2) records how many connected components in X2 connect to +components in X1 synchronously. In our previous work, we show that local regions with high 0-local merging numbers +are likely to be more joint parts of the ambient space and have the potential to analyze handwritten text with texture +data [30, 29]. In these works, we focus on local merging numbers in dimension 1 and (2, 3) barcodes in short filtrations, +while the geometric meanings of higher dimensional merging numbers and (3, +∞) barcodes are still unknown. +In the following theorem, we show that the number of barcodes (3, +∞) in a short filtration can verify whether X1 and +X2 contribute a hole (with dimension ≥ 1) in X. +Theorem 3. Let F be a field. Let (X, X1, X2) be a local system of topological spaces and q a non-negative integer. +Let Γ be the global section space of the sheaf structure Hq(X1; F) −→ Hq(X; F) ←− Hq(X2; F). Then the number of +(3, +∞) in the PH Pq(G1) : 0 −→ Hq(X1; F) −→ Hq(X1 ∪ X2; F) −→ Hq(X; F) equals +dimF (Hq(X; F)) − dimF (Hq(X1; F)) − dimF (Hq(X2; F)) + dimF (Γ). +9 + +A PREPRINT - JANUARY 16, 2023 +(a) A 6 × 6 image domain +(b) Rectangle R +(c) Boundary B of R +(d) �R = R \ B +(e) Black pixel set X +(f) X1 = X ∩ �R +(g) X2 = X \ R +(h) X1 ∪ X2 +Figure 4: An illustration of the construction of a local system in a 2D binary image. In this example, we have +m0(X1; X2) = 3, o0(X1; X2) = 0, m1(X1; X2) = 0, and o1(X1; X2) = 1. +Proof. Let ρ1 : Hq(X1; F) −→ Hq(X; F) and ρ2 : Hq(X2; F) −→ Hq(X; F) be the canonical linear transformations +that are induced by the inclusions. Define φ = ρ1 − ρ2 : Hq(X1) ⊕ Hq(X2) −→ Hq(X), then +dimF (Γ) = dimF (Hq(X1; F)) + dimF (Hq(X2; F)) − dimF (im(φ)) +(7) +by Theorem 1. Because Hq(X1 ∪ X2; F) is canonically isomorphic to Hq(X1; F) ⊕ Hq(X2; F), the images of ρ1 − ρ2 +and the map Hq(X1 ∪ X2; F) −→ Hq(X; F) in Pq(G1) are equal. Then the number of barcodes (3, +∞) in the +persistent homology Pq(G1) counts the dimension of the space Hq(X; F)/im(φ). Therefore, +#{barcode (3, +∞) in Pq(G1)} = dimF (Hq(X; F)) − dimF (im(φ)). +(8) +By plugging equation (7) into equation (8), the theorem follows. +If c ∈ Zq(X1; F) ⊆ Zq(X; F) is a q-cycle that represents a q-dimensional hole in X1, then c must have a barcode +(1, ⋆) in the persistent homology Pq(G1). On the other hand, c ∈ Zq(X2; F) ⊆ Zq(X; F) representing a hole in X2 +implies that it has a barcode (2, ⋆). In other words, the number of barcodes (3, +∞) in the persistent homology Pq(G1) +records how many q-holes in X are “supported” by both X1 and X2. In particular, removing either X1 or X2 will make +those holes disappear. Intuitively, those holes are constructed by gluing the parts by X1 and X2, and hence we have the +following definition. +Definition 11. Let (X, X1, X2) be a local system of topological spaces and q ≥ 0. We define the q-th local outer- +merging number of X1 and X2 as the numbers of barcodes (3, +∞) in Pq(G1) and denote it by oq(X1; X2). +From the above discussion, it can be seen that the local outer-merging number records the contribution of a specific +local area in the topological space to the hole structure. We present local outer-merging numbers for digital images in +the next section (Section 3.2). In addition, we will analyze the location of holes in the image by segmenting the image +and the local outer-merging number of the corresponding region. +3.2 +Local Systems in Binary Images +Section 3.1 introduces the local system and its persistent homology. Theorem 2 and Theorem 3 tell us that counting +the numbers of barcodes (2, 3) and (3, +∞) in (X, X1, X2) can detect the glue relationship of local objects in X. +Among them, constructing the admissible triad (X, X1, X2) is the most crucial part of the calculation. For an object +X in Rn and a bounded A ⊆ X, one can choose r1, r2 > 0 with r1 < r2 such that A ⊆ B(0, r1) and define +X2 = X ∩ {x ∈ Rn : |x| ≥ r2}. Then, clX(X1) ∩ clX(X2) = ∅. Based on the same idea, the section presents a more +efficient way to build local systems in binary images. +10 + +A PREPRINT - JANUARY 16, 2023 +(a) A 9 × 9 binary image +(b) A bounding box of the loop +(c) A better bounding region +Figure 5: An illustration of bounding regions of the hole structure. (a) A binary image that contains a single 1- +dimensional loop structure. (b) A rectangular bounding box formed by yellow and brown pixels. (c) A more compact +bounding region formed by red and purple pixels. +As we introduced in Section 2.2, a 2-dimensional image is identified as a non-negative real-valued function f : P −→ R≥0 +on a discrete 2D rectangle P = ([a, b] × [c, d]) ∩ Z2, where a, b, c, d are integers with a ≤ b and c ≤ d. In the paper, +we focus on the geometric realization of black pixels of a binary image and compute its homology (see Figure 2(d) and +Figure 4). For a binary image f : P −→ {0, 1}, we consider the black pixel set f −1(0) and denote it by X ⊆ P. We +use a rectangle in P to cover a concerned region of X, say R = ([a1, b1] × [c1, d1]) ∩ Z2 with a ≤ a1 ≤ b1 ≤ b and +c ≤ c1 ≤ d1 ≤ d (see Figure 4(b)). Consider +B = Z2 ∩ +� +({a1} × [c1, d1]) ∪ ({b1} × [c1, d1]) ∪ ([a1, b1] × {c1}) ∪ ([a1, b1] × {d1}) +� +as the boundary of R (see Figure 4(c)), we define �R = R\B (see Figure 4(d)). Defining X1 = X ∩ �R and X2 = X \R +(see Figure 4(e)-(h)), we obtain a triad (X, X1, X2) with the property X1 ∩ X2 = ∅. Because X, X1, and X2 are +subspaces in R2 that are formed by finitely many closed squares in R2, we must have clX(X1) ∩ clX(X2) = ∅. +For example, the local system (X, X1, X2) in Figure 4 has merging and outer-merging numbers m0(X1; X2) = 3, +o0(X1; X2) = 0, m1(X1; X2) = 0, and o1(X1; X2) = 1. In [30], we separated a 2D image into disjoint blocks X(i) +i +(called a local patches) and calculated the local merging numbers m0(X(i) +1 ; X(i) +2 ) to form a heatmap of the image. In +the paper, we mainly focus on the number o1(X1; X2) to approximate the hole positions in a binary image. We show in +Section 4 how to use the local system described in this section to construct local patches in the image and use them to +estimate the 1D holes in the image. +3.3 +Discussion +The organization of the section is as follows. First, we compare the proposed method with possible methods in Section +3.3.1. Second, Section 3.3.2 discusses how to estimate the size and shape of the holes in the topological space through +the local system. Finally, Section 3.3.3 discusses local systems composed of n subspaces and their local sections, which +will be an important future research direction to promote the theory of this paper. +3.3.1 +Comparison with other methods +To detect the local regions that contain pores in an m × n binary image, some naive methods can be used to tackle +this question. For example, one can search every subfigure of the given image and compute whether it contains hole +structures. However, it is generally infeasible to compute all the +�m +2 +� +· +�n +2 +� += (m2 − m)(n2 − n) +4 +, +subfigures for large m and n. Except for the computational complexity, covering an irregular hole costs a large bounding +rectangle and makes the estimation less precise and compact (see Figure 5). +11 + +A PREPRINT - JANUARY 16, 2023 +Following our previous approach [30], we have two strategies to reduce computational complexity through the image’s +local systems and sheaf information. The first is splitting the image into many pairs (X(i) +1 , X(i) +2 ) with disjoint X(i) +1 s +and computing each Pq(Gi,1) on the filtration Gi,1 : ∅ ⊆ X(i) +1 +⊆ X(i) +1 +∪ X(i) +2 +⊆ X. The second is to use a sliding +window technique to cover the entire image and compute short-persistent homology for each local window. The second +strategy computes the persistent homology O(mn) times, and pore locations generated by this strategy are usually more +refined than the first one. In the paper, we mainly follow the second strategy and show in Section 4 that the proposed +barcode and local system framework can detect local holes effectively and with more concise bounding regions than +bounding boxes (Figure 5 (b), (c)). +3.3.2 +Size Issues +As we mentioned in Section 2.2, homological generators generally lack specific geometric properties, such as the +size and stability of pores (Figure 1), which cannot be detected by traditional homology or persistent homology of +sub-level set filtration. In recent years, the shape and size of pore structures have become more and more important +topics in bioinformatics and material science [34, 35, 52, 55, 54, 4]. Recently, some research has shown the potential +and advantage of persistent homology in pore size analysis [33, 58, 32, 11]. +As far as the field of image processing is concerned, the combination of persistent homology and mathematical +morphology has opened up a new research direction for this field [20, 32, 11, 48]. In particular, our approach in [11] +applies morphological opening and closing to measure the spatial information of black and white regions in binary +images. Mathematical morphological operations can estimate the sizes of image pores, and most of the current work +focuses on the global description of such spatial information, such as the number of pores with a specified morphological +size and the average image pore size. However, the location information of pores in images is still limited in present +methods. Through the discussion in this section, we will see that localized systems can capture both the location and +size of pores, providing a richer pore analysis technique. +Theorem 4. Let F be a field. Let (X, X1, X2) be a local system of topological spaces and q a non-negative integer. +Then, non-zero elements in Hq(X1) and Hq(X2) in the persistent homology Pq(G1) : 0 −→ Hq(X1; F) −→ Hq(X1 ∪ +X2; F) −→ Hq(X; F) has birth number < 3. +Proof. Suppose s1 is a non-zero element in Hq(X1), then the birth number of s1 is 1. On the other hand, the assumption +of clX(X1)∩clX(X2) = ∅ forces that Hq(X1 ∪X2; F) ≃ Hq(X1; F)⊕Hq(X2; F) canonically. Then every non-zero +element in Hq(X2) must have barcode 2. +Corollary 1. Let X be a subspace of the n-dimensional Euclidean space Zn. Let F be a field and q a non-negative +integer. For every c ∈ Zq(X; F) with [c] ̸= 0 in Hq(X; F), there is a bounded set X1 ⊆ X and an X2 ⊆ X such that +clX(X1) ∩ clX(X2) = ∅ and c ∈ Hq(X1; F). In particular, [c] has a barcode (1, ⋆) in Pq(G1). +Proof. For a chain c in Sq(X; F), we can write c = �n +i=1 λiσi, where λi ∈ F \ {0} and σi : ∆q −→ X is a continuous +for each i. Recall that the support of c denoted by |c| is defined as the union of the images of the σi. Because ∆q +is compact, and σi is continuous, the support of c is a compact subset of X. In particular, the support of c is closed +and bounded. Therefore, we may choose a positive number r1 such that |c| ⊆ B(0, r1). Choose r2 > r1 and set +X1 = X ∩ B(0, r1) and X2 = X ∩ {x ∈ Rn : |x| ≥ r2}, then clX(X1) ∩ clX(X2) = ∅ and c ∈ Zq(X1; F). By the +proof of Theorem 4, [c] has a barcode (1, ⋆) in Pq(G1). +Definition 12. For convenience, we use iq(X1; X2) to denote the number of barcodes in Pq(G1). +Corollary 1 gives us a way to measure the size of q-holes (q > 0) for any subspace in Rn by choosing the local system +appropriately. More precisely, we can choose a bounded subspace X1 of X that contains the hole and is therefore an +approximation of the size of the hole. Also, since the location of X1 is known, it also keeps track of the hole location. +We also demonstrate in Section 4 an application of Corollary 1 to detect the "largest" holes in images. +3.3.3 +More general systems +In the paper, we focus on a local system consisting of topological spaces X, X1, X2 that satisfy X1 ⊆ X, X2 ⊆ X, +and clX(X1) ∩ clX(X2) = ∅. This system induces a sheaf structure as in Theorem 1, and its global section space +can be computed by the persistent homology of the short filtration. Actually, one can consider a more general case +consisting of n + 1 spaces X, X1, ..., Xn with clX(Xi) ∩ clX(Xj) = ∅ for i ̸= j. We synthesize the above settings +into the following definition and theorem. +12 + +A PREPRINT - JANUARY 16, 2023 +Definition 13. Let X be a topological space and X1, ..., Xn by subspaces of X that satisfy clX(Xi) ∩ clX(Xj) = ∅ +for i ̸= j. The (n + 1)-tuple (X, X1, ..., Xn) is called a local n-system (or an admissible (n+1)-tuple) of topological +spaces. +Furthermore, for an n-system (X1, ..., Xn), we can consider the diagram +Hq(X1) +ρ1 +� +Hq(Xk) +ρk +� Hq(X) +Hq(Xn) +ρn +� +with homologies and induced homomorphisms [28]. As above, we define its global section space by +Γ = +� +(s1, s2, ..., sn) ∈ +n +� +i=1 +Hq(Xi) : ρi(si) = ρj(sj) for i, j ∈ {1, 2, ..., n} +� +. +Theorem 5. Let (X, X1, ..., Xn) be a local n-system of topological spaces. Let R be a commutative ring with identity +and q ≥ 0 a non-negative integer. Consider the sheaf structure +Hq(X1; R) +ρ1 +� +Hq(Xk; R) +ρk +� Hq(X; R) +Hq(Xn; R) +ρn +� +of R-modules and module homomorphisms and the homomorphism +n +� +i=1 +Hq(Xi; R) +φ +−−−→ +n +� +i=2 +Hq(X; R), (si)n +i=1 �−→ (ρ1(s1) − ρi(si))n +i=2. +(9) +Let Γ be the global section space of the sheaf. Then Γ is the kernel of φ. +Proof. An n-tuple (si)n +i=1 is a global section if and only if ρi(si) − ρj(sj) = 0 for every i, j ∈ {1, 2, ..., n}. If it holds, +then ρ1(s1) − ρj(sj) = 0 for every j ∈ {2, ..., n}. Conversely, suppose ρ1(s1) − ρj(sj) = 0 for every j ∈ {2, ..., n}, +then ρi(si) − ρj(sj) = ρi(si) − ρ1(s1) + ρ1(s1) − ρj(sj) = 0 for every i, j ∈ {2, ..., n}, as desired. +Although Theorem 5 provides a way to sculpt the global section space, it still has a limitation in computation. When +n = 2, and R = F is a field, the homomorphism in (9) and Hq(X1 ∪ X2; F) −→ Hq(X; F) have the same image. In +this case, the approximation developed in Theorem 2 and Theorem 3 is available. However, the map in (9) and the +canonical one Hq(X1 ∪ X2; F) −→ Hq(X; F) are not coincident and can not be calculated by counting the barcodes in +the short filtration. Computing global sections through persistence barcodes is one of our future research directions. +4 +Demonstration on Digital Images +Hole structures in images can be subtle and complicated. As shown in Figure 6(a)1, although many white areas appear +in the image as porosity structures or closed voids, many of these areas connect to the white background and thus +1Karen Arnold has released this “Fingerprint Clipart” image under a Public Domain license. +https://www. +publicdomainpictures.net/en/view-image.php?image=462168&picture=fingerprint-clipart +13 + +A PREPRINT - JANUARY 16, 2023 +(a) Input image +(b) Marked holes +(c) Output heatmap +(d) Location estimation +Figure 6: A demonstration of Algorithm 1 on a binary image. (a) An illustration of a 500 × 700 fingerprint image, +where the Betti pair of the image is (β0, β1) = (92, 14). (b) The marked white regions as the 14 holes of the fingerprint +image. (c) The output heatmap by Algorithm 1. (d) The non-zero parts of the output heatmap, form an estimation for +the hole positions. Here we choose the window R as a 30 × 30 square with step k = 15. +are not actual holes. In order to detect the image’s hole positions, we propose Algorithm 1 to approximate the holes’ +geometric locations using the above theory and sliding window technique. Figure 6 is a demonstration of Algorithm 1 +on a binary image. We can see that all the holes in the image are detected by the output heatmap as Figure 6(c). +We note that Line 6 in Algorithm 1 considers both i1(X1; X2) and o1(X1; X2). If i1(X1; X2) ̸= 0, it means that X1 +contains some holes in X and already a bounding box of certain holes (Corollary 1). On the other hand, i1(X1; X2) +records whether (part of) the black pixels in X1 can contribute to hole structures and the number of these structures. +Therefore, the sum of i1(X1; X2) and o1(X1; X2) can estimate whether X1 nears a hole structure in X. +Algorithm 1 The hole structure detection algorithm. +Input: Binary image f : P −→ {0, 1} on a rectangle R, X = f −1(0), an n × n square window R, and a sliding step k. +Output: A function H : P → R as a heatmap of f. The heatmap estimates the hole locations in image f. A point in +P with a high heat value is more possible as a part of a hole. +1: Denote P = ([0, a] × [0, b]) ∩ Z2 and R = ([0, n] × [0, n]) ∩ Z2. Define B and �R as in Section 3.2. +2: Set H : P → R as the zero function. +3: for i ∈ {0, 1, ..., a} and j ∈ {0, 1, ..., b} do +4: +if (i · k, j · k) + R ⊆ P then +5: +Set X1 = ((i, j) + �R) ∩ X and X2 = X \ ((i, j) + R) +6: +Compute M = i1(X1; X2) + o1(X1; X2) +7: +Define H′ : P −→ {0, 1} as follows: +H′(x) = +� +H(x) + M +if x ∈ (i, j) + �R, +0 +otherwise. +8: +H ←− H′ +9: +else +10: +continue +11: +end if +12: end for +13: return H′ · (1 − f) +Apart from the task of detecting the location of holes in an image, recognizing the size or shape of holes is also an +interesting one. As introduced in Section 3.3.2, local windows that contain holes will produce (1, 2) barcodes in +the corresponding short persistent homology. Based on the observation, we can modify Algorithm 1 by considering +14 + +A PREPRINT - JANUARY 16, 2023 +(a) Input image +(b) 50 × 50 +(c) 100 × 100 +(d) 150 × 150 +(e) Sum of (b)-(d) +(f) Estimation +Figure 7: A demonstration of Algorithm 2 on a binary image. (a) An illustration of a 200 × 300 binary image with Betti +pair (β0, β1) = (2, 28). (b), (c), and (d) are the output heatmaps of Algorithm 2 with local windows of sizes 50 × 50, +100 × 100, and 150 × 150, respectively. (e) The sum heatmap of (b), (c), and (d). (e) The non-zero parts of the output +heatmap, form an estimation for the hole positions. Here we choose the step k = 25. +the information on the size of local windows and changing the M value to tackle this task. As follows, we propose +Algorithm 2 as a modification of Algorithm 1. We note here that the we implement these two algorithms in Python +with the Gudhi package [47]. +Algorithm 2 The hole size estimation algorithm. +Input: Binary image f : P −→ {0, 1} on a rectangle R, X = f −1(0), an n × n square window R, and a sliding step k. +Output: A function H : P → R as a heatmap of f. The heatmap estimates the hole locations in image f. A point in +P with a high heat value is more possible as a part of a “large” hole. +1: Denote P = ([0, a] × [0, b]) ∩ Z2 and R = ([0, n] × [0, n]) ∩ Z2. Define B and �R as in Section 3.2. +2: Set H : P → R as the zero function. +3: for i ∈ {0, 1, ..., a} and j ∈ {0, 1, ..., b} do +4: +if (i · k, j · k) + R ⊆ P then +5: +Set X1 = ((i, j) + �R) ∩ X and X2 = X \ ((i, j) + R) +6: +Compute M = vol(R) · (o1(X1; X2) − i1(X1; X2)) +▷ The only difference to Algorithm 1 +7: +Define H′ : P −→ {0, 1} as follows: +H′(x) = +� +H(x) + M +if x ∈ (i, j) + �R, +0 +otherwise. +8: +H ←− H′ +9: +else +10: +continue +11: +end if +12: end for +13: return H′ · (1 − f) +We note that the only difference between Algorithm 1 and Algorithm 2 is the value of M in line 6 of both algorithms. +As we discussed above, i1(X1; X2) > 0 means that some holes are bounded by the area X1. Due to this, we apply +−vol(R) · i1(X1; X2) as the punishment term of the region’s heat value. In addition, since punishment will produce +negative values, if a certain area has many fine holes, the heat function will have a high negative value in this area and +also estimates the location of these holes. +Figure 7 demonstrates Algorithm 2 on a binary image with hole structures in different sizes. Within different local +windows (50 × 50, 100 × 100, and 150 × 150 squares), Algorithm 2 gives different attention to these holes. We notice +that small holes may get more attention from Algorithm 2 (Figure 7(b)) since there are many holes next to each other, +and hence the shared edges will get a higher heat value. Keeping enlarging the size of the local window, we observe +that smaller holes in the image would get more punishment in heat values. The sum of the three heatmaps summarizes +15 + +.A PREPRINT - JANUARY 16, 2023 +the “importances” of black pixels in the image. Finally, we see that Algorithm 2 successfully approximates the position +of the “largest hole” by the thresholding method. +However, real pore structure data may be more complex than the images presented in the paper. Methods to use barcode +information in pore structure analysis, such as the choice of the penalty function, still need research and development, +which is our main development work in the future. +5 +Conclusion +To summarize the paper, we propose a local system and local persistent homology framework to study the local merging +relations in an arbitrary topological space. By using the merging and out-merging numbers of local regions, we propose +an algorithm to detect the sizes and positions of pores in the image. Although the demonstration focuses on digital +images, the framework can be adapted to any topological space with local systems. We also look forward to applying +this framework to point-cloud data, especially its applications in crystalline data analysis. +Acknowledgement +Most of the work in this article was completed by the author during his doctoral study at National Taiwan Normal +University (2016-2022). The author would like to thank Dr. Chun-Chi Lin (NTNU) and Dr. Yu-Min Chung (Eli Lilly +and Company), the author’s doctoral supervisors, for their comments and suggestions on the work. Especially, Dr. +Yu-Min Chung provided many suggestions for studying the geometric meaning of persistent barcodes and the global +sections of the local system, making the discussion more fruitful and rigorous. The author would also like to thank Dr. +Kelin Xia, the author’s postdoctoral supervisor at Nanyang Technological University. The author got a lot of inspiration +from the discussion with Dr. Xia so that this paper can have more research directions, such as a more detailed study of +the geometry of cellular sheaves, and more possible applications. +References +[1] H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta, +and L. Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine +Learning Research, 18, 2017. +[2] N. Akai, T. Hirayama, and H. Murase. Persistent homology in lidar-based ego-vehicle localization. In 2021 IEEE +Intelligent Vehicles Symposium (IV), pages 889–896, 2021. +[3] D. V. Anand, Q. Xu, J. Wee, K. Xia, and T. C. Sum. Topological feature engineering for machine learning based +halide perovskite materials design. npj Computational Materials, 8(1):1–8, 2022. +[4] G. Bassu, M. Laurati, and E. Fratini. Microgel dynamics within the 3d porous structure of transparent peg +hydrogels. Colloids and Surfaces B: Biointerfaces, 221:112938, 2023. +[5] P. Bubenik, G. Carlsson, P. T. Kim, and Z.-M. Luo. Statistical topology via morse theory persistence and +nonparametric estimation. Algebraic methods in statistics and probability II, 516:75–92, 2010. +[6] P. Bubenik et al. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res., +16(1):77–102, 2015. +[7] G. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255–308, 2009. +[8] G. Carlsson and F. Mémoli. Multiparameter hierarchical clustering methods. In Classification as a Tool for +Research, pages 63–70. Springer, 2010. +[9] F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo, A. Rinaldo, and L. Wasserman. Robust topological inference: +Distance to a measure and kernel distance. The Journal of Machine Learning Research, 18(1):5845–5884, 2017. +[10] Y.-M. Chung and S. Day. Topological fidelity and image thresholding: A persistent homology approach. Journal +of Mathematical Imaging and Vision, 60(7):1167–1179, 2018. +[11] Y.-M. Chung, S. Day, and C.-S. Hu. A multi-parameter persistence framework for mathematical morphology. +Scientific reports, 12(1):1–25, 2022. +[12] Y.-M. Chung and A. Lawson. Persistence curves: A canonical framework for summarizing persistence diagrams. +Advances in Computational Mathematics, 48(1):1–42, 2022. +[13] A. De, T. Vo, and M. Wright. Value-offset bifiltrations for digital images. Computational Geometry, 109:101939, +2023. +16 + +A PREPRINT - JANUARY 16, 2023 +[14] V. De Silva and R. Ghrist. Coordinate-free coverage in sensor networks with controlled boundaries via homology. +The International Journal of Robotics Research, 25(12):1205–1222, 2006. +[15] O. Delgado-Friedrichs, V. Robins, and A. Sheppard. Morse theory and persistent homology for topological +analysis of 3d images of complex materials. In 2014 IEEE International Conference on Image Processing (ICIP), +pages 4872–4876. IEEE, 2014. +[16] H. Edelsbrunner. Persistent homology in image processing. In International Workshop on Graph-Based Represen- +tations in Pattern Recognition, pages 182–183. Springer, 2013. +[17] H. Edelsbrunner and J. Harer. Persistent homology-a survey. Contemporary mathematics, 453:257–282, 2008. +[18] H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. American Mathematical Society, 01 +2010. +[19] P. Frosini. Measuring shapes by size functions. In Intelligent Robots and Computer Vision X: Algorithms and +Techniques, volume 1607, pages 122–133. SPIE, 1992. +[20] A. Garin and G. Tauzin. A topological" reading" lesson: Classification of mnist using tda. In 2019 18th IEEE +International Conference On Machine Learning And Applications (ICMLA), pages 1551–1556. IEEE, 2019. +[21] K. Garside, A. Gjoka, R. Henderson, H. Johnson, and I. Makarenko. Event history and topological data analysis. +Biometrika, 108(4):757–773, 2021. +[22] R. Ghrist. Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society, 45(1):61–75, +2008. +[23] R. Ghrist and A. Muhammad. Coverage and hole-detection in sensor networks via homology. In IPSN 2005. +Fourth International Symposium on Information Processing in Sensor Networks, 2005., pages 254–260. IEEE, +2005. +[24] W. Gong, J. Wee, M.-C. Wu, X. Sun, C. Li, and K. Xia. Persistent spectral simplicial complex-based machine +learning for chromosomal structural analysis in cellular differentiation. Briefings in Bioinformatics, 2022. +[25] M. J. Greenberg and J. R. Harper. Algebraic Topology, A First Course. Addison-Wesley Publishing Company, +1980. +[26] D. Gunther, J. Reininghaus, I. Hotz, and H. Wagner. Memory-efficient computation of persistent homology for +3d images using discrete morse theory. In 2011 24th SIBGRAPI Conference on Graphics, Patterns and Images, +pages 25–32. IEEE, 2011. +[27] A. Hatcher. Algebraic topology. Cambridge Univ. Press, Cambridge, 2000. +[28] C.-S. Hu. A brief note for sheaf structures on posets. arXiv preprint arXiv:2010.09651, 2020. +[29] C.-S. Hu. Sheaf Structures on the Multi-parameter Persistent Homology Arising from Mathematical Morphology. +PhD thesis, National Taiwan Normal University, 2022. +[30] C.-S. Hu and Y.-M. Chung. A sheaf and topology approach to detecting local merging relations in digital images. +In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4396–4405, +2021. +[31] C.-S. Hu, A. Lawson, J.-S. Chen, Y.-M. Chung, C. Smyth, and S.-M. Yang. Toporesnet: A hybrid deep learning +architecture and its application to skin lesion classification. Mathematics, 9(22):2924, 2021. +[32] C.-S. Hu, A. Lawson, Y.-M. Chung, and K. Keegan. Two-parameter persistence for images via distance transform. +In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4176–4184, 2021. +[33] S. Ishihara, G. Franks, and J. Kano. Effect of particle packing structure on the elastic modulus of wet powder +compacts analyzed by persistent homology. Advanced Powder Technology, 34(1):103874, 2023. +[34] F. Jiang, T. Tsuji, and T. Shirai. Pore geometry characterization by persistent homology theory. Water Resources +Research, 54(6):4150–4163, 2018. +[35] Y. Jiao, F. Stillinger, and S. Torquato. Modeling heterogeneous materials via two-point correlation functions. ii. +algorithmic details and applications. Physical Review E, 77(3):031135, 2008. +[36] T. Kaczynski, K. Mischaikow, and M. Mrozek. Computational Homology. Applied Mathematical Sciences. +Springer New York, 2004. +[37] H. Kannan, E. Saucan, I. Roy, and A. Samal. Persistent homology of unweighted complex networks via discrete +morse theory. Scientific reports, 9(1):1–18, 2019. +[38] A. D. Keros, V. Nanda, and K. Subr. Dist2cycle: A simplicial neural network for homology localization. In +Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 7133–7142, 2022. +17 + +A PREPRINT - JANUARY 16, 2023 +[39] J. Latschev. Vietoris-rips complexes of metric spaces near a closed riemannian manifold. Archiv der Mathematik, +77(6):522–528, 2001. +[40] K. Mischaikow and V. Nanda. Morse theory for filtrations and efficient computation of persistent homology. +Discrete & Computational Geometry, 50(2):330–353, 2013. +[41] J. R. Munkres. Elements Of Algebraic Topology. CRC Press, 2018. +[42] J. L. Nielson, J. Paquette, A. W. Liu, C. F. Guandique, C. A. Tovar, T. Inoue, K.-A. Irvine, J. C. Gensel, J. Kloke, +T. C. Petrossian, et al. Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain +injury. Nature communications, 6(1):1–12, 2015. +[43] A. Onuchin and O. Kachan. Individual topology structure of eye movement trajectories. In International +Conference on Neuroinformatics, pages 45–55. Springer, 2023. +[44] H. Riihimäki, W. Chachólski, J. Theorell, J. Hillert, and R. Ramanujam. A topological data analysis based +classification method for multiple measurements. BMC bioinformatics, 21(1):1–18, 2020. +[45] V. Robins. Towards computing homology from finite approximations. In Topology proceedings, volume 24, pages +503–532, 1999. +[46] B. Stolz-Pretzer. Global and local persistent homology for the shape and classification of biological data. PhD +thesis, University of Oxford, 2019. +[47] The GUDHI Project. GUDHI User and Reference Manual. GUDHI Editorial Board, 2015. +[48] S. Tymochko, E. Munch, J. Dunion, K. Corbosiero, and R. Torn. Using persistent homology to quantify a diurnal +cycle in hurricanes. Pattern Recognition Letters, 133:137–143, 2020. +[49] M. Usher and J. Zhang. Persistent homology and floer–novikov theory. Geometry & Topology, 20(6):3333–3430, +2016. +[50] R. Vandaele, T. De Bie, and Y. Saeys. Local topological data analysis to uncover the global structure of data +approaching graph-structured topologies. In Joint European Conference on Machine Learning and Knowledge +Discovery in Databases, pages 19–36. Springer, 2019. +[51] J. W. Vick. Homology Theory, A Introduction to Algebraic Topology. Springer-Verlag Publishing Company, +Second Edition, 1973. +[52] H.-J. Vogel and K. Roth. Quantitative morphology and network representation of soil pore structure. Advances in +water resources, 24(3-4):233–242, 2001. +[53] J. Wee and K. Xia. Persistent spectral based ensemble learning (perspect-el) for protein–protein binding affinity +prediction. Briefings in Bioinformatics, 23(2), 2022. +[54] K. Wei, Q. Wang, and C.-P. Huang. The distribution of adsorption energy of u (vi) onto aeptes-functionalized +porous silica with multiple average pore sizes. Chemical Engineering Journal, 451:138716, 2023. +[55] C. Wu, Z. Li, Y. Li, J. Wu, Y. Zhao, and Y. Liao. Effect of starch on pore structure and thermal conductivity of +diatomite-based porous ceramics. Ceramics International, 49(1):383–391, 2023. +[56] K. Xia and G.-W. Wei. Persistent homology analysis of protein structure, flexibility, and folding. International +journal for numerical methods in biomedical engineering, 30(8):814–844, 2014. +[57] X. Xu, J. Cisewski-Kehe, S. B. Green, and D. Nagai. Finding cosmic voids and filament loops using topological +data analysis. Astronomy and Computing, 27:34–52, 2019. +[58] Y. Yamauchi, T. Yatagawa, Y. Ohtake, and H. Suzuki. Bin-scanning: Segmentation of x-ray ct volume of binned +parts using morse skeleton graph of distance transform. Computational Visual Media, 9(2):319–333, 2023. +[59] A. Zomorodian and G. Carlsson. Computing persistent homology. In Proceedings of the twentieth annual +symposium on Computational geometry, pages 347–356, 2004. +18 + diff --git a/utE5T4oBgHgl3EQfLg6S/content/tmp_files/load_file.txt b/utE5T4oBgHgl3EQfLg6S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..823a1444506d68447634bbc60ad373083fdff2ca --- /dev/null +++ b/utE5T4oBgHgl3EQfLg6S/content/tmp_files/load_file.txt @@ -0,0 +1,1404 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf,len=1403 +page_content='LOCATING TOPOLOGICAL STRUCTURES IN DIGITAL IMAGES VIA LOCAL HOMOLOGY A PREPRINT Chuan-Shen Hu School of Physical and Mathematical Sciences Nanyang Technological University 50 Nanyang Avenue 639798, Singapore chuanshen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='hu@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='sg peterbill26@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='com January 16, 2023 ABSTRACT Topological data analysis (TDA) is a rising branch in modern applied mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' It extracts topological structures as features of a given space and uses these features to analyze digital data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology, one of the central tools in TDA, defines persistence barcodes to measure the changes in local topologies among deformations of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Although local spatial changes characterize barcodes, it is hard to detect the locations of corresponding structures of barcodes due to computational limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The paper provides an efficient and concise way to divide the underlying space and applies the local homology of the divided system to approximate the locations of local holes in the based space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We also demonstrate this local homology framework on digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Keywords Topological data analysis · Persistent homology · Local hole structures · Persistence barcodes · Local systems and patches · Short filtrations · Cellular sheaves · Global sections · Merging and outer-merging numbers 1 Introduction Homology is an algebraic description of topological spaces and has become one of the foundations of modern geometry and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' It uses algebra to detect genera in topological spaces, such as loops and high-dimensional voids, and to classify the topological types and shapes of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In addition to its importance in pure mathematics, over the past two decades or so, data scientists have noticed the benefits and potential of homology in numerical data and raised a new field called topological data analysis (TDA) [59, 7, 22, 8, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology plays a central role in TDA, which transforms a sequence of topological spaces linked by continuous functions into a homology chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' By checking the birth and death of elements in the chain, one can understand which homological generator can have a longer lifespan and shows its importance in the continuous process [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology and related techniques have been applied in many data science tasks, such as bioinformatics [44, 42, 31], molecular analysis [56, 24, 3, 53], image processing [11, 10, 16, 13, 43], and material science [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent barcodes (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2) record the lifespans of connected components, loops, and voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Many applications use persistence barcodes and related statistical features as machine learning features [6, 1, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Although persistent homology and persistence barcode has shown their potential in many real applications, it still has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' One is it can only capture the global information of how connected components and holes behave during geometric deformation, while the local merging relations are usually omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This information is theoretically present in the definition of persistent homology and persistent barcodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, for computational efficiency, hole representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', q-circular representation of q-holes) or positions are often buried in the Gaussian elimination of the matrices in the computation of persistence barcodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='05474v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='AT] 13 Jan 2023 A PREPRINT - JANUARY 16, 2023 Recently, some scholars noticed the importance of local information on persistent homology and proposed some interesting works on the local behavior of persistent homology [50, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For example, Vandaele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [50] investigated the local Vietoris-Rips complexes of the point cloud and applied the local Betti pairs to form a global descriptor of the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This descriptor can be viewed as a heatmap of the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Regions with higher heat values usually mean they have more significant topological/geometric information, such as higher local branch numbers or loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Also, Stolz described in her doctoral dissertation [46] how to apply the Mayer–Vietoris sequence to compute the local Vietoris–Rips complex linked from a data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' On the other hand, some of the research also aims to detect the locations of loop or hole structures in the topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For example, Akai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [2] generate persistence barcodes of the Vietoris–Rips complex as inputs of a neural network model and apply them for the ego-vehicle localization application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Similarly, Keros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [38] train on a Hodge Laplacian-based graph neural network to detect the nearest optimal homology as a location representation of homologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Furthermore, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [57] apply the distance measurement (DTM) function [9] to enhance the robustness of Vietoris–Rips complex construction, and apply persistence and distance information to detect holes and voids in point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, while image structures are more regular than point clouds, making it easier to compare local-global attributes, most current methods are designed for point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theoretical assurance methods for localized hole detection are still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The paper provides a theoretically guaranteed framework for hole position detection in arbitrary topological spaces, demonstrated on digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This paper is an extension of our previous work presented as a workshop paper at CVPR 2021 (2021 Conference on Computer Vision and Pattern Recognition) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The work [46] introduces the concept of cellular sheaves and connects them to persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In [46], we define the local merging number can consider its geometric meaning in 0-dimensional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This paper extends the framework in [30] and focuses on 1-dimensional merging relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In addition to theoretical promotion, we have a preliminary demonstration of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' It shows that the 1-dimensional merging relations can estimate the position of holes in the space, which provides a way to analyze the local topological characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1 Organization The organization of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Section 2 quickly recaps the homology, Betti numbers, persistent homology, and barcodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We present the main results in Section 3 and separate the section into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1 introduces how we divide the ambient space by a local region and apply the divided system to compute its persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We also interpret the geometric meaning of the computed barcodes and explain how they detect the cycle locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We also compare the proposed framework with previous methods in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Section 4 shows how to adapt the theory developed in Section 3 on digital images and demonstrates the proposed locating method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Finally, we discuss future directions and summarize the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 2 Persistent Homology and Barcodes We briefly introduce the standard notions and terminologies of singular homology, including its functoriality, Betti numbers, and geometric meanings in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2 focuses on persistent homology and barcodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We will also show in this section typical ways for building filtrations, especially the construction relying on the thresholding technique, which is the foundation of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1 Homology This section briefly recalls the singular homology and related properties of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' One can find these materials in several classic textbooks on algebraic topology [27, 51, 41, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We start the section with the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For any non-negative integer q, we define the geometric q-simplex, denoted by ∆q, as the convex hull of the standard basis {e0, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', eq} for the (q + 1)-dimensional Euclidean space Rq+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' That is, ∆q = conv(e0, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', eq) = � t0e0 + t1e1 + · · · + tqeq : ti ∈ [0, 1] and q � i=0 ti = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For any (q + 1) points x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', xq in Rn we can define the affine map [x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', xq] : ∆q → Rn by (t0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', tq) �−→ t0x0 + · · · + tqxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (1) 2 A PREPRINT - JANUARY 16, 2023 Then [x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', xq] is a continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A continuous function from ∆q to a topological space X is called a singular q-simplex in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, any affine map [x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', xq] : ∆q → Rn is a singular q-simplex in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For q ∈ Z≥0 and i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', q + 1} we define f i q+1 = [e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', �ei, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', eq+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In other words, f i q+1 is a singular (q − 1)-simplex in Rq+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' One can see that the image of f i q+1 is actually the convex hull of the set {e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', �ei, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', eq+1}, which is the i-th (q − 1)-face of the geometric simplex ∆q [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let X be a topological space, R a commutative ring with identity, and q a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We define Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) as the free R-module generated by all continuous maps σ : ∆q → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For convenience, we usually define Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) = 0 for q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The singular simplexes give us a way to express geometric simplexes in arbitrary topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Euclidean spaces, one can explore the faces as boundaries of geometric simplexes by using convex analysis, while it is not applicable in general spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In algebraic topology, we use the following boundary maps to read the boundary data of singular simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let X be a topological space, R a commutative ring with identity, and q ∈ N a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The q-boundary map is the function ∂q : Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) → Sq−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) that extends by the mapping σ �−→ q � i=0 (−1)i · σ ◦ f i q for all continuous σ : ∆q → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Note that ∂q is well-defined since each σ ◦ f i q is a singular (q − 1)-simplex in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) is defined as the zero space for q < 0, we also define ∂q as the zero maps for q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The following proposition is the foundation of homology theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proposition 1 ([25], (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let X, R, q and ∂q be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then ∂q−1 ◦ ∂q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The equation ∂q−1 ◦ ∂q = 0 shows that im(∂q) ⊆ ker(∂q−1) for every q ∈ Z, and hence we can define the q-th singular homology of X as the R-module Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) = ker(∂q) im(∂q+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Notation ([25, 51, 41, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' To simply the notations, for a topological space X and q ≥ 0, we use Zq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) and Bq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) to denote the modules ker(∂q) and im(∂q+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' That is, Zq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) = ker(∂q), Bq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) = im(∂q+1), and Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) = Zq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) Bq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (2) Chains in Zq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) and Bq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) are called the q-cycles and q-boundaries of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Except for sending each topological space X to an R-module Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R), for every continuous map f : X → Y and q ∈ Z≥0 we can define an R-module homomorphism Sq(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) : Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) → Sq(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) that extends the mapping σ �−→ f ◦ σ for all singular q-simplexes σ : ∆q → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Note that the mapping is well-defined since f ◦ σ is also a continuous map from ∆q to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This observation leads to the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proposition 2 ([25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let Top and ModR be the categories of topological spaces and R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For each q ∈ Z≥0, the assignments X ∈ Ob(Top) �→ Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) and f ∈ HomTop(X, Y ) �→ Sq(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) form a functor from Top to ModR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In fact, for a continuous map f : X → Y , one can prove that the rectangles in the ladder · · � Sq+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ∂q+1(X) � Sq+1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='R) � Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ∂q(X) � Sq(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='R) � Sq−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) � Sq−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='R) � · · · · � Sq+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ∂q+1(Y ) � Sq(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ∂q(Y ) � Sq−1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) � · · · of R-modules and R-module homomorphisms are commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Therefore, for every q, this ladder induces an R-module homomorphism Hq(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) : Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) −→ Hq(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) that sends each equivalence class [c] in Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) to the class [Sq(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R)(c)] in Hq(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Furthermore, we can see that the assignment Hq(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) of topological spaces and continuous maps also forms a functor from Top to ModR: 3 A PREPRINT - JANUARY 16, 2023 (a) f = f0 (b) f1 (c) f2 (d) f3 (e) f4 (f) g = g0 (g) g1 (h) g2 (i) g3 (j) g4 Figure 1: Two filtrations of 2-dimensional black pixels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' that is, f −1 0 (0) ⊆ f −1 1 (0) ⊆ f −1 2 (0) ⊆ f −1 3 (0) ⊆ f −1 4 (0) and g−1 0 (0) ⊆ g−1 1 (0) ⊆ g−1 2 (0) ⊆ g−1 3 (0) ⊆ g−1 4 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Although images f and g share the same 1-dimensional homology space Z2, the persistent homologies of these two images depict different lifespans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Indeed, the 1-dimensional hole in (a)-(e) has the barcode (0, 2) while the hole in (f)-(j) has (0, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proposition 3 ([25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let Top and ModR be the categories of topological spaces and R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For each q ∈ Z≥0, the assignments X ∈ Ob(Top) �→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) and f ∈ HomTop(X, Y ) �→ Hq(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) form a functor from Top to ModR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' An important purpose of developing singular homology is to detect holes in a topological space in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This property of singular homology is sometimes called the Poincaré lemma of singular homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We state this lemma as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proposition 4 (Corollary (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='5), [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let n ≥ 1 be a positive integer, and let Sn = {(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', xn+1) ∈ Rn+1 : x2 1 + · · · + x2 n+1 = 1} be the n-sphere in Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then, for every commutative ring R with identity and a non-negative integer q ≥ 0, we have Hq(Sn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ≃ �R if q = n or q = 0, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (3) In particular, for every topological space X, we have H0(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ≃ Rm, where m is the number of path-connected components of X, and each path-connected component of X can be represented by a constant function from [0, 1] to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The Poincaré lemma provides us with a reliable measurement to detect the number of q-dimensional holes in a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This number is called the q-th Betti number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 4 ([25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let R be a PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For any topological space X and integer q ≥ 0, we define the q-th Betti number βq = βq(X) of X to be the rank of the R-module Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, when R = F is a field, we have βq = dimF Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In applications, we often set R as the binary field Z2 = Z/2Z and simplify the notation Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Z2) to Hq(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In the paper, we will focus on homology over Z2 and the singular homology of (binary) images (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2 Prescient Homology Homology detects the hole structure in a given topological space, while it may omit some geometry of the based space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For example, two geometric objects with a single 1-dimensional hole in different sizes share the same first homology group (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As a generalization of homology, persistent homology (PH) concerns sequences of topological spaces and their homologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' It was motivated by the works related to the Morse theory of Patrizio Frosini [19] and Vanessa Robins [45] in the 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Morse theory, a height function f : M → R on a smooth manifold M can form a sublevel 4 A PREPRINT - JANUARY 16, 2023 set filtration of subspaces of M [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The topological changes of such sublevel sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', the changes of Betti numbers) track the shape of M along the direction of the height function and hence a descriptor (or fingerprint) of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology of height functions is now a well-known and fundamental tool in Morse theory and has many applications in theory [40, 5, 49] and data science [10, 15, 26, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' More generally, besides the smooth structures, suppose we have a sequence X1 f1 −→ X2 f2 −→ · · · fn−1 −−−→ Xn of topological spaces and continuous maps, then the functoriality of singular homology shown in Proposition 3 induces a sequence of homologies as follows: Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) Hq(f1) −−−−→ Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) Hq(f2) −−−−→ · · · Hq(fn−1) −−−−−−→ Hq(Xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R), where q is an arbitrary non-negative integer, and Hq(Xi), Hq(fi) are vector spaces and linear transformations over Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because continuous maps can deform the geometry of spaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', sizes, lengths, and connectivity), the changes in homological cycles and Betti numbers depict how the hole structures in the spaces changed among the continuous deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Computing homologies connected by continuous maps is challenging in real applications, so one usually considers a chain of filtered topological spaces with subspace relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A tower of such topological spaces is called a filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We list the formal definition of filtration as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 5 ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A filtration of topological spaces is a sequence ∅ = X0, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', Xn of topological spaces such that Xi is a subspace of Xi+1 for each i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We usually use the chain F : ∅ = X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn of topological spaces to denote a filtration of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because Hq(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) : Top → ModR is a functor, a filtration of topological spaces ∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn and a non-negative integer q ≥ 0 induce a sequence of R-modules and R-module homomorphisms: PHq : 0 = Hq(∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ρ0,1 −−→ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ρ1,2 −−→ Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) → · · · → Hq(Xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) (4) where the R-module homomorphism ρi,j : Hq(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) → Hq(Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) for i ≤ j is induced by the inclusion Xi �→ Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Based on the functoriality of singular homology on the sequence (4), we define ρi,j = ρj−1,j ◦ρj−2,j−1 ◦· · ·◦ρi,i+1 for every i ≤ j in {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', n}, then the ρi,j is also the R-module homomorphism induced by the inclusion map Xi �→ Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 6 ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Suppose F : ∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn is a filtration of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then, for every ring R and q ∈ Z≥0, we call the sequence defined in (4) is the q-th persistent homology of the filtration F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' One of the primary purposes of persistent homology is to track the lifespans of local holes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', the births/deaths of connected components, loops, and higher dimensional voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' To tackle this problem, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Edelsbrunner and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Harer proposed the persistence barcode of persistent homology to detect such topological changes [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We refer to the definition of persistence barcodes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 7 ([17, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Suppose ∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn is a filtration of topological spaces and 0 → Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) → · · · → Hq(Xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) is the induced qth persistent homology over a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let si be an element in Hq(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) (i ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then we have the following definitions: (a) si is said to be born at i if si /∈ im(ρi−1,i), i is called the birth of si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (b) si is said to die at j if ρi,j−1(si) /∈ im(ρi−1,j−1) and ρi,j(si) ∈ im(ρi−1,j), j is called the death of si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' If si is still alive at n, we define the death of si to be +∞ (up to this filtration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The tuple (i, j) of si is called the persistence barcode of the element si ∈ Hq(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The multiset of all persistence barcodes of non-repeated representative generators in all Hq(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) is called the persistence diagram of the filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For example, by considering the geometry of 2D black objects, rows in Figure 1 define two filtrations of subspaces in R2, and the induced first persistent homologies (over Z2) are Z2 idZ2 � Z2 � 0 � 0 � 0 and Z2 idZ2 � Z2 idZ2 � Z2 idZ2 � Z2 � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (5) By definition, the 1-dimensional hole in Figure 1(a)-(e) has the barcode (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' On the other hand, the hole in Figure 1(f)-(j) has barcode (0, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' There are many different ways to construct filtrations and compute their persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A typical one is the Vietoris–Rips complexes for the point-cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For a (finite) set X in the n-dimensional Euclidean space Rn and 5 A PREPRINT - JANUARY 16, 2023 a fixed positive real number ϵ > 0, one explores the intersections of n-dimensional balls centered at points x in X with radius ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Regarding points in X as the vertices of a simplicial complex, higher repeated regions lead to higher dimensional simplexes in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The strategy of the Vietoris–Rips complex is to enlarge the radius to construct a filtration of simplicial complexes [39, 23, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As shown in Figure 1 and equation (5), except for the point-cloud data, one can also construct filtrations of digital images and compute their persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We referred to an m-dimensional digital image as a function f : P −→ R≥0 from a non-empty set P of Zm to the set of all non-negative real numbers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' An image f is called binary if its range is contained in the binary set {0, 1} and called grayscale for otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For a binary image, the primage of zero f −1(0) referred to the set of all black pixels of f, and f −1(1) denotes the set of all white pixels of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Viewing each black pixel as a closed cube in Rm, we regard f −1(0) as a subspace of Rm and consider its topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The first row in Figure 2 provides examples of 2-dimensional grayscale and binary digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As in Figure 1, one can construct filtrations of images by using image processing techniques on a given binary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Another typical method of building filtrations is operating the sub-level sets of a grayscale image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For a image f : P −→ R≥0 and a threshold t ∈ R, we define a binary image ft : P −→ {0, 1} by setting ft(x) = 0 if f(x) ≤ t and ft(x) = 1 for otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then f −1 t1 (0) ⊆ f −1 t2 (0) ⊆ · · · ⊆ f −1 tn (0) for t1 ≤ t2 ≤ · · · ≤ tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The second the third row in Figure 2 illustrate how sub-level sets of a grayscale image form a filtration of black pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, the 0-th and 1-th persistence diagrams of the filtrations are {(0, +∞)} and {(0, 3), (2, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For readers who are interested in persistent homology on digital images, see [36] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In this paper, we focus on 2-dimensional binary images and their local homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We combine image segmentation techniques, local homology, and persistence barcodes to illustrate how to estimate and detect the positions of holes in 2D binary images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The combination of this detection method with more image processing techniques (such as mathematical morphology and sub-level set filtration) will be our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3 Our Approaches The section is separated into three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' First, we quote the definitions of local systems and short persistent homology in our previous work [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Local systems and short persistent homology induce a cellular sheaf structure of topological spaces and can depict the spatial merging relations via their global/local sections [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We discuss the relationship between hole positions, global/local sections, and persistence barcodes on local systems (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Second, we introduce how we adapt the theory to digital images and implement the method (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Finally, we discuss some properties of the proposed framework, such as the relationship between local systems, the location of holes, and image noises (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1 Persistent Homology of Local Systems For a topological space X and a concerned local region A of X, the relative homology Hq(X, A) considers the equivalence classes of cycles in X that do not meet the subspace A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' One can formulate the relative homology of X and A by Hq(X, A) = Zq(X, A)/Bq(X, A), where Zq(X, A) = {c ∈ Sq(X) : ∂q(c) ∈ Sq−1(A)} = ∂−1 q (Sq(A)) is the set of all chains in Sq(X) with boundaries in Sq−1(A), and Bq(X, A) = Bq(X) + Sq(A) is the submodule generated by all q-boundaries of X and q-chains in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Elements in Zq(X, A) and Bq(X, A) are called relative q-cycles and relative q-boundaries of X, respectively [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Roughly speaking, relative homology detects holes in X except for holes that are totally contained in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' More precisely, one can apply the snake lemma on the short exact sequence 0 −→ S•(A) ι• −→ S•(X) π• −→ S•(X)/S•(A) −→ 0 with canonical inclusion and projection to obtain the long exact sequence · · −→ Hq(A) ιq −→ Hq(X) πq −→ Hq(X, A) δq −→ Hq−1(A) ιq−1 −−−→ Hq−1(X) πq−1 −−−→ Hq−1(X, A) −→ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (6) One can use barcode representation to detect hole structures in the spaces H•(A), H•(X), and H•(X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For example, a non-zero element in Hq(X) \\ im(ιq) represents a hole in X that is not totally emerged in the region A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' On the other hand, c ∈ Hq(X) dies at Hq(X, A) if the cycle c does not represent a hole in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because the long exact sequence in (6) is a chain complex, every lifespan b − d of a barcode (b, d) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Relative homology can capture holes contributed by A, X \\ A, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, it is difficult and expensive to implement and compute due to the complicated data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This paper proposes a relatively efficient method to detect hole relations and positions via persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' To achieve this goal, we introduce here two main ideas proposed in our previous work, called local system and short filtration [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 6 A PREPRINT - JANUARY 16,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='(h) Binary image g3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='(i) Binary image g0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='(j) Binary image g1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='(k) Binary image g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='(l) Binary image g3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='Figure 2: First row: a 6 × 6 image domain P in Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' a grayscale image g : P → {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' and a binary image f : P → {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Figures (c) and (d) are two different representations for the image f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In a binary image f as in (d), pixels with a value of 0 represent the black pixels of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Second row: a filtration of binary images made by image g and thresholds 0, 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Third row: the white-black pixel representations of images in the second row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 8 ([30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let X be a topological space and X1, X2 be subspaces of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The triad (X, X1, X2) is called a local system (or an admissible triad) if clX(X1) ∩ clX(X2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For any topological space X and its subspaces X1 and X2, we have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 9 ([30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let (X, X1, X2) be a triad of topological spaces with X1 ⊆ X and X2 ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This triad leads to two filtrations ∅ ⊆ X1 ⊆ X1 ∪ X2 ⊆ X and ∅ ⊆ X2 ⊆ X1 ∪ X2 ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We call them short filtrations of the triad (X, X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Focusing on the first one in Definition 9, the birth information at H•(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) depicts whether X2 contains a homological generator that cannot be represented via generators in H•(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' When clX(X1) ∩ clX(X2) = ∅, the homology H•(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) is canonically isomorphic to the space H•(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ⊕ H•(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) since X1 and X2 are two path-connected components of X1 ∪ X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In this case, every generator s2 in H•(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) is born at H•(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) of the persistent homology 0 −→ H•(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ H•(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ H•(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) and dies at H•(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) if there is an s1 ∈ H•(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) such that s1 and s2 represent the same homological generator in H•(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This property will benefit computing the homological changes of holes in X1, X2, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Furthermore, we will show in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2 that the condition clX(X1) ∩ clX(X2) = ∅ can be easily established in image data through elementary image processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 7 A PREPRINT - JANUARY 16, 2023 Figure 3: Two local systems (X, X1, X2) of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let Γ0 denote the global section space of the sheaf structure H0(X1) −→ H0(X) ←− H0(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then we have the following information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The first row: H0(X1) ≃ Z2, H0(X2) ≃ Z5 2, H0(X1 ∪ X2) ≃ Z6 2, H0(X) ≃ Z2 2, and Γ0 ≃ Z4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The second row: H0(X1) ≃ Z4 2, H0(X2) ≃ Z2 2, H0(X1 ∪ X2) ≃ Z6 2, H0(X) ≃ Z2, and Γ0 ≃ Z5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In [30], we applied the two filtrations of a local system (X, X1, X2) to construct the following cellular sheaf structure: Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ρ1 � Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ρ2 � where q is any non-negative integer, F is a fixed field, and ρ1, ρ2 are the F-linear transformations induced by the inclusions X1 �→ X and X2 �→ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We often call the maps ρ1, ρ2 restriction maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A pair (s1, s2) ∈ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ⊕ Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) is called a global section of the sheaf if ρ1(s1) = ρ2(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We use Γ to denote the subspace of all global sections in Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ⊕ Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), and it can be sculptured by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For the following sheaf structure of F-vector spaces and F-linear maps: V f � P , W g � we define φ : V ⊕W → P by (v, w) �−→ f(v)−g(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then, φ is also an F-linear linear map and (V ⊕W)/Γ ≃ im(φ), where Γ = {(v, w) : f(v) = g(w)} is the space of global sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, dim(Γ) = dim(V ) + dim(W) − dim(im(φ)) if the spaces V, W, P are finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In addition, dim(Γ) = dim(V ) + dim(W) − dim(P) if φ is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' It is evident that φ is F-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because (v, w) ∈ ker(φ) if and only if f(v) = f(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' By the first isomorphism theorem of modules, the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 8 (a) Xi (b) X2 (c) Xi U X2 (d) X 二亚Ⅲ 三 (e) Xi (f) X2 (g) Xi U X2 (h) XA PREPRINT - JANUARY 16, 2023 Let (X, X1, X2) be a local system of topological spaces and Γ the global section space of the sheaf structure Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ←− Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Examples shown in Figure 3 depict that the vector spaces Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), and Γ can be totally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In other words, the global section space provides an additional than the homology of X1, X2, X1 ∪ X2, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Actually, suppose we have a sequence (Xi, Xi1, Xi2) of local systems that satisfy Xi1 ⊆ X(i+1)1, Xi2 ⊆ X(i+1)2, and Xi ⊆ Xi+1, then we have the following commutative diagram: Hq(X11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' � φ11 � Hq(X21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' � φ21 � Hq(X31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) � res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' � · · Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) φ1 � Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) φ2 � Hq(X3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) � · · · Hq(X12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' � φ12 � Hq(X22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' � φ22 � Hq(X32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) � res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' � · · where φij and φi are the F-linear maps induced by the inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' One can check the following sequence is also valid: Γ1 φ11⊕φ12|Γ1 � Γ2 φ21⊕φ22|Γ2 � Γ3 � · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In other words, except for computing single global section spaces, one can also consider the persistent homology of global section spaces induced by any filtered local systems of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorem 1 presents a way to compute global section spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, on many occasions, computing the image of φ in the theorem may be infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' To tackle this, we previously proposed an approximation method using persistent homology [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We quote the method as the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorem 2 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1 [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let R be a commutative ring with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let (X, X1, X2) be a local system of topological spaces and q a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let G1 be the short filtration ∅ ⊆ X1 ⊆ X1 ∪ X2 ⊆ X and s2 ∈ Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) a non-zero element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then the followings are equivalent: (a) There is an s1 ∈ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) such that (s1, s2) ∈ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ⊕ Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) is global section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (b) �s2 := ω2(s2) has barcode (2, 3) in the PH Pq(G1) : 0 −→ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) −→ Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For a local system (X, X1, X2), numbers of barcode (2, 3) in Pq(G1) records how many homological non-zero generators in Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) that merge to a generator in Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In [30], we defined it as the q-th local merging number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 10 ([30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let (X, X1, X2) be a local system of topological spaces and q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We define the q-th local merging number of X1 and X2 as the numbers of barcodes (2, 3) in Pq(G1) and denote it by mq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We use the two pairs in Figure 3 to explain the local merging numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For the first row, we have m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 5 since there are 5 connected components that merge to X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' On the other hand, m0(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X1) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Similarly, the m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) and m0(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X1) of the second row are 2 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, these two examples show the local merging numbers m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) and m0(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X1) are not equal in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Actually, one can prove that max{m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2), m0(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X1)} ≤ dim(Γ) ≤ m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) + m0(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X1) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' When q = 0, the local merging numbers m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) records how many connected components in X2 connect to components in X1 synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In our previous work, we show that local regions with high 0-local merging numbers are likely to be more joint parts of the ambient space and have the potential to analyze handwritten text with texture data [30, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In these works, we focus on local merging numbers in dimension 1 and (2, 3) barcodes in short filtrations, while the geometric meanings of higher dimensional merging numbers and (3, +∞) barcodes are still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In the following theorem, we show that the number of barcodes (3, +∞) in a short filtration can verify whether X1 and X2 contribute a hole (with dimension ≥ 1) in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let F be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let (X, X1, X2) be a local system of topological spaces and q a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let Γ be the global section space of the sheaf structure Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ←− Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then the number of (3, +∞) in the PH Pq(G1) : 0 −→ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) equals dimF (Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)) − dimF (Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)) − dimF (Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)) + dimF (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 9 A PREPRINT - JANUARY 16, 2023 (a) A 6 × 6 image domain (b) Rectangle R (c) Boundary B of R (d) �R = R \\ B (e) Black pixel set X (f) X1 = X ∩ �R (g) X2 = X \\ R (h) X1 ∪ X2 Figure 4: An illustration of the construction of a local system in a 2D binary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In this example, we have m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 3, o0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 0, m1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 0, and o1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let ρ1 : Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) and ρ2 : Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) be the canonical linear transformations that are induced by the inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Define φ = ρ1 − ρ2 : Hq(X1) ⊕ Hq(X2) −→ Hq(X), then dimF (Γ) = dimF (Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)) + dimF (Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)) − dimF (im(φ)) (7) by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) is canonically isomorphic to Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ⊕ Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), the images of ρ1 − ρ2 and the map Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) in Pq(G1) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then the number of barcodes (3, +∞) in the persistent homology Pq(G1) counts the dimension of the space Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)/im(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Therefore, #{barcode (3, +∞) in Pq(G1)} = dimF (Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)) − dimF (im(φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (8) By plugging equation (7) into equation (8), the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' If c ∈ Zq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ⊆ Zq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) is a q-cycle that represents a q-dimensional hole in X1, then c must have a barcode (1, ⋆) in the persistent homology Pq(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' On the other hand, c ∈ Zq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ⊆ Zq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) representing a hole in X2 implies that it has a barcode (2, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In other words, the number of barcodes (3, +∞) in the persistent homology Pq(G1) records how many q-holes in X are “supported” by both X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, removing either X1 or X2 will make those holes disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Intuitively, those holes are constructed by gluing the parts by X1 and X2, and hence we have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let (X, X1, X2) be a local system of topological spaces and q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We define the q-th local outer- merging number of X1 and X2 as the numbers of barcodes (3, +∞) in Pq(G1) and denote it by oq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' From the above discussion, it can be seen that the local outer-merging number records the contribution of a specific local area in the topological space to the hole structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We present local outer-merging numbers for digital images in the next section (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In addition, we will analyze the location of holes in the image by segmenting the image and the local outer-merging number of the corresponding region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2 Local Systems in Binary Images Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1 introduces the local system and its persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorem 2 and Theorem 3 tell us that counting the numbers of barcodes (2, 3) and (3, +∞) in (X, X1, X2) can detect the glue relationship of local objects in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Among them, constructing the admissible triad (X, X1, X2) is the most crucial part of the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For an object X in Rn and a bounded A ⊆ X, one can choose r1, r2 > 0 with r1 < r2 such that A ⊆ B(0, r1) and define X2 = X ∩ {x ∈ Rn : |x| ≥ r2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then, clX(X1) ∩ clX(X2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Based on the same idea, the section presents a more efficient way to build local systems in binary images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 10 A PREPRINT - JANUARY 16, 2023 (a) A 9 × 9 binary image (b) A bounding box of the loop (c) A better bounding region Figure 5: An illustration of bounding regions of the hole structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (a) A binary image that contains a single 1- dimensional loop structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (b) A rectangular bounding box formed by yellow and brown pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (c) A more compact bounding region formed by red and purple pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As we introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2, a 2-dimensional image is identified as a non-negative real-valued function f : P −→ R≥0 on a discrete 2D rectangle P = ([a, b] × [c, d]) ∩ Z2, where a, b, c, d are integers with a ≤ b and c ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In the paper, we focus on the geometric realization of black pixels of a binary image and compute its homology (see Figure 2(d) and Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For a binary image f : P −→ {0, 1}, we consider the black pixel set f −1(0) and denote it by X ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We use a rectangle in P to cover a concerned region of X, say R = ([a1, b1] × [c1, d1]) ∩ Z2 with a ≤ a1 ≤ b1 ≤ b and c ≤ c1 ≤ d1 ≤ d (see Figure 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Consider B = Z2 ∩ � ({a1} × [c1, d1]) ∪ ({b1} × [c1, d1]) ∪ ([a1, b1] × {c1}) ∪ ([a1, b1] × {d1}) � as the boundary of R (see Figure 4(c)), we define �R = R\\B (see Figure 4(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Defining X1 = X ∩ �R and X2 = X \\R (see Figure 4(e)-(h)), we obtain a triad (X, X1, X2) with the property X1 ∩ X2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because X, X1, and X2 are subspaces in R2 that are formed by finitely many closed squares in R2, we must have clX(X1) ∩ clX(X2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For example, the local system (X, X1, X2) in Figure 4 has merging and outer-merging numbers m0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 3, o0(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 0, m1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 0, and o1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In [30], we separated a 2D image into disjoint blocks X(i) i (called a local patches) and calculated the local merging numbers m0(X(i) 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X(i) 2 ) to form a heatmap of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In the paper, we mainly focus on the number o1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) to approximate the hole positions in a binary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We show in Section 4 how to use the local system described in this section to construct local patches in the image and use them to estimate the 1D holes in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3 Discussion The organization of the section is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' First, we compare the proposed method with possible methods in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Second, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2 discusses how to estimate the size and shape of the holes in the topological space through the local system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Finally, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3 discusses local systems composed of n subspaces and their local sections, which will be an important future research direction to promote the theory of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='1 Comparison with other methods To detect the local regions that contain pores in an m × n binary image, some naive methods can be used to tackle this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For example, one can search every subfigure of the given image and compute whether it contains hole structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, it is generally infeasible to compute all the �m 2 � �n 2 � = (m2 − m)(n2 − n) 4 , subfigures for large m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Except for the computational complexity, covering an irregular hole costs a large bounding rectangle and makes the estimation less precise and compact (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 11 A PREPRINT - JANUARY 16, 2023 Following our previous approach [30], we have two strategies to reduce computational complexity through the image’s local systems and sheaf information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The first is splitting the image into many pairs (X(i) 1 , X(i) 2 ) with disjoint X(i) 1 s and computing each Pq(Gi,1) on the filtration Gi,1 : ∅ ⊆ X(i) 1 ⊆ X(i) 1 ∪ X(i) 2 ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The second is to use a sliding window technique to cover the entire image and compute short-persistent homology for each local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The second strategy computes the persistent homology O(mn) times, and pore locations generated by this strategy are usually more refined than the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In the paper, we mainly follow the second strategy and show in Section 4 that the proposed barcode and local system framework can detect local holes effectively and with more concise bounding regions than bounding boxes (Figure 5 (b), (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2 Size Issues As we mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2, homological generators generally lack specific geometric properties, such as the size and stability of pores (Figure 1), which cannot be detected by traditional homology or persistent homology of sub-level set filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In recent years, the shape and size of pore structures have become more and more important topics in bioinformatics and material science [34, 35, 52, 55, 54, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Recently, some research has shown the potential and advantage of persistent homology in pore size analysis [33, 58, 32, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As far as the field of image processing is concerned, the combination of persistent homology and mathematical morphology has opened up a new research direction for this field [20, 32, 11, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, our approach in [11] applies morphological opening and closing to measure the spatial information of black and white regions in binary images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Mathematical morphological operations can estimate the sizes of image pores, and most of the current work focuses on the global description of such spatial information, such as the number of pores with a specified morphological size and the average image pore size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, the location information of pores in images is still limited in present methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Through the discussion in this section, we will see that localized systems can capture both the location and size of pores, providing a richer pore analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let F be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let (X, X1, X2) be a local system of topological spaces and q a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then, non-zero elements in Hq(X1) and Hq(X2) in the persistent homology Pq(G1) : 0 −→ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) has birth number < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Suppose s1 is a non-zero element in Hq(X1), then the birth number of s1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' On the other hand, the assumption of clX(X1)∩clX(X2) = ∅ forces that Hq(X1 ∪X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) ≃ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F)⊕Hq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then every non-zero element in Hq(X2) must have barcode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let X be a subspace of the n-dimensional Euclidean space Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let F be a field and q a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For every c ∈ Zq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) with [c] ̸= 0 in Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), there is a bounded set X1 ⊆ X and an X2 ⊆ X such that clX(X1) ∩ clX(X2) = ∅ and c ∈ Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, [c] has a barcode (1, ⋆) in Pq(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For a chain c in Sq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F), we can write c = �n i=1 λiσi, where λi ∈ F \\ {0} and σi : ∆q −→ X is a continuous for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Recall that the support of c denoted by |c| is defined as the union of the images of the σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Because ∆q is compact, and σi is continuous, the support of c is a compact subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In particular, the support of c is closed and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Therefore, we may choose a positive number r1 such that |c| ⊆ B(0, r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Choose r2 > r1 and set X1 = X ∩ B(0, r1) and X2 = X ∩ {x ∈ Rn : |x| ≥ r2}, then clX(X1) ∩ clX(X2) = ∅ and c ∈ Zq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' By the proof of Theorem 4, [c] has a barcode (1, ⋆) in Pq(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' For convenience, we use iq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) to denote the number of barcodes in Pq(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Corollary 1 gives us a way to measure the size of q-holes (q > 0) for any subspace in Rn by choosing the local system appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' More precisely, we can choose a bounded subspace X1 of X that contains the hole and is therefore an approximation of the size of the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Also, since the location of X1 is known, it also keeps track of the hole location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We also demonstrate in Section 4 an application of Corollary 1 to detect the "largest" holes in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3 More general systems In the paper, we focus on a local system consisting of topological spaces X, X1, X2 that satisfy X1 ⊆ X, X2 ⊆ X, and clX(X1) ∩ clX(X2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' This system induces a sheaf structure as in Theorem 1, and its global section space can be computed by the persistent homology of the short filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Actually, one can consider a more general case consisting of n + 1 spaces X, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', Xn with clX(Xi) ∩ clX(Xj) = ∅ for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We synthesize the above settings into the following definition and theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 12 A PREPRINT - JANUARY 16, 2023 Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let X be a topological space and X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', Xn by subspaces of X that satisfy clX(Xi) ∩ clX(Xj) = ∅ for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The (n + 1)-tuple (X, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', Xn) is called a local n-system (or an admissible (n+1)-tuple) of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Furthermore, for an n-system (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', Xn), we can consider the diagram Hq(X1) ρ1 � Hq(Xk) ρk � Hq(X) Hq(Xn) ρn � with homologies and induced homomorphisms [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As above, we define its global section space by Γ = � (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', sn) ∈ n � i=1 Hq(Xi) : ρi(si) = ρj(sj) for i, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', n} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let (X, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', Xn) be a local n-system of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Let R be a commutative ring with identity and q ≥ 0 a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Consider the sheaf structure Hq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ρ1 � Hq(Xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ρk � Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) Hq(Xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) ρn � of R-modules and module homomorphisms and the homomorphism n � i=1 Hq(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R) φ −−−→ n � i=2 Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R), (si)n i=1 �−→ (ρ1(s1) − ρi(si))n i=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (9) Let Γ be the global section space of the sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Then Γ is the kernel of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' An n-tuple (si)n i=1 is a global section if and only if ρi(si) − ρj(sj) = 0 for every i, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' If it holds, then ρ1(s1) − ρj(sj) = 0 for every j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Conversely, suppose ρ1(s1) − ρj(sj) = 0 for every j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', n}, then ρi(si) − ρj(sj) = ρi(si) − ρ1(s1) + ρ1(s1) − ρj(sj) = 0 for every i, j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', n}, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Although Theorem 5 provides a way to sculpt the global section space, it still has a limitation in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' When n = 2, and R = F is a field, the homomorphism in (9) and Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) have the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In this case, the approximation developed in Theorem 2 and Theorem 3 is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, the map in (9) and the canonical one Hq(X1 ∪ X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) −→ Hq(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F) are not coincident and can not be calculated by counting the barcodes in the short filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Computing global sections through persistence barcodes is one of our future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 4 Demonstration on Digital Images Hole structures in images can be subtle and complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As shown in Figure 6(a)1, although many white areas appear in the image as porosity structures or closed voids, many of these areas connect to the white background and thus 1Karen Arnold has released this “Fingerprint Clipart” image under a Public Domain license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' publicdomainpictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='net/en/view-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='image=462168&picture=fingerprint-clipart 13 A PREPRINT - JANUARY 16, 2023 (a) Input image (b) Marked holes (c) Output heatmap (d) Location estimation Figure 6: A demonstration of Algorithm 1 on a binary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (a) An illustration of a 500 × 700 fingerprint image, where the Betti pair of the image is (β0, β1) = (92, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (b) The marked white regions as the 14 holes of the fingerprint image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (c) The output heatmap by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (d) The non-zero parts of the output heatmap, form an estimation for the hole positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Here we choose the window R as a 30 × 30 square with step k = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' are not actual holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In order to detect the image’s hole positions, we propose Algorithm 1 to approximate the holes’ geometric locations using the above theory and sliding window technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Figure 6 is a demonstration of Algorithm 1 on a binary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We can see that all the holes in the image are detected by the output heatmap as Figure 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We note that Line 6 in Algorithm 1 considers both i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) and o1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' If i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) ̸= 0, it means that X1 contains some holes in X and already a bounding box of certain holes (Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' On the other hand, i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) records whether (part of) the black pixels in X1 can contribute to hole structures and the number of these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Therefore, the sum of i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) and o1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) can estimate whether X1 nears a hole structure in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Algorithm 1 The hole structure detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Input: Binary image f : P −→ {0, 1} on a rectangle R, X = f −1(0), an n × n square window R, and a sliding step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Output: A function H : P → R as a heatmap of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The heatmap estimates the hole locations in image f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A point in P with a high heat value is more possible as a part of a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 1: Denote P = ([0, a] × [0, b]) ∩ Z2 and R = ([0, n] × [0, n]) ∩ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Define B and �R as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 2: Set H : P → R as the zero function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3: for i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', a} and j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', b} do 4: if (i · k, j · k) + R ⊆ P then 5: Set X1 = ((i, j) + �R) ∩ X and X2 = X \\ ((i, j) + R) 6: Compute M = i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) + o1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) 7: Define H′ : P −→ {0, 1} as follows: H′(x) = � H(x) + M if x ∈ (i, j) + �R, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 8: H ←− H′ 9: else 10: continue 11: end if 12: end for 13: return H′ · (1 − f) Apart from the task of detecting the location of holes in an image, recognizing the size or shape of holes is also an interesting one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2, local windows that contain holes will produce (1, 2) barcodes in the corresponding short persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Based on the observation, we can modify Algorithm 1 by considering 14 A PREPRINT - JANUARY 16, 2023 (a) Input image (b) 50 × 50 (c) 100 × 100 (d) 150 × 150 (e) Sum of (b)-(d) (f) Estimation Figure 7: A demonstration of Algorithm 2 on a binary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (a) An illustration of a 200 × 300 binary image with Betti pair (β0, β1) = (2, 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (b), (c), and (d) are the output heatmaps of Algorithm 2 with local windows of sizes 50 × 50, 100 × 100, and 150 × 150, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (e) The sum heatmap of (b), (c), and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' (e) The non-zero parts of the output heatmap, form an estimation for the hole positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Here we choose the step k = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' the information on the size of local windows and changing the M value to tackle this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As follows, we propose Algorithm 2 as a modification of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We note here that the we implement these two algorithms in Python with the Gudhi package [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Algorithm 2 The hole size estimation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Input: Binary image f : P −→ {0, 1} on a rectangle R, X = f −1(0), an n × n square window R, and a sliding step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Output: A function H : P → R as a heatmap of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The heatmap estimates the hole locations in image f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A point in P with a high heat value is more possible as a part of a “large” hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 1: Denote P = ([0, a] × [0, b]) ∩ Z2 and R = ([0, n] × [0, n]) ∩ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Define B and �R as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 2: Set H : P → R as the zero function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 3: for i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', a} and j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', b} do 4: if (i · k, j · k) + R ⊆ P then 5: Set X1 = ((i, j) + �R) ∩ X and X2 = X \\ ((i, j) + R) 6: Compute M = vol(R) · (o1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) − i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2)) ▷ The only difference to Algorithm 1 7: Define H′ : P −→ {0, 1} as follows: H′(x) = � H(x) + M if x ∈ (i, j) + �R, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 8: H ←− H′ 9: else 10: continue 11: end if 12: end for 13: return H′ · (1 − f) We note that the only difference between Algorithm 1 and Algorithm 2 is the value of M in line 6 of both algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' As we discussed above, i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) > 0 means that some holes are bounded by the area X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Due to this, we apply −vol(R) · i1(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' X2) as the punishment term of the region’s heat value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In addition, since punishment will produce negative values, if a certain area has many fine holes, the heat function will have a high negative value in this area and also estimates the location of these holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Figure 7 demonstrates Algorithm 2 on a binary image with hole structures in different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Within different local windows (50 × 50, 100 × 100, and 150 × 150 squares), Algorithm 2 gives different attention to these holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We notice that small holes may get more attention from Algorithm 2 (Figure 7(b)) since there are many holes next to each other, and hence the shared edges will get a higher heat value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Keeping enlarging the size of the local window, we observe that smaller holes in the image would get more punishment in heat values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The sum of the three heatmaps summarizes 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='A PREPRINT - JANUARY 16, 2023 the “importances” of black pixels in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Finally, we see that Algorithm 2 successfully approximates the position of the “largest hole” by the thresholding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' However, real pore structure data may be more complex than the images presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Methods to use barcode information in pore structure analysis, such as the choice of the penalty function, still need research and development, which is our main development work in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 5 Conclusion To summarize the paper, we propose a local system and local persistent homology framework to study the local merging relations in an arbitrary topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' By using the merging and out-merging numbers of local regions, we propose an algorithm to detect the sizes and positions of pores in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Although the demonstration focuses on digital images, the framework can be adapted to any topological space with local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' We also look forward to applying this framework to point-cloud data, especially its applications in crystalline data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Acknowledgement Most of the work in this article was completed by the author during his doctoral study at National Taiwan Normal University (2016-2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The author would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chun-Chi Lin (NTNU) and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Yu-Min Chung (Eli Lilly and Company), the author’s doctoral supervisors, for their comments and suggestions on the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Especially, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Yu-Min Chung provided many suggestions for studying the geometric meaning of persistent barcodes and the global sections of the local system, making the discussion more fruitful and rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The author would also like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kelin Xia, the author’s postdoctoral supervisor at Nanyang Technological University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The author got a lot of inspiration from the discussion with Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Xia so that this paper can have more research directions, such as a more detailed study of the geometry of cellular sheaves, and more possible applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Adams, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Emerson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kirby, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Neville, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Peterson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Shipman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chepushtanova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hanson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Motta, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ziegelmeier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistence images: A stable vector representation of persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Journal of Machine Learning Research, 18, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Akai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hirayama, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Murase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology in lidar-based ego-vehicle localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In 2021 IEEE Intelligent Vehicles Symposium (IV), pages 889–896, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Anand, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Xia, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Topological feature engineering for machine learning based halide perovskite materials design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' npj Computational Materials, 8(1):1–8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Bassu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Laurati, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Fratini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Microgel dynamics within the 3d porous structure of transparent peg hydrogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Colloids and Surfaces B: Biointerfaces, 221:112938, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Bubenik, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Carlsson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kim, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Statistical topology via morse theory persistence and nonparametric estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Algebraic methods in statistics and probability II, 516:75–92, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Bubenik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Statistical topological data analysis using persistence landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', 16(1):77–102, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Carlsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Topology and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Bulletin of the American Mathematical Society, 46(2):255–308, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Carlsson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Mémoli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Multiparameter hierarchical clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Classification as a Tool for Research, pages 63–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Springer, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chazal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Fasy, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Lecci, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Michel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Rinaldo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Rinaldo, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wasserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Robust topological inference: Distance to a measure and kernel distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The Journal of Machine Learning Research, 18(1):5845–5884, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chung and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Topological fidelity and image thresholding: A persistent homology approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Journal of Mathematical Imaging and Vision, 60(7):1167–1179, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Day, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A multi-parameter persistence framework for mathematical morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Scientific reports, 12(1):1–25, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chung and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Lawson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistence curves: A canonical framework for summarizing persistence diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Advances in Computational Mathematics, 48(1):1–42, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' De, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Vo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Value-offset bifiltrations for digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Computational Geometry, 109:101939, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 16 A PREPRINT - JANUARY 16, 2023 [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' De Silva and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ghrist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Coordinate-free coverage in sensor networks with controlled boundaries via homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The International Journal of Robotics Research, 25(12):1205–1222, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [15] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Delgado-Friedrichs, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Robins, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Sheppard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Morse theory and persistent homology for topological analysis of 3d images of complex materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In 2014 IEEE International Conference on Image Processing (ICIP), pages 4872–4876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' IEEE, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Edelsbrunner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In International Workshop on Graph-Based Represen- tations in Pattern Recognition, pages 182–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Edelsbrunner and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Harer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology-a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Contemporary mathematics, 453:257–282, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Edelsbrunner and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Harer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Computational Topology: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' American Mathematical Society, 01 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Frosini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Measuring shapes by size functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Intelligent Robots and Computer Vision X: Algorithms and Techniques, volume 1607, pages 122–133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' SPIE, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Garin and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Tauzin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A topological" reading" lesson: Classification of mnist using tda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pages 1551–1556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Garside, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Gjoka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Henderson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Johnson, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Makarenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Event history and topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Biometrika, 108(4):757–773, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ghrist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Barcodes: the persistent topology of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Bulletin of the American Mathematical Society, 45(1):61–75, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ghrist and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Muhammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Coverage and hole-detection in sensor networks via homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In IPSN 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Fourth International Symposium on Information Processing in Sensor Networks, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=', pages 254–260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' IEEE, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [24] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Gong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Li, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent spectral simplicial complex-based machine learning for chromosomal structural analysis in cellular differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Briefings in Bioinformatics, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Greenberg and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Harper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Algebraic Topology, A First Course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Addison-Wesley Publishing Company, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Gunther, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Reininghaus, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hotz, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wagner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Memory-efficient computation of persistent homology for 3d images using discrete morse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In 2011 24th SIBGRAPI Conference on Graphics, Patterns and Images, pages 25–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' IEEE, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Press, Cambridge, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A brief note for sheaf structures on posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='09651, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Sheaf Structures on the Multi-parameter Persistent Homology Arising from Mathematical Morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' PhD thesis, National Taiwan Normal University, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [30] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A sheaf and topology approach to detecting local merging relations in digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4396–4405, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Lawson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Smyth, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Toporesnet: A hybrid deep learning architecture and its application to skin lesion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Mathematics, 9(22):2924, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Lawson, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chung, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Keegan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Two-parameter persistence for images via distance transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4176–4184, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ishihara, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Franks, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Effect of particle packing structure on the elastic modulus of wet powder compacts analyzed by persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Advanced Powder Technology, 34(1):103874, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Jiang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Tsuji, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Shirai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Pore geometry characterization by persistent homology theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Water Resources Research, 54(6):4150–4163, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [35] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Jiao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Stillinger, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Torquato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Modeling heterogeneous materials via two-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' algorithmic details and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Physical Review E, 77(3):031135, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [36] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kaczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Mischaikow, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Mrozek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Computational Homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Applied Mathematical Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Springer New York, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kannan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Saucan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Roy, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Samal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology of unweighted complex networks via discrete morse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Scientific reports, 9(1):1–18, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Keros, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Nanda, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Subr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Dist2cycle: A simplicial neural network for homology localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 7133–7142, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 17 A PREPRINT - JANUARY 16, 2023 [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Latschev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Vietoris-rips complexes of metric spaces near a closed riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Archiv der Mathematik, 77(6):522–528, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [40] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Mischaikow and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Nanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Morse theory for filtrations and efficient computation of persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Discrete & Computational Geometry, 50(2):330–353, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Munkres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Elements Of Algebraic Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' CRC Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Nielson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Paquette, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Guandique, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Tovar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Inoue, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Irvine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Gensel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kloke, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Petrossian, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Nature communications, 6(1):1–12, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Onuchin and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Kachan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Individual topology structure of eye movement trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In International Conference on Neuroinformatics, pages 45–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Springer, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [44] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Riihimäki, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chachólski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Theorell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Hillert, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ramanujam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' A topological data analysis based classification method for multiple measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' BMC bioinformatics, 21(1):1–18, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [45] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Robins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Towards computing homology from finite approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Topology proceedings, volume 24, pages 503–532, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [46] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Stolz-Pretzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Global and local persistent homology for the shape and classification of biological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' PhD thesis, University of Oxford, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [47] The GUDHI Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' GUDHI User and Reference Manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' GUDHI Editorial Board, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Tymochko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Munch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Dunion, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Corbosiero, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Torn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Using persistent homology to quantify a diurnal cycle in hurricanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Pattern Recognition Letters, 133:137–143, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Usher and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology and floer–novikov theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Geometry & Topology, 20(6):3333–3430, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [50] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Vandaele, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' De Bie, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Saeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Local topological data analysis to uncover the global structure of data approaching graph-structured topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 19–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Vick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Homology Theory, A Introduction to Algebraic Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Springer-Verlag Publishing Company, Second Edition, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [52] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Vogel and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Quantitative morphology and network representation of soil pore structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Advances in water resources, 24(3-4):233–242, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [53] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wee and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent spectral based ensemble learning (perspect-el) for protein–protein binding affinity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Briefings in Bioinformatics, 23(2), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [54] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wei, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' The distribution of adsorption energy of u (vi) onto aeptes-functionalized porous silica with multiple average pore sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Chemical Engineering Journal, 451:138716, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [55] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Zhao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Liao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Effect of starch on pore structure and thermal conductivity of diatomite-based porous ceramics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ceramics International, 49(1):383–391, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [56] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Xia and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Persistent homology analysis of protein structure, flexibility, and folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' International journal for numerical methods in biomedical engineering, 30(8):814–844, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [57] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Cisewski-Kehe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Green, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Nagai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Finding cosmic voids and filament loops using topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Astronomy and Computing, 27:34–52, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [58] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Yamauchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Yatagawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Ohtake, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Bin-scanning: Segmentation of x-ray ct volume of binned parts using morse skeleton graph of distance transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Computational Visual Media, 9(2):319–333, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' [59] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Zomorodian and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Carlsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' Computing persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' In Proceedings of the twentieth annual symposium on Computational geometry, pages 347–356, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE5T4oBgHgl3EQfLg6S/content/2301.05474v1.pdf'} diff --git a/vdE2T4oBgHgl3EQf2gj_/content/2301.04163v1.pdf b/vdE2T4oBgHgl3EQf2gj_/content/2301.04163v1.pdf new file mode 100644 index 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b/vtE3T4oBgHgl3EQf-guw/content/tmp_files/2301.04827v1.pdf.txt @@ -0,0 +1,1770 @@ +arXiv:2301.04827v1 [hep-th] 12 Jan 2023 +5-D thermal field theory, Einstein field equations and spontaneous symmetry breaking +S. Ganesh∗ +It has been shown previously, that the spatial thermal variation of a thermal medium can be recast +as a variation in the Euclidean metric. It is now extended to temporal variations in temperature, +for a non-relativistic thermal bath, which remains in local thermal equilibrium. This is achieved +by examining the thermal field theory in a five-dimensional space-time-temperature. +The bulk +thermodynamic quantity, namely the energy density, is calculated for a neutral scalar field with a +time-dependent Hamiltonian. Furthermore, the concept of recasting thermal variations as a variation +in the metric is extended to thermal systems in a gravitational field. The Einstein field equations, +in the 5-D space-time-temperature, is determined. It is shown that the resulting Ricci scalar can +then lead to spontaneous symmetry breaking, leading to the Higgs mechanism. +In essence, the +asymmetry in the distribution of temperature in space-time can translate to spontaneous symmetry +breaking of particle fields, in a very strong gravitational field. +Keywords : Thermal gradient, 5-D Thermal field theory, Einstein field equation, Gravity, Sponta- +neous symmetry breaking, Higgs +PACS numbers : 11.10.Wx, 04.50.Kd, 11.15.Ex +I. +INTRODUCTION +The thermal field theory incorporates a Euclidean +space-time, obtained by an analytical continuation of +time in Minkowski space-time to an imaginary time, to +model thermal systems [1–4]. +Numerous research has +been done using the framework of thermal field theory in +some form. Some examples include Refs. [5–15]. There +are, however thermal systems, where there are significant +thermal variations in both spatial and temporal dimen- +sions, e.g., the Quark Gluon Plasma (QGP) [16, 17]. +The idea was first mooted in Ref. [18], that spatial ther- +mal variations can be modeled by recasting the variation +in the temperature as a variation in the metric. This idea +was used to determine the quark anti-quark potential in a +thermal system with spatial variations, using AdS-CFT +correspondence. The thermal medium was assumed to +be under local thermal equilibrium. In Ref. [19], the con- +cept that the spatial thermal variations can be recast +as a variation in the Euclidean metric, was placed on +firmer grounds, by analyzing the Polyakov loop, the par- +tition function, the correlation function, and the geodesic +equation. It has been shown that the partition function +for thermal systems with spatial thermal variations, nat- +urally leads to the notion of a curved Euclidean space. +While most of the analysis was carried out for a canoni- +cal ensemble, the framework was also touched upon, for +a grand canonical ensemble. It is now shown, that the +concept of the 5-dimensional space first introduced in +Ref. [19], enables the generalization of the concept to +temporal variations in temperature. The temporal varia- +tions must be sufficiently slow to maintain local thermal +equilibrium. +The 5-D space-time-temperature approach further en- +∗Corresponding author: +Email: gans.phy@gmail.com +ables the extension of the concept to a thermal bath in +a gravitational field. The Einstein field equations are de- +termined in the 5-D space. The resultant field equations +are then expressed in terms of the usual 4-D space-time +covariant operators. This process of dimension reduction +leads to additional terms in the Ricci tensor, and subse- +quently, the Ricci scalar. +There has been significant research on the source of +spontaneous symmetry breaking. Reference [20] has indi- +cated that radiative corrections can be a possible source. +Reference [21] has explored the production of Higgs at +the Large Hadron Collider (LHC). A crucial outcome of +the proposed framework is that the modified Ricci scalar +may be a source of spontaneous symmetry breaking. A +negative Ricci scalar can make the term, (m2 + R)φ2, in +the Lagrangian for a scalar field, become negative, i.e., +(m2 + R)φ2 → −µ2φ2, for some real valued µ. Thus, a +negative Ricci scalar curvature may be a viable candi- +date for the negative potential necessary for spontaneous +symmetry breaking for a scalar field. The spontaneous +symmetry breaking leads to the Higgs mechanism [22– +24]. +For the sake of clarity, let us briefly revisit some of the +relevant concepts that were discussed in Ref. [19]. An +8-D space, βµ × xν, was considered to model thermal +systems. Under conditions that the thermal system is +stationary, i.e., the four-velocity of the thermal medium, +uµ = (1, 0, 0, 0), it reduces to a 5-D space, (iβ, t, x). +Moreover, if the thermal system is time-invariant, it +may suffice to consider the 4-D sub-space, (iβ, x), lead- +ing to the imaginary time formalism. +If the temper- +ature varies in space, with spatial variation s(x), i.e., +β ≡ β(x) = s(x)β0, where β0 is a constant, then it leads +to a curved Euclidean space, with the spatial variation, +h00 = s(x)2. +A 5-D metric tensor along with the h00 +term is shown in Eq. 1. +G(5) +ab = +� h00 +0 +0 +g(4) +µν +� +, +(1) + +2 +where, g(4) +µν , is the usual metric tensor in 4-D space-time, +resulting purely as a result of gravitational fields. The 5- +D space consideration was necessitated to allow separate +metric components, h00, due to thermal variations, and +g00, due to gravity. Furthermore, the 5-D space was again +shown as a necessity, in order to model an external Dirac +spinor, with energy E, traversing a thermal medium. The +3-momentum observable for the Dirac spinor, traversing +a thermal bath, can be made manifest only with a 5-D +representation of the Dirac equation, instead of the 4- +D imaginary time formalism. The conjugate momentum +variables to time and temperature, namely the energy, +E, and the Matsubara frequency, ωn = Cn +h00 , capture the +intrinsic energy, E, of the particle, and the interaction en- +ergy with the thermal medium, iEc = Cn +h00 , respectively. +The curvature of Euclidean space was considered in the +complete absence of gravity. +In [19], the analysis was primarily for a time-invariant +system. However, since time and inverse temperature, +β, are separate dimensions, it is tempting, to extend the +formalism to time-varying systems. We now develop the +formalism for time-varying systems. In addition, in the +current work, the curvature due to the effects of temper- +ature variations and gravity is encapsulated in a com- +mon framework. Furthermore, the criterion for the Ricci +scalar, R, to be negative, under the combined influence +of the thermal variations and gravity, is determined. +The rest of the paper is as follows. Section II, devel- +ops the formalism for a 5-D time-varying thermal system, +that remains in local thermal equilibrium. The expecta- +tion value of energy density, for a time-varying Hamil- +tonian, is then determined in Sec. III. The 5-D Einstein +field equations are developed in Sec. IV. The criterion for +spontaneous symmetry breaking is also developed in this +section. Finally, the results are summarized in Sec. V. +II. +THE TIME VARIATION IN TEMPERATURE +A time-varying thermal system would also have a four- +velocity, uµ, of the thermal medium, leading to an 8-D +modeling as mentioned in [19]. +However, in the non- +relativistic limit, uµ = (1, u) ≈ (1, 0, 0, 0), the analysis +can be approximated by an analysis in 5-D space itself. +The 8-D space can be divided into two 4-D Lorentz invari- +ant sub-manifolds [19]. Thus, a 5-D sub-space may not +lend itself to a full-fledged Lorentz invariant treatment. +A complete Lorentz invariant treatment would require +the entire 8-D space. It is planned to extend the current +framework to a complete Lorentz invariant treatment in +8-D space in the future. +In order to model partition function in a 5-D space- +time-temperature, it would be required to take care that +the system is not acausal or non-local in nature. This +rules out an Hamiltonian of the form H = � Hdtd3x. +Instead, one may analyze the system in each time slice. +A partition function imbibes the statistical properties of +the system. For a time-varying system, it is possible to +take time slices and define the statistical properties in +each time slice. The process of taking time slices, how- +ever, does not construe that each time slice is modeled as +a static system, with the effects of time derivatives being +ignored. The Hamiltonian would now be time-dependent, +and in general, be different from the Hamiltonian of a +static system. The time-dependent Hamiltonian should +capture the non-trivial effects due to the time variations. +Consequently, we define the partition function at each +time slice for a time-varying system, which continues to +be in local thermal equilibrium. We now proceed on sim- +ilar lines as Ref. [19], for each time slice, albeit with a +time-dependent Hamiltonian density. +Let us first develop the field theory for a non- +interacting Lagrangian, in 5-D space-time-temperature. +The Lagrangian density in a 5-D space, for a neutral +scalar field would be, +L(ˆφ, ∂a ˆφ) = 1 +2 +� +∂a ˆφ∂a ˆφ − m2 ˆφ2� +, +(2) +with the index a = 0, 1, 2, 3, 4 corresponding to the di- +mensions [τ, t, x, y, z]. τ is the inverse temperature and +varies from 0 to β. The sign convention used is (-,+,-,-,-). +This gives rise to the equation of motion: +∂a∂a ˆφ + m2 ˆφ = 0 +(3) +The constraining 5 momentum delta function would be, +δ(E2 − ω2 − p2 − m2), and the 5-D integral measure is: +1 +β +� +n +� +d3p +(2π)3 +dE +2π 2πδ(E2 − ω2 +n − p2 − m2) += 1 +β +� +n +� +d3p +(2π)3 +1 +2E , +(4) +where, the Matsubara frequency = ωn = 2nπ +β , for a bo- +son. Let us look at the physical interpretation of the delta +function, δ(E2−ω2 +n−p2−m2). As mentioned in Ref. [19], +E, may be considered the original intrinsic energy of a +particle, and ωn = iEc, can be considered as the interac- +tion energy of the particle with the thermal medium. The +variable, ωn, determines the decay or enhancement of +a particle wave-function with temperature (for example, +the Dirac spinor in Ref. [19]). The magnitude of the to- +tal energy is then, |E + iEc| = +� +E2 + E2c = +� +E2 − ω2n. +It is intuitive, that a particle’s 3-momentum would be +affected by both E and Ec, and not just E. Thus, one +may consider E2 − ω2 +n = p2 + m2. A portion of the par- +ticle’s original energy, E, is lost due to interaction with +the thermal medium. This provides an intuition behind +the delta function, δ(E2 − ω2 +n − p2 − m2). +The operator for a neutral scalar field in 5-D space- +time-temperature is, +ˆφ(x, τ, t) = 1 +β +� +n +� +d3p +(2π)3 +1 +� +2Ep +� +s +� +a† +p,ωne−ipxe−iωnτ + ap,ωneipxeiωnτ� +(5) + +3 +The operator, +a† +p,ωn, +creates a particle with 3- +momentum p, and Matsubara frequency ωn. +One may premise the below commutation relation: +[ap1,ωn1, a† +p2,ωn2] = (2π)3δ3(p1 − p2)ζ(β)δn1,n2, +(6) +where, ζ(β) is a scalar normalization function, and needs +to be determined. Let us define, +ap(τ) = +� +n +f(ωn)ap,ωn, +a† +p(τ) = +� +n +f ∗(ωn)a† +p,ωn. +(7) +Equation 7 can be interpreted in the following way. When +a momentum state |p⟩ is created, then |p⟩ itself can be +treated as a superposition of the momentum-Matsubara +eigenstates |p, ωn⟩, with probability amplitudes f(ωn). +Since f(ωn) is a probability amplitude, � +n f 2(ωn) = 1. +Then, the equal τ commutator, +[ap1(τ), a† +p2(τ)] += +� +n1 +� +n2 +[ap1,ωn1, a† +p2,ωn2]f(ωn1)f ∗(ωn2) += +� +n1 +� +n2 +(2π)3δ3(p1 − p2)ζ(β)δn1,n2f(ωn1)f ∗(ωn2) += +� +n1 +(2π)3ζ(β)δ3(p1 − p2)f 2(ωn1). +(8) +Since � +n1 f 2(ωn1) = 1, let us assign ζ(β) = 1, in Eq. 8, +to obtain, +[ap1(τ), a† +p2(τ)] = (2π)3δ3(p1 − p2). +(9) +Thus, the usual commutation relation between the 3- +momentum annihilation and creation operator is recov- +ered. For large β, i.e., β → ∞, one may follow a similar +procedure as above, but in the continuous domain. The +commutation relation in Eq. 6, can be seen to be, +[ap1,ωn1, a† +p2,ωn2] = (2π)4δ3(p1 − p2)δ(ωn2 − ωn1), (10) +with ζ(β) = 2π. +The relation between Eq. 6 and +Eq. 10 can be understood by noting that the 4-D space- +temperature manifold is a R3 × S1 manifold. As β → ∞, +R3 × S1 → R4. +We now proceed to determine the conjugate momenta +and the Hamiltonian. There can be a conjugate momenta +w.r.t. either the time variable or the temperature vari- +able, i.e., +ˆπt = δL +δ ∂ ˆφ +∂t +; +ˆπβ = i δL +δ ∂ ˆφ +∂τ +. +(11) +The corresponding Hamiltonian densities are: +Ht = ˆπt +∂ ˆφ +∂t − L; +Hβ = −iˆπβ +∂ ˆφ +∂τ − L. +(12) +They would obey the evolution equations: +i∂ ˆφ +∂t = [ˆφ, Ht]; +∂ ˆφ +∂τ = [ˆφ, Hβ]. +(13) +Since we are modeling a thermal system, and are in- +terested in evolution in τ, the main object of interest +would be Hβ and πβ. For convenience, we now drop the +subscript β. In the rest of the paper, unless otherwise +mentioned, H and ˆπ refer to Hβ and ˆπβ. We now follow +a similar procedure as Ref [19, 25], albeit modified for a +5-D space with thermal variations. +Let φ(x) and |φ(x)⟩ be the eigenfunction and the eigen- +ket of the Schrodinger picture field operator ˆφ(x, 0, 0), +while, π(x) and |π(x)⟩ be the eigenfunction and the +eigenket of the conjugate momentum field operator +ˆπ(x, 0, 0). In other words, +ˆφ(x, 0, 0)|φ⟩ = φ(x)|φ⟩, +ˆπ(x, 0, 0)|π⟩ = π(x)|π⟩. +(14) +The eigenkets, |φ⟩ and |π⟩, obey the following relation: +⟨φ|π⟩ = exp +� +i +� +d3xπ(x)φ(x) +� +. +(15) +For a time-dependent system, the time-dependent +Hamiltonian can be written as: +H(t) = H0 + HI(t), +(16) +where H0 is the time independent part, and HI(t) be +the time dependent part. As mentioned earlier, HI(t) +should capture the non-trivial effects of time variation +in a time-varying system. H0 is written in terms of the +Schrodinger picture operators ˆπ(x, 0, 0) and ˆφ(x, 0, 0). +H0 = +� +d3xH0(ˆπ(x, 0, 0), ˆφ(x, 0, 0)) ≡ +� +d3xH0(x). +(17) +For simplicity of notation, the form, H0(x), is used as a +representation of H0(ˆπ(x, 0, 0), ˆφ(x, 0, 0)). +The free Hamiltonian density, corresponding to the La- +grangian in Eq. 2, would then be, +H0 = 1 +2 + +ˆπ2 + (∇ˆφ)2 − +� +∂ ˆφ +∂t +�2 ++ m2 ˆφ2 + + . +(18) +But, +∂ ˆφ +∂t = −iE ˆφ. +In a gas composed of scalar fields, +which is equilibrated, E → 0, as only the ensemble in- +teraction energy, captured by the Matsubara frequency, +ωn, is non zero [19]. +We are then left with the stan- +dard imaginary time formalism. On similar lines, in a +vacuum, as β → ∞, ωn → 0. Then, the only energy +left is the particle energy, E. The Lagrangian in Eq. 2, +then boils down to the normal 4-D space-time Quantum +Field Theory (QFT). When both E and ωn are non-zero, +Eq. 2 can be used to model particles that are not yet fully + +4 +equilibrated with the thermal medium. A case in point is +the external Dirac spinor traversing a thermal medium, +which was modeled in Ref. [19]. +The external Dirac +Spinor (not equilibrated with the thermal medium), will +have its intrinsic energy, E, as well as a non-zero ωn, due +to interaction with the thermal medium. With, E → 0, +the Hamiltonian density in Eq. 18 becomes, +H0 = 1 +2 +� +ˆπ2 + (∇ˆφ)2 + m2 ˆφ2� +. +(19) +Equations 9 and 19, indicate that the generalizations to +5-D, characterized by Eqs. 5, 6, 11, 12, are backward +compatible with the existing 4-D imaginary time formal- +ism. A fairly generic time-dependent Hamiltonian den- +sity can be written as: +HI(t) = +� +HI(x, t)d3x, +(20) +where, +HI(x, t) = +� +i +ci(x, t)fi(ˆφ, ∂µ ˆφ, ˆπ, ∂µˆπ), +(21) +and ci(x, t) are arbitrary scalar functions. However, vari- +ations in ci(x, t), should not be sharp enough to throw +the system out of local thermal equilibrium. In this pa- +per, the thermodynamic properties are evaluated for the +below specific cases: +1. HI(x, t) = V (x, t)ˆφ2, +2. HI(x, t) = λ(x, t)ˆφ4. +In the case of a thermal bath composed of scalar particles, +V (x, t)ˆφ2 can represent the coupling of an external field, +along with its derivatives, with ˆφ2. +The evolution operator U(Hβ, β, t) provides the evolu- +tion w.r.t. β, i.e., +Uβ(Hβ, β, t) = exp +� +− +� � β(x,t) +0 +Hβ(x, t)dτd3x +� +. +(22) +As mentioned earlier, we drop the subscript β from U, +H and H, and obtain, +U(H, β, t) = exp +� +− +� � β(x,t) +0 +H(x, t)dτd3x +� +, += exp +� +− +� � β0 +0 +s(x, t)H(x, t)dτd3x +� +, +(23) +where, H(x, t) = H0(x) + HI(x, t). The integrand in the +exponent of Eq. 23, indicates a volume element dτd3x +of a 4-D slice, at time t, within the 5-D space, with +metric, diag[−s(x, t)2, 1, −1, −1, −1], and determinant, +√g5 = s(x, t). Thus, it describes a curved 5-D space- +time-temperature. +We can then define the partition function Z(t), at a +time slice, t, in 5-D space-time-temperature as: +Z(β0, t) = tr +� +⟨φf|U(H, β0, t)|φ0⟩ +� +, +(24) +with |φ0⟩ and |φf⟩ being the eigenkets at τ = 0 and +τ = β0 respectively. +The procedure to evaluate the partition function, +Z(β0, t), is now straight forward and similar to Ref. [19]. +We first evaluate U(H, β, t). +Let β0 be sliced into N +slices, i.e., β0 = N∆β, with N → ∞ and ∆β → 0. This +gives, +U(H, β0, t) = +lim +∆β→0 exp +� +− +� +n +�� +s(x, t)H(x, t)d3x +� +∆β +� +(25) += lim +∆β→0 +� +n +exp +� +− +�� +s(x, t)H(x, t)d3x +� +∆β +� +. +(26) +Let |πj⟩ be the β based conjugate momentum eigen- +state. Inserting a complete set of eigenstates: +I = +� +|φj⟩⟨φj|dφj × +� +|πj−1⟩⟨πj−1|dπj−1 +2π +, +between each product term in Eq. 26, and +I = +� +|πN−1⟩⟨πN−1|dπN−1 +2π +; +I = +� +|φ1⟩⟨φ1|dφ1, +in the beginning and the end of the R.H.S. of Eq. 26, +respectively, we get the expression for K(φf, φ0, β0, t) = +⟨φf|U(H, β0, t)|φ0⟩: +K(φf, φ0, β0, t) = lim +∆β→0 +� +N−1 +� +j=1 +⟨φj+1|πj⟩ +× ⟨πj| exp +� +− +� +s(x, t)H(x, t)∆βd3x +� +|φj⟩ +× ⟨φ1|φ0⟩dφj +dπj +2π . +(27) +It is possible to evaluate the expression in Eq. 27, using +the relations: +• +⟨φj+1|πj⟩ = exp +� +i +� +d3xπj(x)φj+1(x) +� +, +(28) +• +⟨πj| exp +� +−i +� +s(x, t)Hd3x∆β +� +|φj⟩ += ⟨πj|φj⟩ exp +� +−i +� +s(x, t)Hjd3x∆β +� +, +(29) +with, +Hj += +H(πj, φj, t) += +H0(πj, φj) + +HI(πj, φj, t), + +5 +• and +⟨φ1|φ0⟩ = +� +x +δ (φ1(x) − φ0(x)) . +(30) +It is to be noted that the form of HI(x, t), used in Eq. 21, +would satisfy the relation specified in Eq. 29. +This is +because, the operator arguments of HI, namely, ˆφ(x) and +ˆπ(x), continue to be Schrodinger picture operators, and +the temporal aspect is captured by the scalar functions, +ci(x, t). If a different form of HI(x, t) is used, then Eq. 29 +needs to be checked for validity. Inserting Eqs. 28, 29 +and 30 in Eq. 27, we obtain, +K(φf, φ0, β0, t) = lim +∆β→0 +� � +j +dφj +dπj +2π exp +� � +d3x +× +� +− iπj(x) +� +φj+1(x) − φj(x) +� +− s(x, t)Hj∆β +�� +(31) += lim +∆β→0 +�  +� +j +dφj +dπj +2π + + exp +� � +k +� +∆β +� +d3x +× s(x, t) +� +−iπk(x){φk+1(x) − φk(x)} +s(x, t)∆β +− Hk +� �� +. +(32) +We recollect that, s(x, t) = +� +h00(x, t) (Eq. 1). +Fur- +ther, for the metric, G5 = diag[−s(x, t)2, 1, −1, −1, −1], +we have, +� +|G5| = s(x, t). We denote the determinant +of the Euclidean metric, |G5|, by g5, and obtain in the +continuum limit: +K(φf, φ0, β0, t) = +� +Dφ +� Dπ +2π exp +� � β0 +0 +dτ +� +d3x +� +g5(x, t) +× +� +− iπ(x, τ) +1 +� +h00(x, t) +Dτφ(x, τ) +− H(π(x, τ), φ(x, τ), t) +�� +, +(33) +where, Dτ is the covariant derivative w.r.t. +τ. +For a +scalar φ(x, τ), Dτφ(x, τ) = ∂φ(x,τ) +∂τ +. Finally, the partition +function becomes, +Z(β0, t) = trK(φf, φ0, β0, t) = +� +periodic +Dφ +� Dπ +2π exp +� � β0 +0 +dτ +� +d3x +� +g5(x, t) +× +� +iπ(x, τ) +1 +� +h00(x, t) +Dτφ(x, τ) +− H(π(x, τ), φ(x, τ), t) +�� +. +(34) +III. +EVALUATION OF ⟨E(t)⟩ +In this section, the expectation value of the energy +density at different instances of time, ⟨E(t)⟩, is evaluated +for HI = V (t)φ2, and HI = λ(t)φ4. +A. +⟨E(t)⟩ when HI = V (t)φ2 +Since only φ2 is involved in the partition function, this +is easily evaluated as Gaussian integrals. Using the pro- +cedure outlined in [19], the expression for the energy ex- +pectation value, ⟨E(t)⟩ is: +⟨E(t)⟩ = −∂ ln(Z) +∂β0 += V +� +d3p +(2π)3 +� +ωp +2 coth +�β0ωp +2 +� ++ +� +⟨η(x, t)⟩ + +1 +2ω2p +⟨s(x, t)V (x, t)⟩ +� +× +�ωp +2 coth +�β0ωp +2 +� ++ β0ω2 +p +4 +cosech2 +�β0ωp +2 +� �� +, +(35) +where, ⟨s(x, t)V (t)⟩ = +� d3xs(x,t)V (t) +� +d3x +, s(x, t) = 1+η(x, t), +⟨η(x, t)⟩ = +� +d3xη(x,t) +� +d3x +and ωp = +� +p2 + m2. +B. +⟨E(t)⟩ when HI = λ(x, t)φ4 +The Lagrangian density for a neutral scalar field would +now read: +L = 1 +2 +� +∂µ ˆφ∂µ ˆφ − m2 ˆφ2� +− λ(x, t)ˆφ4. +(36) +The corresponding interaction Hamiltonian is, HI += +λ(x, t)ˆφ4. For small λ(x, t), the partition function can +be written as: +Z(β0, t) = +� +periodic +Dφ +� Dπ +2π +× +� +1 + +� +r +1 +r! +�� +dτd3x√g5λ(x, t)φ4 +�r� +× exp +� � β0 +0 +dτ +� +d3x√g5 +� +− iπ(x, τ) +1 +� +h00(x) +∂φ(x, τ) +∂τ +−1 +2 +� +π2 + (∇φ(x, τ))2 + m2φ(x, τ)2� �� +. +(37) +Then, one may represent the above equation as: +ln Z(β0, t) = ln(Z0(β0, t)) + ln +� +1 + +� +r +1 +r! +Zr +I (β0, t) +Z0(β0, t) +� +. +(38) + +6 +2 +pv, n1 +pl, n2 +pu, n1 +pk, n +FIG. 1: Feynman diagram depicting the first order ˆφ4 inter- +action. +The calculation of ln(Z0(β0, t)), at time slice t, is identical +to the calculation of ln(Z0(β0)), in Ref. [19]. +For r = 1, the expression for Z1 +I , corresponding to the +Feynman diagram in Fig. 1 is: +Z1 +I (β0, t) = +� +periodic +Dφ +� Dπ +2π +�� +dτd3x√g5λ(x, t)φ4 +� +× exp +� � β0 +0 +dτ +� +d3x√g5 +� +− iπ(x, τ) +1 +� +h00(x) +∂φ(x, τ) +∂τ +−1 +2 +� +π2 + (∇φ(x, τ))2 + m2φ(x, τ)2� �� +. +(39) +The factor +Z1 +I +Z0 , is now calculated. +The +� +Dπ integrals +can be done in exactly the same manner as the non- +interacting case. We consider a space of volume V, dis- +cretize it, and divide it into M 3 cubes of length ∆a. Fur- +ther, express φ(x) in terms of its momentum domain rep- +resentation, φn,p [19], +φ(x, τ) = +� +β0 +V +� +i +∞ +� +n=−∞ +eiCnτeipi.xφn,pi, +(40) +where, Cn is related to ωn. We follow a procedure sim- +ilar to the non-interacting case in Ref [19]. Then, after +replacing +� +∆p +2π +�3 +as V in the exponent, one may obtain +Z1 +I as: +Z1 +I = +� +N +� 1 +2π +�NM3/2 � +× +� +Dφn,p +�� +dτd3xs(x, t)λ(x, t)φ(x)4 +� +× exp +� +−1 +2 +� +i +� +j +∞ +� +n=−∞ +φ∗ +n,pj +� +β2 +0 +� +(C2 +n + pipj + m2)δij ++ ηf(pj − pi, t) 1 +V(−C2 +n + pipj + m2) +�� +φn,pi +� +, +(41) +where, we have used √g5 = s(x, t), and N is an incon- +sequential multiplication factor and does not affect the +dynamics of the system [25, 26]. The length, ∆a, gets +absorbed in N. The variable, ηf(p, t), is the 3-D Fourier +transform of η(x, t). Based on Eq. 40, φ(x)4 can be rep- +resented in terms of its Fourier components. +� +dτd3xs(x, t)λ(x, t)φ4(x) = +�� +β0 +V +�4 +× +� +pu,nu +� +pv,nv +� +pk,nk +� +pl,nl +β0δ (Cnu + Cnv + Cnk + Cnl) +×sλf(−pu − pv − pk − pl, t)φnu,puφnv,pvφnk,pkφnl,pl, +(42) +where, sλf(p, t) is the momentum domain representation +of the product term, s(x, t)λ(x, t). The delta function +forces, nu = −nv = n1(say), and nk = −nl = n2(say). +Secondly, we make the transformation pv → −pv and +pl → −pl. With these, Eq. 42, becomes +� +dτd3xs(x, t)φ4(x) = 3 β3 +0 +V2 +� +n1 +� +n2 +� +u +� +v +� +k +� +l +� +sλ,f (pv − pu + pl − pk, t) φ∗ +n1,pvφn1,puφ∗ +n2,plφn2,pk +� +. +(43) +Let the exponent enclosed between φ∗ +n,pj and φn,pi in +Eq. 41 be Mn, i.e., +Mn = +� +β2 +0 +� +(C2 +n + pipj + m2)δij ++ηf(pj − pi, t)‘ 1 +V(−C2 +n + pipj + m2) +�� +. +(44) +Let Λn diagonalize Mn. Then Mn = Λ−1 +n DnΛn, where +Dn is a diagonal matrix. Let ψn,a = � +i Λn,a,iφn,pi. This +means φn,pi = � +a Λ−1 +n,i,aψn,a. +Inserting the above transformation of φ in Eq. 43, and +making use of the fact that (Λ−1 +n,i,aψn,a)∗ = ψ∗ +n,aΛt +n,i,a, +� +dτd3x s(x, t)λ(x, t)φ4(x) += 3 β3 +0 +V2 +� +n1 +� +n2 +� +b +� +a +� +d +� +c +ψ∗ +n1,bψ∗ +n2,d +× +� � +v +� +u +� +l +� +k +Λn1,b,vΛn2,d,l +× sλ,f(pv − pu + pl − pk, t)Λ−1 +n1,u,aΛ−1 +n2,k,c +� +ψn1,aψn2,c. +(45) + +7 +Or, +� +dτd3x s(x, t)λ(x, t)φ4(x) += 3 β3 +0 +V2 +� +n1 +� +n2 +� +b +� +a +� +d +� +c +× +� +ψ∗ +n1,bψ∗ +n2,dΛs +bdacn1n2ψn1,aψn2,c +� +. +(46) +Equation 46 is same as Eq. 45, but with a more compact +notation, by introducing, Λs +bdacn1n2, for the term enclosed +within the [ ] brackets. +Also, we have, +� +i +� +j +φ∗ +n,pjMn,j,iφn,pi = +� +l +ψ∗ +n,lDn,l,lψn,l. +(47) +Since the transformation, φ → ψ, is unitary, we can re- +place the measure +� +Dφ by +� +Dψ. Substituting Eq, 46, +and Eq. 47, in Eq. 41, we obtain: +Z1 +I = λ(t) +� +N +� 1 +2π +�NM3/2 � +× +� +n1 +� +n2 +� +b +� +a +� +d +� +c +� +3 β3 +0 +V2 +� +× +� +Dψn,i +� +ψ∗ +n1,bψ∗ +n2,dΛs +bdacn1n2ψn1,aψn2,c +� +× exp +� +− 1 +2 +� +i +∞ +� +n=−∞ +ψ∗ +n,iDn,i,iψn,i +� +. +(48) +The functional integrals where, b ̸= a and d ̸= c, are +zero since the integrands are odd. The path integral then +simplifies to: +Z1 +I = +� +N +� 1 +2π +�NM3/2 � +× +� +n1 +� +n2 +� +a +� +c +Λs +acacn1n2 +� +3 β3 +0 +V2 +� +× +� +dψn1,aψ2 +n1,a exp +� +−1 +2ψ∗ +n1,aDn,a,aψn1,a +� +× +� +dψn2,cψ2 +n2,c exp +� +−1 +2ψ∗ +n2,cDn,c,cψn2,c +� +× +� +remaining +Dψn,i exp +� +−1 +2 +� +(n,i)̸= +(n1,a),(n2,c) +ψ∗ +n,iDn,i,iψn,i +� +. +(49) +This finally gives, +Z1 +I = +� +N +� 1 +2π +�NM3/2 � +× +� +3 β3 +0 +V2 +� � +n1 +� +n2 +� +a +� +c +� +Λs +acacn1n2 +× +� +√ +2π +2D3/2 +n1,a,a +� � +√ +2π +2D3/2 +n2,c,c +� +� +(n,i)̸= +(n1,a),(n2,c) +� √ +2π +2D1/2 +n,i,i +� � +, +(50) +Since, ΠkDn,k,k = det(Dn) = det(Mn), it is possible to +write: +Z1 +I +Z0 += 3 β3 +0 +V2 +� +n1 +� +n2 +� +a +� +c +� +Λs +acacn1n2 +× +� +1 +2Dn1,a,a +� � +1 +2Dn2,c,c +� � +. +(51) +We now estimate, Λs +acacn1n2, to the first level of approxi- +mation. Better approximations using numerical methods +or superior techniques could lead to higher accuracy. For +matrices, Mn, An and En, where Mn = An + En, and if +eigenvalues, λA, of An is known, it is possible to estimate +the eigenvalues, λM, of Mn using the relation [27]: +λM = λA + xtEnx +xtx ++ O(∥E∥2), +(52) +where, xt is eigenvector of An. +Since, O(∥E∥) +∼ +O(∥nf∥), this should estimate eigenvalues of Mn to order +O(∥nf∥2). The matrix, Mn, is given in Eq. 44. Let us +take An and En to be the matrices: +An,i,j = β2 +0(C2 +n + pipj + m2)δij, +(53) +En,i,j = β2 +0ηf(pj − pi, t) 1 +V(−C2 +n + pipj + m2). +(54) +Since An is diagonal, λA = β2 +0(C2 +n + p2 + m2), and the +eigenvectors, xt, are unit vectors with only one non-zero +entry. This gives, +λM;n,i +≈ β2 +0 +� +(C2 +n + p2 +i + m2) + ηf(0, t) +V +(−C2 +n + p2 +i + m2) +� +. +(55) +Thus, +Dn,a,a = λM;n,a +≈ β2 +0 +� +(C2 +n + p2 +a + m2) + ηf(0, t) +V +(−C2 +n + p2 +a + m2) +� +. +(56) +This approximation has extracted the diagonal elements +of E as the contribution from E to λM. Consequently, + +8 +the estimated eigenvector of Mn continues to be x, at this +level of approximation. Ergo, Λn,a = Λn,c ≈ I, leading +to, Λs +bdacn1n2 ≈ sλf(0, t). With all this, Eq. 51 becomes, +Z1 +I +Z0 += 3 β3 +0 +V2 sλf(0, t) +��� +a +� +n1 +1 +2Dn1,a,a +��2 +. +(57) +The value of Dn,a,a in Eq. 56, can be simplified, by rec- +ognizing that +• ηf(0,t) +V += ⟨η(x, t)⟩, +• 1 + ⟨η(x, t)⟩ = ⟨s(x, t)⟩ and, +• 1 − ⟨η(x, t)⟩ ≈ +1 +⟨s(x,t)⟩. +Substituting the value of Dn,a,a thus obtained, in Eq. 57, +Z1 +I +Z0 += 3 β3 +0 +V2 sλf(0, t) +× + +� +a +� +n +1 +β2 +0 +� +( +C2 +n +⟨s(x,t)⟩ + ⟨s(x, t)⟩ (p2a + m2)) +� + + +2 +. +(58) +Finally, +Z1 +I +Z0 += 3 +4(β0V)⟨s(x, t)λ(x, t)⟩ +× +� � +d3p +(2π)3 +1 +ωp +coth +�ωpβ0⟨s(x, t)⟩ +2 +� �2 +, +(59) +where, ωp = +� +p2 + m2 and ⟨s(x, t)λ(x, t)⟩ = sλf(0, t). +For, r += 1, one may then determine, ⟨E(t)⟩ += +∂ ln(Z(t)) +∂β0 +. +IV. +THE RICCI TENSOR AND NEGATIVE +POTENTIAL +The Einstein field equations are now determined in the +5-D space-time-temperature. We use the letters, a, b, as +indices for the 5-D space-time, i.e., a, b = 0, 1, 2, 3, 4, +with the index 0 referring to the temperature dimension, +while, the index 1 refers to the time dimension, We use +the letters, µ, ν, as indices for the 4-D Lorentzian space- +time, i.e., µ, ν = 1, 2, 3, 4. A superscript, (N), within +brackets, refers to N dimensional space. For example, +∇(4) +µ , refers to the covariant derivative in 4-D space-time. +Let us consider the Lagrangian, +L(φ, ∂aφ) = 1 +2 +� +−∂aφ∂aφ − m2φ2 − ζRφ2� +, +with the sign convention (+,-,+,+,+), and the corre- +sponding action, +S = +� � +−g(5)L(φ, ∂aφ)d5x. +(60) +Let us consider the 5-D metric, g(5) +ab , as: +g(5) +ab = +� +s2(g(4) +µν (x, t), x, t) +0 +0 +g(4) +µν +� +, +(61) +where, g(4) +µν , is the usual metric tensor in 4-D space- +time, and is purely due to gravitational fields. It is also +noted that s would now additionally be a function of +g(4) +µν also. As an example, based on the Ehrenfest Tol- +man effect [28, 29], in the case of a uniform temperature +(system in global equilibrium), in a gravitational field, +s(g(4) +µν (x, t), x, t) = +� +gµνζµζν, where ζµ is the Killing +vector in 4-D space-time. We speculate that, the func- +tion, s(), can be factored, i.e., +s(gµν(x, t), x, t) = s1(gµν(x, t)) s2(x, t), +(62) +where, s1(gµν(x, t)) is due to the Ehrenfest Tolman effect, +and s2(x, t) is due to thermal gradients. It is the simplest +function that has the following limits: +1. When the thermal system is in global equilibrium, +s(gµν(x, t), x, t) → s1(gµν(x, t)), and, +2. when gµν → ηµν, s(gµν(x, t), x, t) → s2(x, t). +It is, however desirable to prove or improvise upon Eq. 62, +based on first principles. +For the metric, g(5) +ab , the Einstein field equations in the +5-D space become: +R(5) +ab − 1 +2g(5) +ab R(5) = 8πG +c4 T (5) +ab , +(63) +where, +T (5) +ab = − +2 +� +g(5) +δS +δg(5) +ab +. +(64) +The 5-D Ricci tensor, R(5) +ab , can be expressed in terms of +the 4-D covariant derivative operator, ∇(4) +µ , and the 4-D +Ricci tensor, R(4) +µν , as: +R(5) +ab = +� Rββ +0 +0 +R(4) +µν − 1 +s∇(4) +µ ∇(4) +ν s +� +. +(65) +where, Rββ = s∇(4)µ∇(4) +µ s. +The treatment of the +geodesic equation in Ref. [19], gives an insight into the +physical interpretation of how thermal gradients, effects +an apparent curvature in the Lorentzian space-time. It +can be easily seen that the expression for R(5) +ab reduces to +the 4-D Ricci tensor, for the uniform temperature case, +i.e., ∂µs → 0. Another thing to note is that in the metric +s(x, t)2dβ2 + gµνdxµxν, the temperature dimension can +be eliminated by taking the limit, s → 0. Thus, for a +vacuum, one may take the limits ∂µs → 0, followed by +s → 0, in which case, the Einstein field equations (Eq. 63) + +9 +reduce to the usual 4-D space-time equations. Finally the +5-D Ricci scalar, R(5) can be expressed as: +R(5) = R(5)a +a = R(4) − 2 +s∇(4)µ∇(4) +µ s, +(66) +where, R(4) is the conventional 4-D Ricci scalar, in the +Lorentzian space, due to gravitational fields, in the ab- +sence of any temperature gradients. When the 5-D Ricci +scalar is sufficiently negative, it can lead to spontaneous +symmetry breaking, i.e., when, m2 + ζR(5) < 0. This +translates to: +R(5) < −m2 +ζ , +⇒ 2 +s∇(4)µ∇(4) +µ s > R(4) + m2 +ζ . +(67) +It is to be noted that the quantity, 2 +s∇(4)µ∇(4) +µ s, is scale- +invariant, i.e., the absolute temperature is inconsequen- +tial. Only the variations in temperature matter. Since, +s(x, t) is a scalar, 2 +s∇(4)µ∇(4) +µ s simplifies to 2 +s∇(4)µ∂µs. +In the case of a thermal medium formed in a collider, +e.g., like the QGP at the LHC, the gravitational effects +can be neglected. This gives R(4) = 0. Equation 67, then +simplifies to: +2 +s∂µ∂µs > m2 +ζ . +(68) +One may, however, consider the Eq. 68, with caution. For +Eq. 68 to be satisfied, 2 +s∂µ∂µs needs to be significantly +large. But if thermal gradients are very large, the thermal +medium may cease to be in local thermal equilibrium, +and the current formalism ceases to be valid. +The scenario in Eq. 67 is however different. To see why, +the value of 2 +s∇(4)µ∇(4) +µ s is affected by three factors: +1. The thermal gradients, ∂µs2(x, t) (refer Eq. 62 for +the factorization of s = s1s2), +2. The +Ehrenfest +Tolman +effect, +represented +by +s1(x, t), and, +3. The terms containing the Christoffel symbols, aris- +ing as part of the covariant derivative. +Whereas the first factor may be small for a medium in +local thermal equilibrium, the second and third factors +depend on the strength of the gravitational field. +V. +SUMMARY +In this work, it is seen that the concept of 5-D space- +time-temperature, permits the incorporation of time- +dependent temperature effects, for systems that continue +to be in local thermal equilibrium. The bulk thermody- +namic property, namely, the energy density expectation +value, has been calculated perturbatively, for a thermal +bath with both spatial and temporal variations, in the ab- +sence of gravity. The Einstein field equations have been +determined in 5-D space-time-temperature, which incor- +porates both gravitational and thermal effects. +In the +presence of a thermal variation, both temperature gra- +dients and gravity determine the net scalar curvature. +The net curvature in the 5-D space can be negative. If +the d’Alembertian of the temperature variation is suffi- +ciently high, spontaneous symmetry breaking may occur +in a scalar field, particularly in the presence of a strong +gravitational field. +As mentioned in Ref. [19], the curvature of the 5-D +space due to temperature variations can be validated ex- +perimentally, in terrestrial experiments, due to the much +lower energy scales involved. Thus, if the concept of the +curved 5-D space is confirmed, then it may have impor- +tant consequences, such as the spontaneous symmetry +breaking as described in this work. +This current for- +malism may be useful in situations where both gravity +and thermal variations are present, like in the interior +of a neutron star [30, 31]. The rotation curves of galax- +ies have been a subject of intensive research [32, 33]. It +might even be possible to consider a galaxy as a thermal +medium, with the stars as point particles. However, a +galaxy is not in equilibrium, which might pose challenges +for such an approach. +[1] Matsubara, T. A New Approach to Quantum-Statistical +Mechanics. Progress of Theoretical Physics, +14, +4, +351–378 (1955). +[2] Martin, P. C., Schwinger, J. Physical Review, 115(6), +1342–1373 (1959). +[3] A.A. Abrikosov, L.P. Gor’kov, I.E. Dzyaloshinskii JETP, +Vol. 9, No. 3, p. 636 (1959) +[4] Arnold, P., Vokos, S., Bedaque, P., Das, A. Physical Re- +view D, 47(10), 4698 (1993). +[5] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Sz- +abo, Nature 443, 675-678 (2006), arXiv:hep-lat/0611014. +[6] Kenji Fukushima and Tetsuo Hatsuda, Rept. Prog. Phys. +74, 014001 (2011), arXiv:1005.4814. +[7] Kraemmer and Rebhan, Rept.Prog.Phys. 67 (2004). +[8] Kraemmer, Rebhan and H. Hchulz, Annals. of Phys. 238 +(1995). +[9] Heng-Tong Ding, Frithjof Karsch, and Swagato Mukher- +jee, +Int. +J. +Mod. +Phys. +E24, +1530007 +(2015), +arXiv:1504.05274. +[10] Blaizot J. P. and Iancu E., Phys. Rep., 359 355–528 +(2002); arxiv hep-ph/0101103. +[11] David E. Morrissey and Michael J. Ramsey-Musolf, New +J. Phys. 14, 125003 (2012), arXiv:1206.2942. +[12] Andrew G. Cohen, D. B. Kaplan, and A. E. Nel- +son, +Ann. Rev. Nucl. Part. Sci. 43, +27-70 (1993), +arXiv:hep-ph/9302210. + +10 +[13] V. A. Rubakov and M. E. Shaposhnikov, Usp. Fiz. Nauk +166, 493-537 (1996), arXiv:hep-ph/9603208. +[14] Yasushi Takahashi, Hiroomi Umezawa, +International +journal of Modern Physics B, 10 (1996). +[15] Torbjorn Lundberg and Roman Pasechnik, European +Physical Journal A, 57, (2021). +[16] P. Braun-Munzinger and Johanna Stachel, Nature 448, +302-309 (2007). +[17] PHENIX Collaboration, +Nature Physics 12, +(2018); +arxiv:nucl-ex/1805.02973 (2018). +[18] S. Ganesh and M. Mishra, Progress of Theoretical and +Experimental Physics, 2021, 1, 013B09 (2021). +[19] S. Ganesh , Int. J. Mod. Phys. A, 37, 17, 2250125 (2022); +arXiv:hep-th/2206.13324 (2022). +[20] S. Coleman, E. Weinberg, Phys. Rev. D, 7, 4 (1973). +[21] J. Baglio, A. Djouadi, JHEP, 2011, 55 (2011); arXiv: +hep-ph/1012.0530 (2012). +[22] Peter Higgs, Physics Letters 12, 132 (1964). +[23] Peter W. Higgs, Phys. Rev. Lett. 13, 508 (1964). +[24] J. Goldstone, A Salam and S Weinberg, Physical Review, +127, 965 (1962). +[25] J. Kapusta, C. Gale, ”Finite Temperature Field The- +ory and Applications”, Cambridge University Press, 2e +(2006). +[26] C. W. Bernard, Phys. Rev. D, 9(12), 3312 (1974). +[27] Yuji Nakatsukasa, Lecture Notes in Computer Science, +EPASA 2015, pp 233-249. +[28] R. C. Tolman, Phys. Rev. 35 904–924 (1930). +[29] R. C. Tolman and P. Ehrenfest, Phys. Rev. 36, 12, +1791–1798 (1930). +[30] N Andersson, GL Comer, and K Glampedakis. Nuclear +Physics A, 763, 212–229 (2005), +[31] Petarpa Boonserm, Matt Visser, and Silke Weinfurtner. +Phys. Rev. D, 76, 4, 044024 (2007), +[32] Y. Sofue et. al., ”The Astrophysical Journal”, 523, 136- +146 (1999). +[33] Kyu-Hyun Chae et. al., The Astrophysical Journal, 904 +(2020). + diff --git a/vtE3T4oBgHgl3EQf-guw/content/tmp_files/load_file.txt b/vtE3T4oBgHgl3EQf-guw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d020b87ec264c544e19f9171ec226c590a51e0c --- /dev/null +++ b/vtE3T4oBgHgl3EQf-guw/content/tmp_files/load_file.txt @@ -0,0 +1,527 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf,len=526 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='04827v1 [hep-th] 12 Jan 2023 5-D thermal field theory, Einstein field equations and spontaneous symmetry breaking S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Ganesh∗ It has been shown previously, that the spatial thermal variation of a thermal medium can be recast as a variation in the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is now extended to temporal variations in temperature, for a non-relativistic thermal bath, which remains in local thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This is achieved by examining the thermal field theory in a five-dimensional space-time-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The bulk thermodynamic quantity, namely the energy density, is calculated for a neutral scalar field with a time-dependent Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Furthermore, the concept of recasting thermal variations as a variation in the metric is extended to thermal systems in a gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The Einstein field equations, in the 5-D space-time-temperature, is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is shown that the resulting Ricci scalar can then lead to spontaneous symmetry breaking, leading to the Higgs mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In essence, the asymmetry in the distribution of temperature in space-time can translate to spontaneous symmetry breaking of particle fields, in a very strong gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Keywords : Thermal gradient, 5-D Thermal field theory, Einstein field equation, Gravity, Sponta- neous symmetry breaking, Higgs PACS numbers : 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='Wx, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='Kd, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='Ex I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' INTRODUCTION The thermal field theory incorporates a Euclidean space-time, obtained by an analytical continuation of time in Minkowski space-time to an imaginary time, to model thermal systems [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Numerous research has been done using the framework of thermal field theory in some form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Some examples include Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [5–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' There are, however thermal systems, where there are significant thermal variations in both spatial and temporal dimen- sions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', the Quark Gluon Plasma (QGP) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The idea was first mooted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [18], that spatial ther- mal variations can be modeled by recasting the variation in the temperature as a variation in the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This idea was used to determine the quark anti-quark potential in a thermal system with spatial variations, using AdS-CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The thermal medium was assumed to be under local thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19], the con- cept that the spatial thermal variations can be recast as a variation in the Euclidean metric, was placed on firmer grounds, by analyzing the Polyakov loop, the par- tition function, the correlation function, and the geodesic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It has been shown that the partition function for thermal systems with spatial thermal variations, nat- urally leads to the notion of a curved Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' While most of the analysis was carried out for a canoni- cal ensemble, the framework was also touched upon, for a grand canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is now shown, that the concept of the 5-dimensional space first introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19], enables the generalization of the concept to temporal variations in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The temporal varia- tions must be sufficiently slow to maintain local thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The 5-D space-time-temperature approach further en- ∗Corresponding author: Email: gans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='phy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='com ables the extension of the concept to a thermal bath in a gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The Einstein field equations are de- termined in the 5-D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The resultant field equations are then expressed in terms of the usual 4-D space-time covariant operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This process of dimension reduction leads to additional terms in the Ricci tensor, and subse- quently, the Ricci scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' There has been significant research on the source of spontaneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Reference [20] has indi- cated that radiative corrections can be a possible source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Reference [21] has explored the production of Higgs at the Large Hadron Collider (LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A crucial outcome of the proposed framework is that the modified Ricci scalar may be a source of spontaneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A negative Ricci scalar can make the term, (m2 + R)φ2, in the Lagrangian for a scalar field, become negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', (m2 + R)φ2 → −µ2φ2, for some real valued µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Thus, a negative Ricci scalar curvature may be a viable candi- date for the negative potential necessary for spontaneous symmetry breaking for a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The spontaneous symmetry breaking leads to the Higgs mechanism [22– 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For the sake of clarity, let us briefly revisit some of the relevant concepts that were discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' An 8-D space, βµ × xν, was considered to model thermal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Under conditions that the thermal system is stationary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', the four-velocity of the thermal medium, uµ = (1, 0, 0, 0), it reduces to a 5-D space, (iβ, t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Moreover, if the thermal system is time-invariant, it may suffice to consider the 4-D sub-space, (iβ, x), lead- ing to the imaginary time formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' If the temper- ature varies in space, with spatial variation s(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', β ≡ β(x) = s(x)β0, where β0 is a constant, then it leads to a curved Euclidean space, with the spatial variation, h00 = s(x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A 5-D metric tensor along with the h00 term is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' G(5) ab = � h00 0 0 g(4) µν � , (1) 2 where, g(4) µν , is the usual metric tensor in 4-D space-time, resulting purely as a result of gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The 5- D space consideration was necessitated to allow separate metric components, h00, due to thermal variations, and g00, due to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Furthermore, the 5-D space was again shown as a necessity, in order to model an external Dirac spinor, with energy E, traversing a thermal medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The 3-momentum observable for the Dirac spinor, traversing a thermal bath, can be made manifest only with a 5-D representation of the Dirac equation, instead of the 4- D imaginary time formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The conjugate momentum variables to time and temperature, namely the energy, E, and the Matsubara frequency, ωn = Cn h00 , capture the intrinsic energy, E, of the particle, and the interaction en- ergy with the thermal medium, iEc = Cn h00 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The curvature of Euclidean space was considered in the complete absence of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In [19], the analysis was primarily for a time-invariant system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' However, since time and inverse temperature, β, are separate dimensions, it is tempting, to extend the formalism to time-varying systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We now develop the formalism for time-varying systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In addition, in the current work, the curvature due to the effects of temper- ature variations and gravity is encapsulated in a com- mon framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Furthermore, the criterion for the Ricci scalar, R, to be negative, under the combined influence of the thermal variations and gravity, is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The rest of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Section II, devel- ops the formalism for a 5-D time-varying thermal system, that remains in local thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The expecta- tion value of energy density, for a time-varying Hamil- tonian, is then determined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The 5-D Einstein field equations are developed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The criterion for spontaneous symmetry breaking is also developed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Finally, the results are summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' THE TIME VARIATION IN TEMPERATURE A time-varying thermal system would also have a four- velocity, uµ, of the thermal medium, leading to an 8-D modeling as mentioned in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' However, in the non- relativistic limit, uµ = (1, u) ≈ (1, 0, 0, 0), the analysis can be approximated by an analysis in 5-D space itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The 8-D space can be divided into two 4-D Lorentz invari- ant sub-manifolds [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Thus, a 5-D sub-space may not lend itself to a full-fledged Lorentz invariant treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A complete Lorentz invariant treatment would require the entire 8-D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is planned to extend the current framework to a complete Lorentz invariant treatment in 8-D space in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In order to model partition function in a 5-D space- time-temperature, it would be required to take care that the system is not acausal or non-local in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This rules out an Hamiltonian of the form H = � Hdtd3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Instead, one may analyze the system in each time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A partition function imbibes the statistical properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For a time-varying system, it is possible to take time slices and define the statistical properties in each time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The process of taking time slices, how- ever, does not construe that each time slice is modeled as a static system, with the effects of time derivatives being ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The Hamiltonian would now be time-dependent, and in general, be different from the Hamiltonian of a static system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The time-dependent Hamiltonian should capture the non-trivial effects due to the time variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Consequently, we define the partition function at each time slice for a time-varying system, which continues to be in local thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We now proceed on sim- ilar lines as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19], for each time slice, albeit with a time-dependent Hamiltonian density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let us first develop the field theory for a non- interacting Lagrangian, in 5-D space-time-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The Lagrangian density in a 5-D space, for a neutral scalar field would be, L(ˆφ, ∂a ˆφ) = 1 2 � ∂a ˆφ∂a ˆφ − m2 ˆφ2� , (2) with the index a = 0, 1, 2, 3, 4 corresponding to the di- mensions [τ, t, x, y, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' τ is the inverse temperature and varies from 0 to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The sign convention used is (-,+,-,-,-).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This gives rise to the equation of motion: ∂a∂a ˆφ + m2 ˆφ = 0 (3) The constraining 5 momentum delta function would be, δ(E2 − ω2 − p2 − m2), and the 5-D integral measure is: 1 β � n � d3p (2π)3 dE 2π 2πδ(E2 − ω2 n − p2 − m2) = 1 β � n � d3p (2π)3 1 2E , (4) where, the Matsubara frequency = ωn = 2nπ β , for a bo- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let us look at the physical interpretation of the delta function, δ(E2−ω2 n−p2−m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' As mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19], E, may be considered the original intrinsic energy of a particle, and ωn = iEc, can be considered as the interac- tion energy of the particle with the thermal medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The variable, ωn, determines the decay or enhancement of a particle wave-function with temperature (for example, the Dirac spinor in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The magnitude of the to- tal energy is then, |E + iEc| = � E2 + E2c = � E2 − ω2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is intuitive, that a particle’s 3-momentum would be affected by both E and Ec, and not just E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Thus, one may consider E2 − ω2 n = p2 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A portion of the par- ticle’s original energy, E, is lost due to interaction with the thermal medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This provides an intuition behind the delta function, δ(E2 − ω2 n − p2 − m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The operator for a neutral scalar field in 5-D space- time-temperature is, ˆφ(x, τ, t) = 1 β � n � d3p (2π)3 1 � 2Ep � s � a† p,ωne−ipxe−iωnτ + ap,ωneipxeiωnτ� (5) 3 The operator, a† p,ωn, creates a particle with 3- momentum p, and Matsubara frequency ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' One may premise the below commutation relation: [ap1,ωn1, a† p2,ωn2] = (2π)3δ3(p1 − p2)ζ(β)δn1,n2, (6) where, ζ(β) is a scalar normalization function, and needs to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let us define, ap(τ) = � n f(ωn)ap,ωn, a† p(τ) = � n f ∗(ωn)a† p,ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (7) Equation 7 can be interpreted in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' When a momentum state |p⟩ is created, then |p⟩ itself can be treated as a superposition of the momentum-Matsubara eigenstates |p, ωn⟩, with probability amplitudes f(ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Since f(ωn) is a probability amplitude, � n f 2(ωn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Then, the equal τ commutator, [ap1(τ), a† p2(τ)] = � n1 � n2 [ap1,ωn1, a† p2,ωn2]f(ωn1)f ∗(ωn2) = � n1 � n2 (2π)3δ3(p1 − p2)ζ(β)δn1,n2f(ωn1)f ∗(ωn2) = � n1 (2π)3ζ(β)δ3(p1 − p2)f 2(ωn1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (8) Since � n1 f 2(ωn1) = 1, let us assign ζ(β) = 1, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 8, to obtain, [ap1(τ), a† p2(τ)] = (2π)3δ3(p1 − p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (9) Thus, the usual commutation relation between the 3- momentum annihilation and creation operator is recov- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For large β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', β → ∞, one may follow a similar procedure as above, but in the continuous domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The commutation relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 6, can be seen to be, [ap1,ωn1, a† p2,ωn2] = (2π)4δ3(p1 − p2)δ(ωn2 − ωn1), (10) with ζ(β) = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The relation between Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 6 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 10 can be understood by noting that the 4-D space- temperature manifold is a R3 × S1 manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' As β → ∞, R3 × S1 → R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We now proceed to determine the conjugate momenta and the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' There can be a conjugate momenta w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' either the time variable or the temperature vari- able, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', ˆπt = δL δ ∂ ˆφ ∂t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' ˆπβ = i δL δ ∂ ˆφ ∂τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (11) The corresponding Hamiltonian densities are: Ht = ˆπt ∂ ˆφ ∂t − L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Hβ = −iˆπβ ∂ ˆφ ∂τ − L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (12) They would obey the evolution equations: i∂ ˆφ ∂t = [ˆφ, Ht];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' ∂ ˆφ ∂τ = [ˆφ, Hβ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (13) Since we are modeling a thermal system, and are in- terested in evolution in τ, the main object of interest would be Hβ and πβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For convenience, we now drop the subscript β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In the rest of the paper, unless otherwise mentioned, H and ˆπ refer to Hβ and ˆπβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We now follow a similar procedure as Ref [19, 25], albeit modified for a 5-D space with thermal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let φ(x) and |φ(x)⟩ be the eigenfunction and the eigen- ket of the Schrodinger picture field operator ˆφ(x, 0, 0), while, π(x) and |π(x)⟩ be the eigenfunction and the eigenket of the conjugate momentum field operator ˆπ(x, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In other words, ˆφ(x, 0, 0)|φ⟩ = φ(x)|φ⟩, ˆπ(x, 0, 0)|π⟩ = π(x)|π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (14) The eigenkets, |φ⟩ and |π⟩, obey the following relation: ⟨φ|π⟩ = exp � i � d3xπ(x)φ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (15) For a time-dependent system, the time-dependent Hamiltonian can be written as: H(t) = H0 + HI(t), (16) where H0 is the time independent part, and HI(t) be the time dependent part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' As mentioned earlier, HI(t) should capture the non-trivial effects of time variation in a time-varying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' H0 is written in terms of the Schrodinger picture operators ˆπ(x, 0, 0) and ˆφ(x, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' H0 = � d3xH0(ˆπ(x, 0, 0), ˆφ(x, 0, 0)) ≡ � d3xH0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (17) For simplicity of notation, the form, H0(x), is used as a representation of H0(ˆπ(x, 0, 0), ˆφ(x, 0, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The free Hamiltonian density, corresponding to the La- grangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 2, would then be, H0 = 1 2 \uf8eb \uf8edˆπ2 + (∇ˆφ)2 − � ∂ ˆφ ∂t �2 + m2 ˆφ2 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (18) But, ∂ ˆφ ∂t = −iE ˆφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In a gas composed of scalar fields, which is equilibrated, E → 0, as only the ensemble in- teraction energy, captured by the Matsubara frequency, ωn, is non zero [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We are then left with the stan- dard imaginary time formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' On similar lines, in a vacuum, as β → ∞, ωn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Then, the only energy left is the particle energy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The Lagrangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 2, then boils down to the normal 4-D space-time Quantum Field Theory (QFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' When both E and ωn are non-zero, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 2 can be used to model particles that are not yet fully 4 equilibrated with the thermal medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A case in point is the external Dirac spinor traversing a thermal medium, which was modeled in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The external Dirac Spinor (not equilibrated with the thermal medium), will have its intrinsic energy, E, as well as a non-zero ωn, due to interaction with the thermal medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' With, E → 0, the Hamiltonian density in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 18 becomes, H0 = 1 2 � ˆπ2 + (∇ˆφ)2 + m2 ˆφ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (19) Equations 9 and 19, indicate that the generalizations to 5-D, characterized by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 5, 6, 11, 12, are backward compatible with the existing 4-D imaginary time formal- ism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A fairly generic time-dependent Hamiltonian den- sity can be written as: HI(t) = � HI(x, t)d3x, (20) where, HI(x, t) = � i ci(x, t)fi(ˆφ, ∂µ ˆφ, ˆπ, ∂µˆπ), (21) and ci(x, t) are arbitrary scalar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' However, vari- ations in ci(x, t), should not be sharp enough to throw the system out of local thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In this pa- per, the thermodynamic properties are evaluated for the below specific cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' HI(x, t) = V (x, t)ˆφ2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' HI(x, t) = λ(x, t)ˆφ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In the case of a thermal bath composed of scalar particles, V (x, t)ˆφ2 can represent the coupling of an external field, along with its derivatives, with ˆφ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The evolution operator U(Hβ, β, t) provides the evolu- tion w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', Uβ(Hβ, β, t) = exp � − � � β(x,t) 0 Hβ(x, t)dτd3x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (22) As mentioned earlier, we drop the subscript β from U, H and H, and obtain, U(H, β, t) = exp � − � � β(x,t) 0 H(x, t)dτd3x � , = exp � − � � β0 0 s(x, t)H(x, t)dτd3x � , (23) where, H(x, t) = H0(x) + HI(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The integrand in the exponent of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 23, indicates a volume element dτd3x of a 4-D slice, at time t, within the 5-D space, with metric, diag[−s(x, t)2, 1, −1, −1, −1], and determinant, √g5 = s(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Thus, it describes a curved 5-D space- time-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We can then define the partition function Z(t), at a time slice, t, in 5-D space-time-temperature as: Z(β0, t) = tr � ⟨φf|U(H, β0, t)|φ0⟩ � , (24) with |φ0⟩ and |φf⟩ being the eigenkets at τ = 0 and τ = β0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The procedure to evaluate the partition function, Z(β0, t), is now straight forward and similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We first evaluate U(H, β, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let β0 be sliced into N slices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', β0 = N∆β, with N → ∞ and ∆β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This gives, U(H, β0, t) = lim ∆β→0 exp � − � n �� s(x, t)H(x, t)d3x � ∆β � (25) = lim ∆β→0 � n exp � − �� s(x, t)H(x, t)d3x � ∆β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (26) Let |πj⟩ be the β based conjugate momentum eigen- state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Inserting a complete set of eigenstates: I = � |φj⟩⟨φj|dφj × � |πj−1⟩⟨πj−1|dπj−1 2π , between each product term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 26, and I = � |πN−1⟩⟨πN−1|dπN−1 2π ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' I = � |φ1⟩⟨φ1|dφ1, in the beginning and the end of the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 26, respectively, we get the expression for K(φf, φ0, β0, t) = ⟨φf|U(H, β0, t)|φ0⟩: K(φf, φ0, β0, t) = lim ∆β→0 � N−1 � j=1 ⟨φj+1|πj⟩ × ⟨πj| exp � − � s(x, t)H(x, t)∆βd3x � |φj⟩ × ⟨φ1|φ0⟩dφj dπj 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (27) It is possible to evaluate the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 27, using the relations: ⟨φj+1|πj⟩ = exp � i � d3xπj(x)φj+1(x) � , (28) ⟨πj| exp � −i � s(x, t)Hd3x∆β � |φj⟩ = ⟨πj|φj⟩ exp � −i � s(x, t)Hjd3x∆β � , (29) with, Hj = H(πj, φj, t) = H0(πj, φj) + HI(πj, φj, t), 5 and ⟨φ1|φ0⟩ = � x δ (φ1(x) − φ0(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (30) It is to be noted that the form of HI(x, t), used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 21, would satisfy the relation specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This is because, the operator arguments of HI, namely, ˆφ(x) and ˆπ(x), continue to be Schrodinger picture operators, and the temporal aspect is captured by the scalar functions, ci(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' If a different form of HI(x, t) is used, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 29 needs to be checked for validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 28, 29 and 30 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 27, we obtain, K(φf, φ0, β0, t) = lim ∆β→0 � � j dφj dπj 2π exp � � d3x × � − iπj(x) � φj+1(x) − φj(x) � − s(x, t)Hj∆β �� (31) = lim ∆β→0 � \uf8eb \uf8ed� j dφj dπj 2π \uf8f6 \uf8f8 exp � � k � ∆β � d3x × s(x, t) � −iπk(x){φk+1(x) − φk(x)} s(x, t)∆β − Hk � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (32) We recollect that, s(x, t) = � h00(x, t) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Fur- ther, for the metric, G5 = diag[−s(x, t)2, 1, −1, −1, −1], we have, � |G5| = s(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We denote the determinant of the Euclidean metric, |G5|, by g5, and obtain in the continuum limit: K(φf, φ0, β0, t) = � Dφ � Dπ 2π exp � � β0 0 dτ � d3x � g5(x, t) × � − iπ(x, τ) 1 � h00(x, t) Dτφ(x, τ) − H(π(x, τ), φ(x, τ), t) �� , (33) where, Dτ is the covariant derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For a scalar φ(x, τ), Dτφ(x, τ) = ∂φ(x,τ) ∂τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Finally, the partition function becomes, Z(β0, t) = trK(φf, φ0, β0, t) = � periodic Dφ � Dπ 2π exp � � β0 0 dτ � d3x � g5(x, t) × � iπ(x, τ) 1 � h00(x, t) Dτφ(x, τ) − H(π(x, τ), φ(x, τ), t) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (34) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' EVALUATION OF ⟨E(t)⟩ In this section, the expectation value of the energy density at different instances of time, ⟨E(t)⟩, is evaluated for HI = V (t)φ2, and HI = λ(t)φ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' ⟨E(t)⟩ when HI = V (t)φ2 Since only φ2 is involved in the partition function, this is easily evaluated as Gaussian integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Using the pro- cedure outlined in [19], the expression for the energy ex- pectation value, ⟨E(t)⟩ is: ⟨E(t)⟩ = −∂ ln(Z) ∂β0 = V � d3p (2π)3 � ωp 2 coth �β0ωp 2 � + � ⟨η(x, t)⟩ + 1 2ω2p ⟨s(x, t)V (x, t)⟩ � × �ωp 2 coth �β0ωp 2 � + β0ω2 p 4 cosech2 �β0ωp 2 � �� , (35) where, ⟨s(x, t)V (t)⟩ = � d3xs(x,t)V (t) � d3x , s(x, t) = 1+η(x, t), ⟨η(x, t)⟩ = � d3xη(x,t) � d3x and ωp = � p2 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' ⟨E(t)⟩ when HI = λ(x, t)φ4 The Lagrangian density for a neutral scalar field would now read: L = 1 2 � ∂µ ˆφ∂µ ˆφ − m2 ˆφ2� − λ(x, t)ˆφ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (36) The corresponding interaction Hamiltonian is, HI = λ(x, t)ˆφ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For small λ(x, t), the partition function can be written as: Z(β0, t) = � periodic Dφ � Dπ 2π × � 1 + � r 1 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' �� dτd3x√g5λ(x, t)φ4 �r� × exp � � β0 0 dτ � d3x√g5 � − iπ(x, τ) 1 � h00(x) ∂φ(x, τ) ∂τ −1 2 � π2 + (∇φ(x, τ))2 + m2φ(x, τ)2� �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (37) Then, one may represent the above equation as: ln Z(β0, t) = ln(Z0(β0, t)) + ln � 1 + � r 1 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Zr I (β0, t) Z0(β0, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (38) 6 2 pv, n1 pl, n2 pu, n1 pk, n FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 1: Feynman diagram depicting the first order ˆφ4 inter- action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The calculation of ln(Z0(β0, t)), at time slice t, is identical to the calculation of ln(Z0(β0)), in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For r = 1, the expression for Z1 I , corresponding to the Feynman diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 1 is: Z1 I (β0, t) = � periodic Dφ � Dπ 2π �� dτd3x√g5λ(x, t)φ4 � × exp � � β0 0 dτ � d3x√g5 � − iπ(x, τ) 1 � h00(x) ∂φ(x, τ) ∂τ −1 2 � π2 + (∇φ(x, τ))2 + m2φ(x, τ)2� �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (39) The factor Z1 I Z0 , is now calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The � Dπ integrals can be done in exactly the same manner as the non- interacting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We consider a space of volume V, dis- cretize it, and divide it into M 3 cubes of length ∆a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Fur- ther, express φ(x) in terms of its momentum domain rep- resentation, φn,p [19], φ(x, τ) = � β0 V � i ∞ � n=−∞ eiCnτeipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='xφn,pi, (40) where, Cn is related to ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We follow a procedure sim- ilar to the non-interacting case in Ref [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Then, after replacing � ∆p 2π �3 as V in the exponent, one may obtain Z1 I as: Z1 I = � N � 1 2π �NM3/2 � × � Dφn,p �� dτd3xs(x, t)λ(x, t)φ(x)4 � × exp � −1 2 � i � j ∞ � n=−∞ φ∗ n,pj � β2 0 � (C2 n + pipj + m2)δij + ηf(pj − pi, t) 1 V(−C2 n + pipj + m2) �� φn,pi � , (41) where, we have used √g5 = s(x, t), and N is an incon- sequential multiplication factor and does not affect the dynamics of the system [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The length, ∆a, gets absorbed in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The variable, ηf(p, t), is the 3-D Fourier transform of η(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 40, φ(x)4 can be rep- resented in terms of its Fourier components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' � dτd3xs(x, t)λ(x, t)φ4(x) = �� β0 V �4 × � pu,nu � pv,nv � pk,nk � pl,nl β0δ (Cnu + Cnv + Cnk + Cnl) ×sλf(−pu − pv − pk − pl, t)φnu,puφnv,pvφnk,pkφnl,pl, (42) where, sλf(p, t) is the momentum domain representation of the product term, s(x, t)λ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The delta function forces, nu = −nv = n1(say), and nk = −nl = n2(say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Secondly, we make the transformation pv → −pv and pl → −pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' With these, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 42, becomes � dτd3xs(x, t)φ4(x) = 3 β3 0 V2 � n1 � n2 � u � v � k � l � sλ,f (pv − pu + pl − pk, t) φ∗ n1,pvφn1,puφ∗ n2,plφn2,pk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (43) Let the exponent enclosed between φ∗ n,pj and φn,pi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 41 be Mn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', Mn = � β2 0 � (C2 n + pipj + m2)δij +ηf(pj − pi, t)‘ 1 V(−C2 n + pipj + m2) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (44) Let Λn diagonalize Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Then Mn = Λ−1 n DnΛn, where Dn is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let ψn,a = � i Λn,a,iφn,pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This means φn,pi = � a Λ−1 n,i,aψn,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Inserting the above transformation of φ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 43, and making use of the fact that (Λ−1 n,i,aψn,a)∗ = ψ∗ n,aΛt n,i,a, � dτd3x s(x, t)λ(x, t)φ4(x) = 3 β3 0 V2 � n1 � n2 � b � a � d � c ψ∗ n1,bψ∗ n2,d × � � v � u � l � k Λn1,b,vΛn2,d,l × sλ,f(pv − pu + pl − pk, t)Λ−1 n1,u,aΛ−1 n2,k,c � ψn1,aψn2,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (45) 7 Or, � dτd3x s(x, t)λ(x, t)φ4(x) = 3 β3 0 V2 � n1 � n2 � b � a � d � c × � ψ∗ n1,bψ∗ n2,dΛs bdacn1n2ψn1,aψn2,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (46) Equation 46 is same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 45, but with a more compact notation, by introducing, Λs bdacn1n2, for the term enclosed within the [ ] brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Also, we have, � i � j φ∗ n,pjMn,j,iφn,pi = � l ψ∗ n,lDn,l,lψn,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (47) Since the transformation, φ → ψ, is unitary, we can re- place the measure � Dφ by � Dψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Substituting Eq, 46, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 47, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 41, we obtain: Z1 I = λ(t) � N � 1 2π �NM3/2 � × � n1 � n2 � b � a � d � c � 3 β3 0 V2 � × � Dψn,i � ψ∗ n1,bψ∗ n2,dΛs bdacn1n2ψn1,aψn2,c � × exp � − 1 2 � i ∞ � n=−∞ ψ∗ n,iDn,i,iψn,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (48) The functional integrals where, b ̸= a and d ̸= c, are zero since the integrands are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The path integral then simplifies to: Z1 I = � N � 1 2π �NM3/2 � × � n1 � n2 � a � c Λs acacn1n2 � 3 β3 0 V2 � × � dψn1,aψ2 n1,a exp � −1 2ψ∗ n1,aDn,a,aψn1,a � × � dψn2,cψ2 n2,c exp � −1 2ψ∗ n2,cDn,c,cψn2,c � × � remaining Dψn,i exp � −1 2 � (n,i)̸= (n1,a),(n2,c) ψ∗ n,iDn,i,iψn,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (49) This finally gives, Z1 I = � N � 1 2π �NM3/2 � × � 3 β3 0 V2 � � n1 � n2 � a � c � Λs acacn1n2 × � √ 2π 2D3/2 n1,a,a � � √ 2π 2D3/2 n2,c,c � � (n,i)̸= (n1,a),(n2,c) � √ 2π 2D1/2 n,i,i � � , (50) Since, ΠkDn,k,k = det(Dn) = det(Mn), it is possible to write: Z1 I Z0 = 3 β3 0 V2 � n1 � n2 � a � c � Λs acacn1n2 × � 1 2Dn1,a,a � � 1 2Dn2,c,c � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (51) We now estimate, Λs acacn1n2, to the first level of approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Better approximations using numerical methods or superior techniques could lead to higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For matrices, Mn, An and En, where Mn = An + En, and if eigenvalues, λA, of An is known, it is possible to estimate the eigenvalues, λM, of Mn using the relation [27]: λM = λA + xtEnx xtx + O(∥E∥2), (52) where, xt is eigenvector of An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Since, O(∥E∥) ∼ O(∥nf∥), this should estimate eigenvalues of Mn to order O(∥nf∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The matrix, Mn, is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let us take An and En to be the matrices: An,i,j = β2 0(C2 n + pipj + m2)δij, (53) En,i,j = β2 0ηf(pj − pi, t) 1 V(−C2 n + pipj + m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (54) Since An is diagonal, λA = β2 0(C2 n + p2 + m2), and the eigenvectors, xt, are unit vectors with only one non-zero entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This gives, λM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='n,i ≈ β2 0 � (C2 n + p2 i + m2) + ηf(0, t) V (−C2 n + p2 i + m2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (55) Thus, Dn,a,a = λM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='n,a ≈ β2 0 � (C2 n + p2 a + m2) + ηf(0, t) V (−C2 n + p2 a + m2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (56) This approximation has extracted the diagonal elements of E as the contribution from E to λM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Consequently, 8 the estimated eigenvector of Mn continues to be x, at this level of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Ergo, Λn,a = Λn,c ≈ I, leading to, Λs bdacn1n2 ≈ sλf(0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' With all this, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 51 becomes, Z1 I Z0 = 3 β3 0 V2 sλf(0, t) ��� a � n1 1 2Dn1,a,a ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (57) The value of Dn,a,a in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 56, can be simplified, by rec- ognizing that ηf(0,t) V = ⟨η(x, t)⟩, 1 + ⟨η(x, t)⟩ = ⟨s(x, t)⟩ and, 1 − ⟨η(x, t)⟩ ≈ 1 ⟨s(x,t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Substituting the value of Dn,a,a thus obtained, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 57, Z1 I Z0 = 3 β3 0 V2 sλf(0, t) × \uf8ee \uf8f0� a � n 1 β2 0 � ( C2 n ⟨s(x,t)⟩ + ⟨s(x, t)⟩ (p2a + m2)) � \uf8f9 \uf8fb 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (58) Finally, Z1 I Z0 = 3 4(β0V)⟨s(x, t)λ(x, t)⟩ × � � d3p (2π)3 1 ωp coth �ωpβ0⟨s(x, t)⟩ 2 � �2 , (59) where, ωp = � p2 + m2 and ⟨s(x, t)λ(x, t)⟩ = sλf(0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For, r = 1, one may then determine, ⟨E(t)⟩ = ∂ ln(Z(t)) ∂β0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' THE RICCI TENSOR AND NEGATIVE POTENTIAL The Einstein field equations are now determined in the 5-D space-time-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We use the letters, a, b, as indices for the 5-D space-time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', a, b = 0, 1, 2, 3, 4, with the index 0 referring to the temperature dimension, while, the index 1 refers to the time dimension, We use the letters, µ, ν, as indices for the 4-D Lorentzian space- time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', µ, ν = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A superscript, (N), within brackets, refers to N dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For example, ∇(4) µ , refers to the covariant derivative in 4-D space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Let us consider the Lagrangian, L(φ, ∂aφ) = 1 2 � −∂aφ∂aφ − m2φ2 − ζRφ2� , with the sign convention (+,-,+,+,+), and the corre- sponding action, S = � � −g(5)L(φ, ∂aφ)d5x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (60) Let us consider the 5-D metric, g(5) ab , as: g(5) ab = � s2(g(4) µν (x, t), x, t) 0 0 g(4) µν � , (61) where, g(4) µν , is the usual metric tensor in 4-D space- time, and is purely due to gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is also noted that s would now additionally be a function of g(4) µν also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' As an example, based on the Ehrenfest Tol- man effect [28, 29], in the case of a uniform temperature (system in global equilibrium), in a gravitational field, s(g(4) µν (x, t), x, t) = � gµνζµζν, where ζµ is the Killing vector in 4-D space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' We speculate that, the func- tion, s(), can be factored, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', s(gµν(x, t), x, t) = s1(gµν(x, t)) s2(x, t), (62) where, s1(gµν(x, t)) is due to the Ehrenfest Tolman effect, and s2(x, t) is due to thermal gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is the simplest function that has the following limits: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' When the thermal system is in global equilibrium, s(gµν(x, t), x, t) → s1(gµν(x, t)), and, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' when gµν → ηµν, s(gµν(x, t), x, t) → s2(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It is, however desirable to prove or improvise upon Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 62, based on first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For the metric, g(5) ab , the Einstein field equations in the 5-D space become: R(5) ab − 1 2g(5) ab R(5) = 8πG c4 T (5) ab , (63) where, T (5) ab = − 2 � g(5) δS δg(5) ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (64) The 5-D Ricci tensor, R(5) ab , can be expressed in terms of the 4-D covariant derivative operator, ∇(4) µ , and the 4-D Ricci tensor, R(4) µν , as: R(5) ab = � Rββ 0 0 R(4) µν − 1 s∇(4) µ ∇(4) ν s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (65) where, Rββ = s∇(4)µ∇(4) µ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The treatment of the geodesic equation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19], gives an insight into the physical interpretation of how thermal gradients, effects an apparent curvature in the Lorentzian space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It can be easily seen that the expression for R(5) ab reduces to the 4-D Ricci tensor, for the uniform temperature case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', ∂µs → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Another thing to note is that in the metric s(x, t)2dβ2 + gµνdxµxν, the temperature dimension can be eliminated by taking the limit, s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Thus, for a vacuum, one may take the limits ∂µs → 0, followed by s → 0, in which case, the Einstein field equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 63) 9 reduce to the usual 4-D space-time equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Finally the 5-D Ricci scalar, R(5) can be expressed as: R(5) = R(5)a a = R(4) − 2 s∇(4)µ∇(4) µ s, (66) where, R(4) is the conventional 4-D Ricci scalar, in the Lorentzian space, due to gravitational fields, in the ab- sence of any temperature gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' When the 5-D Ricci scalar is sufficiently negative, it can lead to spontaneous symmetry breaking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', when, m2 + ζR(5) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This translates to: R(5) < −m2 ζ , ⇒ 2 s∇(4)µ∇(4) µ s > R(4) + m2 ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (67) It is to be noted that the quantity, 2 s∇(4)µ∇(4) µ s, is scale- invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', the absolute temperature is inconsequen- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Only the variations in temperature matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Since, s(x, t) is a scalar, 2 s∇(4)µ∇(4) µ s simplifies to 2 s∇(4)µ∂µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In the case of a thermal medium formed in a collider, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', like the QGP at the LHC, the gravitational effects can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This gives R(4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Equation 67, then simplifies to: 2 s∂µ∂µs > m2 ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' (68) One may, however, consider the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 68, with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' For Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 68 to be satisfied, 2 s∂µ∂µs needs to be significantly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' But if thermal gradients are very large, the thermal medium may cease to be in local thermal equilibrium, and the current formalism ceases to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The scenario in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 67 is however different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' To see why, the value of 2 s∇(4)µ∇(4) µ s is affected by three factors: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The thermal gradients, ∂µs2(x, t) (refer Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 62 for the factorization of s = s1s2), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The Ehrenfest Tolman effect, represented by s1(x, t), and, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The terms containing the Christoffel symbols, aris- ing as part of the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Whereas the first factor may be small for a medium in local thermal equilibrium, the second and third factors depend on the strength of the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' SUMMARY In this work, it is seen that the concept of 5-D space- time-temperature, permits the incorporation of time- dependent temperature effects, for systems that continue to be in local thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The bulk thermody- namic property, namely, the energy density expectation value, has been calculated perturbatively, for a thermal bath with both spatial and temporal variations, in the ab- sence of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The Einstein field equations have been determined in 5-D space-time-temperature, which incor- porates both gravitational and thermal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' In the presence of a thermal variation, both temperature gra- dients and gravity determine the net scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The net curvature in the 5-D space can be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' If the d’Alembertian of the temperature variation is suffi- ciently high, spontaneous symmetry breaking may occur in a scalar field, particularly in the presence of a strong gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' As mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19], the curvature of the 5-D space due to temperature variations can be validated ex- perimentally, in terrestrial experiments, due to the much lower energy scales involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Thus, if the concept of the curved 5-D space is confirmed, then it may have impor- tant consequences, such as the spontaneous symmetry breaking as described in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' This current for- malism may be useful in situations where both gravity and thermal variations are present, like in the interior of a neutron star [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' The rotation curves of galax- ies have been a subject of intensive research [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' It might even be possible to consider a galaxy as a thermal medium, with the stars as point particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' However, a galaxy is not in equilibrium, which might pose challenges for such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [1] Matsubara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A New Approach to Quantum-Statistical Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Progress of Theoretical Physics, 14, 4, 351–378 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [2] Martin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', Schwinger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Physical Review, 115(6), 1342–1373 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Abrikosov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Gor’kov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Dzyaloshinskii JETP, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 9, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 636 (1959) [4] Arnold, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', Vokos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', Bedaque, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', Das, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Physical Re- view D, 47(10), 4698 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Aoki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Endrodi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Fodor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Katz, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Sz- abo, Nature 443, 675-678 (2006), arXiv:hep-lat/0611014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [6] Kenji Fukushima and Tetsuo Hatsuda, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 74, 014001 (2011), arXiv:1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='4814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [7] Kraemmer and Rebhan, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 67 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [8] Kraemmer, Rebhan and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Hchulz, Annals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' of Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 238 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [9] Heng-Tong Ding, Frithjof Karsch, and Swagato Mukher- jee, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' E24, 1530007 (2015), arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='05274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [10] Blaizot J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' and Iancu E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', 359 355–528 (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' arxiv hep-ph/0101103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [11] David E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Morrissey and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Ramsey-Musolf, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 14, 125003 (2012), arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='2942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [12] Andrew G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Cohen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Kaplan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Nel- son, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 43, 27-70 (1993), arXiv:hep-ph/9302210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 10 [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rubakov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Shaposhnikov, Usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Nauk 166, 493-537 (1996), arXiv:hep-ph/9603208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [14] Yasushi Takahashi, Hiroomi Umezawa, International journal of Modern Physics B, 10 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [15] Torbjorn Lundberg and Roman Pasechnik, European Physical Journal A, 57, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Braun-Munzinger and Johanna Stachel, Nature 448, 302-309 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [17] PHENIX Collaboration, Nature Physics 12, (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' arxiv:nucl-ex/1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='02973 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Ganesh and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Mishra, Progress of Theoretical and Experimental Physics, 2021, 1, 013B09 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Ganesh , Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' A, 37, 17, 2250125 (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' arXiv:hep-th/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='13324 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Coleman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Weinberg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' D, 7, 4 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Baglio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Djouadi, JHEP, 2011, 55 (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' arXiv: hep-ph/1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content='0530 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [22] Peter Higgs, Physics Letters 12, 132 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [23] Peter W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Higgs, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 13, 508 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Goldstone, A Salam and S Weinberg, Physical Review, 127, 965 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Kapusta, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Gale, ”Finite Temperature Field The- ory and Applications”, Cambridge University Press, 2e (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Bernard, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' D, 9(12), 3312 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [27] Yuji Nakatsukasa, Lecture Notes in Computer Science, EPASA 2015, pp 233-249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Tolman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 35 904–924 (1930).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Tolman and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Ehrenfest, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' 36, 12, 1791–1798 (1930).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [30] N Andersson, GL Comer, and K Glampedakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Nuclear Physics A, 763, 212–229 (2005), [31] Petarpa Boonserm, Matt Visser, and Silke Weinfurtner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' D, 76, 4, 044024 (2007), [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' Sofue et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', ”The Astrophysical Journal”, 523, 136- 146 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' [33] Kyu-Hyun Chae et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE3T4oBgHgl3EQf-guw/content/2301.04827v1.pdf'} +page_content=', The Astrophysical Journal, 904 (2020).' metadata={'source': 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Inria, CNRS, IRISA +3Univ Rennes, INSA Rennes, CNRS, Inria, IRISA +4Univ Rennes, Inria, CNRS, IRISA +lenaig.cornanguer@irisa.fr +Abstract +To get a good understanding of a dynamical system, it is con- +venient to have an interpretable and versatile model of it. +Timed discrete event systems are a kind of model that respond +to these requirements. However, such models can be inferred +from timestamped event sequences but not directly from nu- +merical data. To solve this problem, a discretization step must +be done to identify events or symbols in the time series. Per- +sist is a discretization method that intends to create persisting +symbols by using a score called persistence score. This allows +to mitigate the risk of undesirable symbol changes that would +lead to a too complex model. After the study of the persis- +tence score, we point out that it tends to favor excessive cases +making it miss interesting persisting symbols. To correct this +behavior, we replace the metric used in the persistence score, +the Kullback-Leibler divergence, with the Wasserstein dis- +tance. Experiments show that the improved persistence score +enhances Persist’s ability to capture the information of the +original time series and that it makes it better suited for dis- +crete event systems learning. +Introduction +With the ever-growing type and variety of available sen- +sors, more and more systems can be monitored in real-time. +This monitoring results in the collection of multiple time +series, i.e. sequences of numerical values captured by the +sensors. For example, in a smart city, streets can be moni- +tored through pedestrian counting as well as pluviometers, +to study the effect of weather conditions and the time of the +day on the number of people walking. +The classical use of the sensor time series is to feed them +to machine learning models for tasks such as classifica- +tion or anomaly detection. However, one may also be in- +terested to understand better the dynamical system being +monitored by the sensors. For such goal, an interpretable +model is required. A good solution is to produce a model in +the form of a timed Discrete Event System from the sensor +data. A first difficulty is that there are globally very few ap- +proaches learning timed Discrete Event Systems from data. +The learning of one of these Discrete Event Systems for- +malisms, Timed Automata (TA), has been well studied with +algorithms like RTI+ and TAG (Verwer, de Weerdt, and Wit- +teveen 2010; Cornanguer et al. 2022). However these algo- +rithms take as input sequences of timestamped events, and +not time series. +In order to use these approaches with sensor data, a solu- +tion is to discretize the time series before using the automata +learning algorithm. There is a large literature on time series +discretization. However, the proposed discretization meth- +ods are not designed for automata learning: they may exhibit +too frequent consecutive changes of symbols, which would +lead to needlessly large and complex automata. With the idea +of minimizing the number of symbol changes, M¨orchen at +Ultsch (M¨orchen and Ultsch 2005) proposed the Persist ap- +proach. Persist is based on the assumption that the time se- +ries reflect the dynamics of an underlying system composed +of recurring persisting states and aims at recovering these +states in the form of a sequence of symbols issued from the +time series discretization. +While the idea of Persist is great, using it in practice +for automata learning reveals that Persist may often “go +too far” by focusing on extreme values leading to a dis- +cretized sequence with hardly any symbol changes. In this +paper, through a thorough analysis of the decision criterion +of Persist, the persistence score, we identify the source of +this behavior leading to a more balanced distribution of the +symbols. Our experiments on numerous real and synthetic +datasets demonstrate that the improved persistence score al- +lows to better capture the information of the original time se- +ries, and can help in producing interpretable timed automata. +Motivation and State-of-the-Art +We consider the problem of converting a numerical time se- +ries, a sequence of values measured at regular time inter- +vals (Lin et al. 2003), into a symbolic representation, a se- +quence of symbols from a finite alphabet. Given a time se- +ries, x = {xi|xi ∈ R, i = 1, ..., n}, the purpose is to pro- +vide its discretized version y = {yi|yi ∈ Σ, i = 1, ..., n} +where each symbol yi is an element of a finite alphabet +Σ. Symbolic representation is an efficient way to deal with +the inherent dimensionality of time series so that they can +be used in a low-dimensional space with data-mining and +machine-learning algorithms. To address this problem, many +approaches have been proposed. The simplest discretization +methods are equal-width (EW) and equal-frequency (EF) in- +terval binning which subdivide continuous ranges into inter- +vals through user specification of width or frequency. SAX +(Symbolic Aggregate approXimation) (Lin et al. 2003) is an- +other simple and widely implemented method based on the +arXiv:2301.05041v1 [cs.AI] 12 Jan 2023 + +piecewise aggregate approximation (PAA) technique. The +time series is divided into equal-size time intervals for which +only the mean value is kept. Given the assumption that time +series follow a normal distribution, the Gaussian curve is +then divided through breakpoints producing equiprobable +symbols. SAX requires two parameters, the number of time +intervals and the number of symbols. SAX suffers from +limitations such as the normal distribution assumption, the +lossy compression, and user parameters which impact the +quality of the results. Several variants have been proposed +to attempt to overcome these limitations. Some other ap- +proaches like ABBA and fABBA (Chen and G¨uttel 2022) +have been developed to capture the shape and the trend +of the time series. These methods are motivated by differ- +ent purposes and appear to be best suited for applications +dedicated to the analysis of time series such as trend pre- +diction or anomaly detection. Yet, no discretization method +has been specifically proposed with the end goal of analyz- +ing or learning behaviors of dynamical systems. However, +M¨orchen and Ultsch (M¨orchen and Ultsch 2005) have pro- +posed a discretization algorithm, Persist, based on an inter- +esting property over time series, called persistence. Persist +attempts to produce discretized time series with persisting +symbols, which would be an advantage for the learning of a +timed discrete event system. Our motivation in this paper is +to improve Persist to obtain a more accurate symbolic repre- +sentation suited for the description of dynamical systems. +Persist +Persist (M¨orchen and Ultsch 2005) is a method for unsu- +pervised discretization of univariate time series proposed +by M¨orchen and Ultsch. It was employed as preprocess- +ing step to find patterns in time series in a language called +Time Series Knowledge Representation (TSKR) (M¨orchen +and Ultsch 2007). Persist is based on the assumption that +the time series are the reflection of an underlying process +that consists of recurring persisting states and it aims to re- +store these states in the form of symbols in a discretized ver- +sion of the time series. M¨orchen and Ultsch state that “if +there is no temporal structure in the time series, the symbols +[in its discretized version] can be interpreted as independent +observations of a random variable according to the marginal +distribution of symbols”. Thus, the idea is to look for states +showing a persisting behavior by creating symbols whose +probability of repetition will be much higher than their prob- +ability of appearance. +To create these symbols, Persist produces a set of break- +points creating intervals in the value space. Each interval is +associated with a symbol s that will replace the numerical +values falling within it in the discretized time series. +The breakpoints are iteratively chosen in a set of candi- +date breakpoints (candidates in Algorithm 1) according to +a score called persistence score (described in the next sec- +tion). The set of candidates is initialized by an equal fre- +quency binning (with a number of bins fixed to 100 by de- +fault). At each iteration, the function best bp individu- +ally tests every candidate breakpoint added to the already +selected breakpoints (bps). The candidate increasing persis- +tence score the most is returned with its score. Persist stops +p +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Divergence +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Figure 1: Symmetric KL divergence between two probabil- +ity distributions with two possible outcomes P = (p, 1 − p) +and Q = (q, 1 − q). +when no more candidate breakpoint increases the persis- +tence score, finding thereby automatically an adequate num- +ber of symbols. +Algorithm 1: Persist +Require: univariate time series ts +Return: a set of breakpoints bps +bps = ∅ +candidates = equal frequency binning(ts, 100) +score = 0 +new score = 0 +(new bp, new score) = best bp(ts, bps, candidates) +while new score < score do +score = new score +bps = bps ∪ new bp +candidates = candidates − new bp +(new bp, new score) = best bp(ts, bps, candidates) +end while +return bps +Persistence score +The persistence score measures the persisting behavior of +the symbols created by the discretization. It is based on +the Kullback-Leibler (KL) divergence. The KL divergence +(Kullback and Leibler 1951) measures how a probability dis- +tribution P is different from another probability distribution +Q. For discrete probability distributions defined on X, the +KL divergence is defined as follows: +DKL(P||Q) = +� +x∈X +P(x) log(P(x) +Q(x)) +This +divergence +is +not +symmetric +(DKL(P||Q) +̸= +DKL(Q||P)). This is why M¨orchen and Ultsch use a sym- +metric version obtained as follows: +SKL(P, Q) = 1 +2(DKL(P||Q) + DKL(Q||P)) + +p +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +0.0 +0.2 +0.4 +0.6 +0.8 +Figure 2: Wasserstein distance between two probability dis- +tributions with two possible outcomes P = (p, 1 − p) and +Q = (q, 1 − q). +s +P(s) +Pr(s) +s1 +0. 97 +0.99 +s2 +0.03 +0.62 +(a) Breakpoint 1 +s +P(s) +Pr(s) +s1 +0.54 +0.92 +s2 +0.47 +0.94 +(b) Breakpoint 2 +Table 1: Two candidate breakpoints, each creating two sym- +bols (s1 and s2). The KL divergence will give a better score +to breakpoint 1 while the Wasserstein distance will give a +better score to breakpoint 2. +In Persist, the probability distributions P and Q are based +on the probability of appearance of the symbols (P(s)) and +their probability of repetition (Pr(s)): P = (P(s), 1−P(s)) +and Q = (Pr(s), 1 − Pr(s)). The persistence score is com- +puted as follows: +Persistence(s) = sgn(Pr(s) − P(s))SKL(P, Q) +The first element of the equation (sgn(Pr(s)−P(s))) allows +to favor only the cases when the probability of repetition is +superior to the probability of appearance, otherwise, it con- +tributes negatively to the persistence score. +The symmetric KL divergence between two probability +distributions with two possible outcomes as in our case is +represented in Figure 1. One of the properties of the KL +divergence is that it has no upper bound, a property inher- +ited by the persistence score. The shape of the surface pro- +duced by this divergence is also particular. The symmetric +KL divergence is null when the probability distributions are +equal and increases non-linearly as the difference between +the distribution grows. To achieve a high value of symmet- +ric KL, p or q (i. e. Pr(s) or P(s)) have to be close to 0 +or 1. The direct consequence of these observations is that +the persistence score based on the KL divergence will fo- +cus on extreme cases. Table 1 and Figure 3 illustrate this +phenomenon. In this example, at the beginning of the al- +gorithm, the first breakpoint will be selected to create two +symbols. Two candidate breakpoints are examined. The first +0 +20 +40 +60 +80 +100 +120 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +Breakpoint 1 +Breakpoint 2 +Figure 3: Two candidate breakpoints, each creating two +symbols (s1 and s2). +breakpoint (Table 1a) will create a first symbol that covers +almost the entire discretized time series and thus has a prob- +ability of appearance and repetition close to 1, and a sec- +ond symbol that almost never appears and doesn’t show a +particularly recurring behavior. The second breakpoint (Ta- +ble 1b) will create two symbols about equally probable and +with very high probabilities of repetition (greater than 0.90). +Persist based on the KL divergence will choose breakpoint +1. The discretized version of the time series in Figure 3 will +consist of the succession of about 90 “s1”, then a few “s2” +and again “s1” until the end, while it would have consisted +of an alternation of persistent “s1” and “s2” if breakpoint 2 +had been chosen. +Improving Persist +The Wasserstein distance (Kantorovich 1939), also called +Kantorovitch distance, Kantorovitch - Rubinstein distance, +or earth mover’s distance, is another measure of difference +between probability distributions. It corresponds to the mini- +mal cost to transform a distribution P in another distribution +Q in the same space. The Wasserstein p-distance between +two probability distributions P and Q is defined by the fol- +lowing equation where Γ(P, Q) are all the possible joint dis- +tributions for (X, Y ) with marginal probability distributions +P and Q: +Wp(P, Q) = +inf +γ∈Γ(P,Q)(E(x,y)∼γd(x, y)p)1/p +In the case of discrete probability distributions with only +two possible outcomes, the Wasserstein distance becomes a +simple subtraction and is defined as follows: +W(P, Q) = |P(x1) − Q(y1)| +This distance is symmetric, bounded, easier to compute +than the KL divergence, and it increases linearly as the dif- +ference between the distributions grows (Figure 2). There- +fore, we use the Wasserstein distance to measure how the +probability of appearance of the symbols and their probabil- +ity of repetition are different in the score of persistence in +place of the KL divergence: +PersistenceW (s) = sgn(Pr(s) − P(s))W(P, Q) +In front of the choice presented in table 1, the persistence +score computed with the KL divergence will be higher for + +TA +TA TA +Train +Test +Discretization +(Persist) +Discretization +TA +learner +Accepted +Not +accepted +time series (ℝ) +discretized time series (Σ) +One color per class +Figure 4: Time series classification using timed automata +learned after a discretization step. +breakpoint 1 while the persistence score computed with the +Wasserstein distance will be higher for breakpoint 2. The +Wasserstein distance leads here to a discretized time series +with more persisting symbols, better respecting the initial +intuition of the persistence score. +Finally, the initialization of the candidate breakpoints +based on an equal frequency (EF) binning allows to have +more possible breakpoints in high-density regions. However, +some time series such as electrocardiograms have a structure +that could be missed with this kind of binning. In such cases, +an equal-width (EW) binning can be preferable. It is then im- +portant to let the user choose in function of the structure of +its data. +We re-implemented Persist (originally coded for MAT- +LAB) in Python with the possibility to choose between the +KL divergence and the Wasserstein distance, and between +an equal-frequency or equal-width binning. It is available +online1. +Experiments +The experiment is in two parts. First, we want to evaluate +the raw discretization quality provided by Persist. Then, we +look at its qualities to discretize time series for dynamical +systems modeling in the form of discrete event models. +Experimental setups +We first want to measure the information retained in the data +after the discretization. To allow a quantitative evaluation, +we choose to evaluate the discretization through a classifica- +tion task. If a good part of information is retained in the dis- +cretized time series, the classification performance should +be high. We performed this evaluation on 111 datasets of the +Time Series Classification Repository2 (univariate datasets +1Link to the repository of Persist re-implementation in Python: +https://gitlab.inria.fr/lcornang/persist discretization +2Anthony Bagnall, Jason Lines, William Vickers and Eamonn +Keogh, The UEA & UCR Time Series Classification Repository, +www.timeseriesclassification.com +only). For each dataset, Persist has produced a set of break- +points from the train subset, used for the discretization. We +trained a Random Forest classifier with 100 trees with the +discretized train subset. The classification was then per- +formed on the discretized test subset. We measured the clas- +sification performance using the accuracy, i.e. the rate of +good classification. We tested Persist using either the KL +divergence or the Wasserstein distance, and either an equal +frequency binning or an equal width binning for the can- +didate breakpoints initialization. We also performed the ex- +periment using SAX to have a performance reference. Un- +like Persist, a number of symbol must be given for SAX. We +used a number of symbols ranging from 2 to 10 and a time +interval width of 2 and we report all these results. +We are interested in Persist to obtain a discrete event +model of the dynamical system at the origin of the time se- +ries. Hence, to evaluate its improvement, we need to mea- +sure how good the models are. As discrete event model, +we choose timed automata (Alur and Dill 1994) which is +a common and well-studied formalism for dynamical sys- +tems. Timed automata (TA) are used to model systems in +which time influences the behavior. A timed automaton de- +fines states connected by transitions and the transitions from +one state to another are conditioned by events and timing +constraints. The model can then be used for many purposes +such as to check properties of the system (e.g. safety prop- +erties), or to perform anomaly detection in new data. The +modeling of a system by a timed automaton can be real- +ized thanks to expert knowledge or automatically from ex- +ecution data of the system. If the execution data takes the +form of logs, a learning process can be applied to produce +a timed automaton where the transitions labels correspond +to the events present in the logs. However, if the execu- +tion takes the form of time series (e.g. sensor data), a pre- +processing step is needed to identify events in the numerical +data, the discretization. Figure 4 illustrates the experimen- +tal setup. As in the first experiment, Persist and SAX were +used to obtain the discretized time series. However, instead +of using the discretized data to train a classifier, it was here +used to learn one discrete event model per class. For each +class, the corresponding discretized train time series were +given to a timed automata learner called TAG (Cornanguer +et al. 2022), which produced a timed automaton accepting +all the input sequences (i.e. there exists a path in the au- +tomaton for the sequence). Then the discretized test time se- +ries were injected in the timed automata. Each automaton re- +ceived the discretized test time series of its class and as many +discretized test time series of other classes. An automaton +should accept the data of its own class and reject the oth- +ers. The accuracy corresponds to the good acceptance rate +for the automaton. The Time Series Classification repository +gathers time series of various types (motion, sensor, traffic, +image, spectrographs...). The image type, as spectrographs, +differs from the others as it consists of shapes converted into +pseudo time series. As it doesn’t belong to the problem of +modeling dynamical systems, these datasets were excluded +from the experiment. + +Persist +KL +EF +Persist +KL +EW +Persist +Wasserstein +EF +Persist +Wasserstein +EW +SAX +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Accuracy +0.668 +0.542 +0.717 +0.709 +0.691 +Figure 5: Classification accuracy with random forest for the +different discretization strategies. The diamond indicates the +mean value. EF: equal-frequency, EW: equal-width. +Persist +KL +EF +Persist +KL +EW +Persist +Wasserstein +EF +Persist +Wasserstein +EW +SAX +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Accuracy +0.614 +0.602 +0.607 +0.645 +0.598 +Figure 6: Classification accuracy with timed automata for +the different discretization strategies. +Results +We first present the results for the classification task using +Random Forest. Figure 5 displays the accuracy achieved ac- +cording to the discretization method. The classification per- +formance is increased by the replacement of the Kullback- +Leibler divergence by the Wasserstein distance. The initial- +ization of the candidate breakpoints by an equal frequency +binning leads generally to a better performance, however, +it depends on the dataset which confirms our hypothesis. +Thanks to the Wasserstein distance, using Persist globally +leads to better results than using SAX in this setup. This in- +dicates that Persist using the Wasserstein distance allows a +good information retention in the discretized data. +We now move on to the results of the second experiment. +Using automata to perform a classification task is unusual +and not optimal. Indeed, each automaton is meant to rep- +0 +5 +10 +15 +20 +Hour +0 +250 +500 +750 +1000 +1250 +1500 +Pedestrian count +Breakpoint +(a) Weekdays. +0 +5 +10 +15 +20 +Hour +0 +250 +500 +750 +1000 +1250 +1500 +Pedestrian count +Breakpoint +(b) Weekends. +Figure 7: Instances of time series from the Chinatown +dataset and breakpoints selected by Persist (on the whole +train set) with the Wasserstein distance and an equal-width +binning. +Discretization method +Accuracy +Persist (KL, EF) +0.686 +Persist (Wasserstein, EF) +0.687 +Persist (KL, EW) +0.780 +Persist (Wasserstein, EW) +0.819 +SAX +0.652 +Table 2: Classification accuracy with timed automata for the +Chinatown dataset. +resent a normal global behavior. There is no emphasis for +the modeling on what makes the data of the different classes +singular. For this reason, we cannot expect as good perfor- +mances as while using a real classifier. Nevertheless, it is in- +teresting to compare the classification performance accord- +ing to the discretization method. If the discretization method +is pertinent for discrete event modeling, a good part of the +information contained in the time series would be retained in +the models and therefore leading to good classification per- +formance. Figure 6 displays the classification accuracy using +timed automata. When using SAX, the classification perfor- +mance suffers the most from the use of timed automata. Per- +sist, in particular while using the Wasserstein distance and +an equal-width binning, preserves a better classification ac- +curacy. This confirms the interest in using an improved ver- +sion of Persist to preprocess time series in order to obtain a +discrete event model of a dynamical system. +To provide an insight into what can be obtained with this + +Weekend +low +[0, 0] +Weekday +low +[0, 0] +p=0.4 +very low +[0, 0] +p=0.6 +very low +[3, 4] +low +[4, 6] +high +[1, 9] +very high +[1, 7] +p=0.58 +low +[8, 11] +p=0.42 +very +low +[1, 2] +low +[6, 8] +high +[3, 5] +low +[9, 11] +p=0.8 +very high +[6, 7] +p=0.2 +high +[2, 3] +Figure 8: Discrete event model learned for each class of the +Chinatown dataset. +method, we show the discretization and the discrete event +models obtained for one dataset (Chinatown dataset). It con- +sists of the pedestrian traffic along the day in a street of Mel- +bourne. The goal is to classify the days between weekend +and weekday. Figure 7 shows instances of time series from +this dataset. The best accuracy using timed automata for the +classification was obtained using the breakpoints from Per- +sist with the Wasserstein distance and an equal-width bin- +ning (accuracy results in Table 2). These breakpoints are +shown in Figure 7 and the intervals they create can be associ- +ated with a symbol (very low to very high). Figure 8 displays +the timed discrete event models obtained with the timed au- +tomata learner for each class. A circle represents a state and +the transitions from one state to another are labeled with a +symbol, an interval of accepted delay since the last event, +and a probability. One can note that the activity in the street +in generally higher during the night (until 3 or 4 a.m.) on +weekends than on weekdays. The street also shows a more +pronounced affluence during the weekend than in the week- +days afternoons. On weekdays, the end of the day is either +calm, or more animated than during weekend days (with a +lower probability, so probably one specific day of the week). +Conclusion +This work studies Persist, a state-of-the-art discretization al- +gorithm originally conceived as pre-processing step for pat- +tern mining in time series. Persist is based on the notion of +persistence which is interesting for the modeling of dynam- +ical systems in the form of discrete event models. We re- +placed the metric used to compute the score of persistence, +originally the Kullback-Leibler divergence, with the Wasser- +stein distance, to avoid an undesirable over-emphasis on the +extreme cases. We also suggested a different initialization +strategy for the algorithm. Our experiments based on a clas- +sification task have shown that the metric substitution en- +ables a better information retention in the discretized time +series, and that Persist is better suited than the state-of-the- +art symbolic data representation SAX when the purpose of +the discretization is to learn a model formalized as discrete +event systems. Classification is not the most common use +of discrete event systems and future work will thus focus +on applications for which discrete event systems are usually +used such as anomaly detection or model-checking. Find- +ing a criterion to determine automatically the best binning +for the data would also be convenient. Another perspective +could be to associate the persistence score with other quality +scores such as the reconstruction error. +References +Alur, R.; and Dill, D. L. 1994. A theory of timed automata. +Theoretical Computer Science, 126(2): 183–235. +Chen, X.; and G¨uttel, S. 2022. An Efficient Aggregation +Method for the Symbolic Representation of Temporal Data. +ACM Trans. Knowl. Discov. Data. +Cornanguer, L.; Largou¨et, C.; Roz´e, L.; and Termier, A. +2022. TAG: Learning Timed Automata from Logs. Pro- +ceedings of the AAAI Conference on Artificial Intelligence, +36(4): 3949–3958. +Kantorovich, L. V. 1939. Mathematical Methods of Orga- +nizing and Planning Production. Management Science, 6(4): +366–422. +Kullback, S.; and Leibler, R. A. 1951. On Information and +Sufficiency. The Annals of Mathematical Statistics, 22(1): +79–86. +Lin, J.; Keogh, E.; Lonardi, S.; and Chiu, B. 2003. A sym- +bolic representation of time series, with implications for +streaming algorithms. In Proceedings of the 8th SIGMOD +DMKD workshop, 2. ACM Press. +M¨orchen, F.; and Ultsch, A. 2005. Optimizing time series +discretization for knowledge discovery. In Grossman, R.; +Bayardo, R. J.; and Bennett, K. P., eds., Proceedings of the +Eleventh ACM SIGKDD Conference, USA, 660–665. ACM. +M¨orchen, F.; and Ultsch, A. 2007. Efficient mining of un- +derstandable patterns from multivariate interval time series. +Data Mining and Knowledge Discovery, 15(2): 181–215. +Verwer, S.; de Weerdt, M.; and Witteveen, C. 2010. +A +likelihood-ratio test for identifying probabilistic determin- +istic real-time automata from positive data. In International +Colloquium on Grammatical Inference, 203–216. Springer. + diff --git a/wdE4T4oBgHgl3EQfXgwy/content/tmp_files/load_file.txt b/wdE4T4oBgHgl3EQfXgwy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d7c5b9f011aa213c09cf652d6e0fdc52ea302b9 --- /dev/null +++ b/wdE4T4oBgHgl3EQfXgwy/content/tmp_files/load_file.txt @@ -0,0 +1,386 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf,len=385 +page_content='Persistence-Based Discretization for Learning Discrete Event Systems from Time Series L´ena¨ıg Cornanguer1, Christine Largou¨et2, Laurence Roz´e3, Alexandre Termier4 1Inria, Univ Rennes, CNRS, IRISA 2Institut Agro, Univ Rennes, Inria, CNRS, IRISA 3Univ Rennes, INSA Rennes, CNRS, Inria, IRISA 4Univ Rennes, Inria, CNRS, IRISA lenaig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='cornanguer@irisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='fr Abstract To get a good understanding of a dynamical system, it is con- venient to have an interpretable and versatile model of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Timed discrete event systems are a kind of model that respond to these requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However, such models can be inferred from timestamped event sequences but not directly from nu- merical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' To solve this problem, a discretization step must be done to identify events or symbols in the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Per- sist is a discretization method that intends to create persisting symbols by using a score called persistence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' This allows to mitigate the risk of undesirable symbol changes that would lead to a too complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' After the study of the persis- tence score, we point out that it tends to favor excessive cases making it miss interesting persisting symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' To correct this behavior, we replace the metric used in the persistence score, the Kullback-Leibler divergence, with the Wasserstein dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Experiments show that the improved persistence score enhances Persist’s ability to capture the information of the original time series and that it makes it better suited for dis- crete event systems learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Introduction With the ever-growing type and variety of available sen- sors, more and more systems can be monitored in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' This monitoring results in the collection of multiple time series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' sequences of numerical values captured by the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' For example, in a smart city, streets can be moni- tored through pedestrian counting as well as pluviometers, to study the effect of weather conditions and the time of the day on the number of people walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The classical use of the sensor time series is to feed them to machine learning models for tasks such as classifica- tion or anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However, one may also be in- terested to understand better the dynamical system being monitored by the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' For such goal, an interpretable model is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A good solution is to produce a model in the form of a timed Discrete Event System from the sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A first difficulty is that there are globally very few ap- proaches learning timed Discrete Event Systems from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The learning of one of these Discrete Event Systems for- malisms, Timed Automata (TA), has been well studied with algorithms like RTI+ and TAG (Verwer, de Weerdt, and Wit- teveen 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Cornanguer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However these algo- rithms take as input sequences of timestamped events, and not time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In order to use these approaches with sensor data, a solu- tion is to discretize the time series before using the automata learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' There is a large literature on time series discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However, the proposed discretization meth- ods are not designed for automata learning: they may exhibit too frequent consecutive changes of symbols, which would lead to needlessly large and complex automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' With the idea of minimizing the number of symbol changes, M¨orchen at Ultsch (M¨orchen and Ultsch 2005) proposed the Persist ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist is based on the assumption that the time se- ries reflect the dynamics of an underlying system composed of recurring persisting states and aims at recovering these states in the form of a sequence of symbols issued from the time series discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' While the idea of Persist is great, using it in practice for automata learning reveals that Persist may often “go too far” by focusing on extreme values leading to a dis- cretized sequence with hardly any symbol changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In this paper, through a thorough analysis of the decision criterion of Persist, the persistence score, we identify the source of this behavior leading to a more balanced distribution of the symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Our experiments on numerous real and synthetic datasets demonstrate that the improved persistence score al- lows to better capture the information of the original time se- ries, and can help in producing interpretable timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Motivation and State-of-the-Art We consider the problem of converting a numerical time se- ries, a sequence of values measured at regular time inter- vals (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2003), into a symbolic representation, a se- quence of symbols from a finite alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Given a time se- ries, x = {xi|xi ∈ R, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=', n}, the purpose is to pro- vide its discretized version y = {yi|yi ∈ Σ, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=', n} where each symbol yi is an element of a finite alphabet Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Symbolic representation is an efficient way to deal with the inherent dimensionality of time series so that they can be used in a low-dimensional space with data-mining and machine-learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' To address this problem, many approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The simplest discretization methods are equal-width (EW) and equal-frequency (EF) in- terval binning which subdivide continuous ranges into inter- vals through user specification of width or frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' SAX (Symbolic Aggregate approXimation) (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2003) is an- other simple and widely implemented method based on the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='05041v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='AI] 12 Jan 2023 piecewise aggregate approximation (PAA) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The time series is divided into equal-size time intervals for which only the mean value is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Given the assumption that time series follow a normal distribution, the Gaussian curve is then divided through breakpoints producing equiprobable symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' SAX requires two parameters, the number of time intervals and the number of symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' SAX suffers from limitations such as the normal distribution assumption, the lossy compression, and user parameters which impact the quality of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Several variants have been proposed to attempt to overcome these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Some other ap- proaches like ABBA and fABBA (Chen and G¨uttel 2022) have been developed to capture the shape and the trend of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' These methods are motivated by differ- ent purposes and appear to be best suited for applications dedicated to the analysis of time series such as trend pre- diction or anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Yet, no discretization method has been specifically proposed with the end goal of analyz- ing or learning behaviors of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However, M¨orchen and Ultsch (M¨orchen and Ultsch 2005) have pro- posed a discretization algorithm, Persist, based on an inter- esting property over time series, called persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist attempts to produce discretized time series with persisting symbols, which would be an advantage for the learning of a timed discrete event system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Our motivation in this paper is to improve Persist to obtain a more accurate symbolic repre- sentation suited for the description of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist Persist (M¨orchen and Ultsch 2005) is a method for unsu- pervised discretization of univariate time series proposed by M¨orchen and Ultsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' It was employed as preprocess- ing step to find patterns in time series in a language called Time Series Knowledge Representation (TSKR) (M¨orchen and Ultsch 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist is based on the assumption that the time series are the reflection of an underlying process that consists of recurring persisting states and it aims to re- store these states in the form of symbols in a discretized ver- sion of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' M¨orchen and Ultsch state that “if there is no temporal structure in the time series, the symbols [in its discretized version] can be interpreted as independent observations of a random variable according to the marginal distribution of symbols”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Thus, the idea is to look for states showing a persisting behavior by creating symbols whose probability of repetition will be much higher than their prob- ability of appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' To create these symbols, Persist produces a set of break- points creating intervals in the value space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Each interval is associated with a symbol s that will replace the numerical values falling within it in the discretized time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The breakpoints are iteratively chosen in a set of candi- date breakpoints (candidates in Algorithm 1) according to a score called persistence score (described in the next sec- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The set of candidates is initialized by an equal fre- quency binning (with a number of bins fixed to 100 by de- fault).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' At each iteration, the function best bp individu- ally tests every candidate breakpoint added to the already selected breakpoints (bps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The candidate increasing persis- tence score the most is returned with its score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist stops p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 Divergence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 Figure 1: Symmetric KL divergence between two probabil- ity distributions with two possible outcomes P = (p, 1 − p) and Q = (q, 1 − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' when no more candidate breakpoint increases the persis- tence score, finding thereby automatically an adequate num- ber of symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Algorithm 1: Persist Require: univariate time series ts Return: a set of breakpoints bps bps = ∅ candidates = equal frequency binning(ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 100) score = 0 new score = 0 (new bp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' new score) = best bp(ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' bps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' candidates) while new score < score do score = new score bps = bps ∪ new bp candidates = candidates − new bp (new bp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' new score) = best bp(ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' bps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' candidates) end while return bps Persistence score The persistence score measures the persisting behavior of the symbols created by the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' It is based on the Kullback-Leibler (KL) divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The KL divergence (Kullback and Leibler 1951) measures how a probability dis- tribution P is different from another probability distribution Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' For discrete probability distributions defined on X, the KL divergence is defined as follows: DKL(P||Q) = � x∈X P(x) log(P(x) Q(x)) This divergence is not symmetric (DKL(P||Q) ̸= DKL(Q||P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' This is why M¨orchen and Ultsch use a sym- metric version obtained as follows: SKL(P, Q) = 1 2(DKL(P||Q) + DKL(Q||P)) p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 Distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 Figure 2: Wasserstein distance between two probability dis- tributions with two possible outcomes P = (p, 1 − p) and Q = (q, 1 − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' s P(s) Pr(s) s1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='99 s2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='62 (a) Breakpoint 1 s P(s) Pr(s) s1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='92 s2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='94 (b) Breakpoint 2 Table 1: Two candidate breakpoints, each creating two sym- bols (s1 and s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The KL divergence will give a better score to breakpoint 1 while the Wasserstein distance will give a better score to breakpoint 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In Persist, the probability distributions P and Q are based on the probability of appearance of the symbols (P(s)) and their probability of repetition (Pr(s)): P = (P(s), 1−P(s)) and Q = (Pr(s), 1 − Pr(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The persistence score is com- puted as follows: Persistence(s) = sgn(Pr(s) − P(s))SKL(P, Q) The first element of the equation (sgn(Pr(s)−P(s))) allows to favor only the cases when the probability of repetition is superior to the probability of appearance, otherwise, it con- tributes negatively to the persistence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The symmetric KL divergence between two probability distributions with two possible outcomes as in our case is represented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' One of the properties of the KL divergence is that it has no upper bound, a property inher- ited by the persistence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The shape of the surface pro- duced by this divergence is also particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The symmetric KL divergence is null when the probability distributions are equal and increases non-linearly as the difference between the distribution grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' To achieve a high value of symmet- ric KL, p or q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Pr(s) or P(s)) have to be close to 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The direct consequence of these observations is that the persistence score based on the KL divergence will fo- cus on extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Table 1 and Figure 3 illustrate this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In this example, at the beginning of the al- gorithm, the first breakpoint will be selected to create two symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Two candidate breakpoints are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The first 0 20 40 60 80 100 120 0 250 500 750 1000 1250 1500 1750 Breakpoint 1 Breakpoint 2 Figure 3: Two candidate breakpoints, each creating two symbols (s1 and s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' breakpoint (Table 1a) will create a first symbol that covers almost the entire discretized time series and thus has a prob- ability of appearance and repetition close to 1, and a sec- ond symbol that almost never appears and doesn’t show a particularly recurring behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The second breakpoint (Ta- ble 1b) will create two symbols about equally probable and with very high probabilities of repetition (greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist based on the KL divergence will choose breakpoint 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The discretized version of the time series in Figure 3 will consist of the succession of about 90 “s1”, then a few “s2” and again “s1” until the end, while it would have consisted of an alternation of persistent “s1” and “s2” if breakpoint 2 had been chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Improving Persist The Wasserstein distance (Kantorovich 1939), also called Kantorovitch distance, Kantorovitch - Rubinstein distance, or earth mover’s distance, is another measure of difference between probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' It corresponds to the mini- mal cost to transform a distribution P in another distribution Q in the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The Wasserstein p-distance between two probability distributions P and Q is defined by the fol- lowing equation where Γ(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Q) are all the possible joint dis- tributions for (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Y ) with marginal probability distributions P and Q: Wp(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Q) = inf γ∈Γ(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='Q)(E(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='y)∼γd(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' y)p)1/p In the case of discrete probability distributions with only two possible outcomes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' the Wasserstein distance becomes a simple subtraction and is defined as follows: W(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Q) = |P(x1) − Q(y1)| This distance is symmetric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' easier to compute than the KL divergence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and it increases linearly as the dif- ference between the distributions grows (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' There- fore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' we use the Wasserstein distance to measure how the probability of appearance of the symbols and their probabil- ity of repetition are different in the score of persistence in place of the KL divergence: PersistenceW (s) = sgn(Pr(s) − P(s))W(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Q) In front of the choice presented in table 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' the persistence score computed with the KL divergence will be higher for TA TA TA Train Test Discretization (Persist) Discretization TA learner Accepted Not accepted time series (ℝ) discretized time series (Σ) One color per class Figure 4: Time series classification using timed automata learned after a discretization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' breakpoint 1 while the persistence score computed with the Wasserstein distance will be higher for breakpoint 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The Wasserstein distance leads here to a discretized time series with more persisting symbols, better respecting the initial intuition of the persistence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Finally, the initialization of the candidate breakpoints based on an equal frequency (EF) binning allows to have more possible breakpoints in high-density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However, some time series such as electrocardiograms have a structure that could be missed with this kind of binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In such cases, an equal-width (EW) binning can be preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' It is then im- portant to let the user choose in function of the structure of its data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We re-implemented Persist (originally coded for MAT- LAB) in Python with the possibility to choose between the KL divergence and the Wasserstein distance, and between an equal-frequency or equal-width binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' It is available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Experiments The experiment is in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' First, we want to evaluate the raw discretization quality provided by Persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Then, we look at its qualities to discretize time series for dynamical systems modeling in the form of discrete event models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Experimental setups We first want to measure the information retained in the data after the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' To allow a quantitative evaluation, we choose to evaluate the discretization through a classifica- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' If a good part of information is retained in the dis- cretized time series, the classification performance should be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We performed this evaluation on 111 datasets of the Time Series Classification Repository2 (univariate datasets 1Link to the repository of Persist re-implementation in Python: https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='fr/lcornang/persist discretization 2Anthony Bagnall, Jason Lines, William Vickers and Eamonn Keogh, The UEA & UCR Time Series Classification Repository, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='timeseriesclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='com only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' For each dataset, Persist has produced a set of break- points from the train subset, used for the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We trained a Random Forest classifier with 100 trees with the discretized train subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The classification was then per- formed on the discretized test subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We measured the clas- sification performance using the accuracy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' the rate of good classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We tested Persist using either the KL divergence or the Wasserstein distance, and either an equal frequency binning or an equal width binning for the can- didate breakpoints initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We also performed the ex- periment using SAX to have a performance reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Un- like Persist, a number of symbol must be given for SAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We used a number of symbols ranging from 2 to 10 and a time interval width of 2 and we report all these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We are interested in Persist to obtain a discrete event model of the dynamical system at the origin of the time se- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Hence, to evaluate its improvement, we need to mea- sure how good the models are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' As discrete event model, we choose timed automata (Alur and Dill 1994) which is a common and well-studied formalism for dynamical sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Timed automata (TA) are used to model systems in which time influences the behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A timed automaton de- fines states connected by transitions and the transitions from one state to another are conditioned by events and timing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The model can then be used for many purposes such as to check properties of the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' safety prop- erties), or to perform anomaly detection in new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The modeling of a system by a timed automaton can be real- ized thanks to expert knowledge or automatically from ex- ecution data of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' If the execution data takes the form of logs, a learning process can be applied to produce a timed automaton where the transitions labels correspond to the events present in the logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However, if the execu- tion takes the form of time series (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' sensor data), a pre- processing step is needed to identify events in the numerical data, the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Figure 4 illustrates the experimen- tal setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' As in the first experiment, Persist and SAX were used to obtain the discretized time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' However, instead of using the discretized data to train a classifier, it was here used to learn one discrete event model per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' For each class, the corresponding discretized train time series were given to a timed automata learner called TAG (Cornanguer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2022), which produced a timed automaton accepting all the input sequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' there exists a path in the au- tomaton for the sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Then the discretized test time se- ries were injected in the timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Each automaton re- ceived the discretized test time series of its class and as many discretized test time series of other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' An automaton should accept the data of its own class and reject the oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The accuracy corresponds to the good acceptance rate for the automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The Time Series Classification repository gathers time series of various types (motion, sensor, traffic, image, spectrographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The image type, as spectrographs, differs from the others as it consists of shapes converted into pseudo time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' As it doesn’t belong to the problem of modeling dynamical systems, these datasets were excluded from the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist KL EF Persist KL EW Persist Wasserstein EF Persist Wasserstein EW SAX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='691 Figure 5: Classification accuracy with random forest for the different discretization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The diamond indicates the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' EF: equal-frequency, EW: equal-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist KL EF Persist KL EW Persist Wasserstein EF Persist Wasserstein EW SAX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='0 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='598 Figure 6: Classification accuracy with timed automata for the different discretization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Results We first present the results for the classification task using Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Figure 5 displays the accuracy achieved ac- cording to the discretization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The classification per- formance is increased by the replacement of the Kullback- Leibler divergence by the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The initial- ization of the candidate breakpoints by an equal frequency binning leads generally to a better performance, however, it depends on the dataset which confirms our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Thanks to the Wasserstein distance, using Persist globally leads to better results than using SAX in this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' This in- dicates that Persist using the Wasserstein distance allows a good information retention in the discretized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We now move on to the results of the second experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Using automata to perform a classification task is unusual and not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Indeed, each automaton is meant to rep- 0 5 10 15 20 Hour 0 250 500 750 1000 1250 1500 Pedestrian count Breakpoint (a) Weekdays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 0 5 10 15 20 Hour 0 250 500 750 1000 1250 1500 Pedestrian count Breakpoint (b) Weekends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Figure 7: Instances of time series from the Chinatown dataset and breakpoints selected by Persist (on the whole train set) with the Wasserstein distance and an equal-width binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Discretization method Accuracy Persist (KL, EF) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='686 Persist (Wasserstein, EF) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='687 Persist (KL, EW) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='780 Persist (Wasserstein, EW) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='819 SAX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='652 Table 2: Classification accuracy with timed automata for the Chinatown dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' resent a normal global behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' There is no emphasis for the modeling on what makes the data of the different classes singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' For this reason, we cannot expect as good perfor- mances as while using a real classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Nevertheless, it is in- teresting to compare the classification performance accord- ing to the discretization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' If the discretization method is pertinent for discrete event modeling, a good part of the information contained in the time series would be retained in the models and therefore leading to good classification per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Figure 6 displays the classification accuracy using timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' When using SAX, the classification perfor- mance suffers the most from the use of timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Per- sist, in particular while using the Wasserstein distance and an equal-width binning, preserves a better classification ac- curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' This confirms the interest in using an improved ver- sion of Persist to preprocess time series in order to obtain a discrete event model of a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' To provide an insight into what can be obtained with this Weekend low [0, 0] Weekday low [0, 0] p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='4 very low [0, 0] p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='6 very low [3, 4] low [4, 6] high [1, 9] very high [1, 7] p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='58 low [8, 11] p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='42 very low [1, 2] low [6, 8] high [3, 5] low [9, 11] p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='8 very high [6, 7] p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='2 high [2, 3] Figure 8: Discrete event model learned for each class of the Chinatown dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' method, we show the discretization and the discrete event models obtained for one dataset (Chinatown dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' It con- sists of the pedestrian traffic along the day in a street of Mel- bourne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The goal is to classify the days between weekend and weekday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Figure 7 shows instances of time series from this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The best accuracy using timed automata for the classification was obtained using the breakpoints from Per- sist with the Wasserstein distance and an equal-width bin- ning (accuracy results in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' These breakpoints are shown in Figure 7 and the intervals they create can be associ- ated with a symbol (very low to very high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Figure 8 displays the timed discrete event models obtained with the timed au- tomata learner for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A circle represents a state and the transitions from one state to another are labeled with a symbol, an interval of accepted delay since the last event, and a probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' One can note that the activity in the street in generally higher during the night (until 3 or 4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=') on weekends than on weekdays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The street also shows a more pronounced affluence during the weekend than in the week- days afternoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' On weekdays, the end of the day is either calm, or more animated than during weekend days (with a lower probability, so probably one specific day of the week).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Conclusion This work studies Persist, a state-of-the-art discretization al- gorithm originally conceived as pre-processing step for pat- tern mining in time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Persist is based on the notion of persistence which is interesting for the modeling of dynam- ical systems in the form of discrete event models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We re- placed the metric used to compute the score of persistence, originally the Kullback-Leibler divergence, with the Wasser- stein distance, to avoid an undesirable over-emphasis on the extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' We also suggested a different initialization strategy for the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Our experiments based on a clas- sification task have shown that the metric substitution en- ables a better information retention in the discretized time series, and that Persist is better suited than the state-of-the- art symbolic data representation SAX when the purpose of the discretization is to learn a model formalized as discrete event systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Classification is not the most common use of discrete event systems and future work will thus focus on applications for which discrete event systems are usually used such as anomaly detection or model-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Find- ing a criterion to determine automatically the best binning for the data would also be convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Another perspective could be to associate the persistence score with other quality scores such as the reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' References Alur, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Dill, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A theory of timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Theoretical Computer Science, 126(2): 183–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and G¨uttel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' An Efficient Aggregation Method for the Symbolic Representation of Temporal Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Knowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Discov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Cornanguer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Largou¨et, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Roz´e, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Termier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' TAG: Learning Timed Automata from Logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Pro- ceedings of the AAAI Conference on Artificial Intelligence, 36(4): 3949–3958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Kantorovich, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 1939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Mathematical Methods of Orga- nizing and Planning Production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Management Science, 6(4): 366–422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Kullback, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Leibler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' On Information and Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' The Annals of Mathematical Statistics, 22(1): 79–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Lin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Keogh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Lonardi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Chiu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A sym- bolic representation of time series, with implications for streaming algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In Proceedings of the 8th SIGMOD DMKD workshop, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' ACM Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' M¨orchen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Ultsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Optimizing time series discretization for knowledge discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In Grossman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Bayardo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Bennett, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=', Proceedings of the Eleventh ACM SIGKDD Conference, USA, 660–665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' M¨orchen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Ultsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Efficient mining of un- derstandable patterns from multivariate interval time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Data Mining and Knowledge Discovery, 15(2): 181–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Verwer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' de Weerdt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' and Witteveen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' A likelihood-ratio test for identifying probabilistic determin- istic real-time automata from positive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' In International Colloquium on Grammatical Inference, 203–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE4T4oBgHgl3EQfXgwy/content/2301.05041v1.pdf'} diff --git a/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf b/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d3ff563310da982e1731d17daa03b7f307328066 --- /dev/null +++ b/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7d6adf87663d89ab643f25f1929e2979e2e5d645436fb98529152d2c7e2baa30 +size 1201239 diff --git a/xtE2T4oBgHgl3EQfhQc5/vector_store/index.pkl b/xtE2T4oBgHgl3EQfhQc5/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3cb205156191748aa159251a2a7ad75afa363222 --- /dev/null +++ b/xtE2T4oBgHgl3EQfhQc5/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1548ed7ea3238985c577fe153f681ddffd2ad632f51d2234d64d79e5d9768e97 +size 132012 diff --git a/zNAzT4oBgHgl3EQfePy_/content/tmp_files/2301.01434v1.pdf.txt b/zNAzT4oBgHgl3EQfePy_/content/tmp_files/2301.01434v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1c9f2d23217eb4bf22d694388dd34805de2949e --- /dev/null +++ b/zNAzT4oBgHgl3EQfePy_/content/tmp_files/2301.01434v1.pdf.txt @@ -0,0 +1,2180 @@ +arXiv:2301.01434v1 [cs.LG] 4 Jan 2023 +Online Learning of Smooth Functions +Jesse Geneson and Ethan Zhou +January 5, 2023 +Abstract +In this paper, we study the online learning of real-valued functions where the hidden function +is known to have certain smoothness properties. Specifically, for q ≥ 1, let Fq be the class of +absolutely continuous functions f : [0, 1] → R such that ∥f ′∥q ≤ 1. For q ≥ 1 and d ∈ Z+, +let Fq,d be the class of functions f : [0, 1]d → R such that any function g : [0, 1] → R formed +by fixing all but one parameter of f is in Fq. For any class of real-valued functions F and +p > 0, let optp(F) be the best upper bound on the sum of pth powers of absolute prediction +errors that a learner can guarantee in the worst case. In the single-variable setup, we find new +bounds for optp(Fq) that are sharp up to a constant factor. We show for all ε ∈ (0, 1) that +opt1+ε(F∞) = Θ(ε− 1 +2 ) and opt1+ε(Fq) = Θ(ε− 1 +2 ) for all q ≥ 2. We also show for ε ∈ (0, 1) that +opt2(F1+ε) = Θ(ε−1). In addition, we obtain new exact results by proving that optp(Fq) = 1 +for q ∈ (1, 2) and p ≥ 2 + +1 +q−1. In the multi-variable setup, we establish inequalities relating +optp(Fq,d) to optp(Fq) and show that optp(F∞,d) is infinite when p < d and finite when p > d. +We also obtain sharp bounds on learning F∞,d for p < d when the number of trials is bounded. +1 +Introduction +Consider a learner that wants to predict the next day’s temperature range at a given location +based on inputs such as the current day’s temperature range, humidity, atmospheric pressure, +precipitation, wind speed, solar radiation, location, and time of year. In our model, this learner +is tested daily. On a given day, the learner gets inputs for that day, which it uses to output a +prediction for the next day’s temperature range; when the next day arrives, it sees the correct +temperature range, then uses this feedback to update future predictions. As this is repeated, the +learner accumulates information to help it make better predictions. A natural question arises: can +the learner guarantee that its predictions become better over time, and if so, how quickly? +We investigate a model of online learning of real-valued functions previously studied in [9, +12, 13, 1, 10, 11] where an algorithm A learns a real-valued function f from some class F in +trials. +Past research on this model focused on functions of one input, for example, predicting +the temperature range solely based on the time of year. The research showed that, as long as the +function is sufficiently smooth, the learner can become a good predictor fairly rapidly. Suppose that +F consists of functions f : S → R for some set S, and fix some f ∈ F. In each trial t = 0, . . . , m, +A receives an input st ∈ S, guesses ˆyt for the value of f(st), and receives the actual value of f(st). +Following [9], we focus on an error function which measures how difficult it is for a learner to +predict functions accurately in the worst case. The error function depends on two parameters, p +and q, which determine how harshly the learner is punished for errors and the types of functions +that the learner might encounter, respectively. Small values of p and q are more difficult for the +1 + +1 +2 +1 +2 +1 +O +� +1 +p−1 +� +p +q +Figure 1: Exact values and bounds on optp(Fq) for p, q > 1 prior to the results in this paper +learner, leading to higher values of the error function. For each algorithm A, p > 0, f ∈ F, and +σ = (s0, . . . , sm) ∈ Sm+1, define +Lp(A, f, σ) = +m +� +t=1 +|ˆyt − f(st)|p. +When f and σ are clear from the context, we refer to Lp(A, f, σ) as the total p-error of A. Define +Lp(A, F) = +sup +f∈F,σ∈∪m∈Z+Sm Lp(A, f, σ) +and optp(F) = inf +A Lp(A, F). Note that unlike the definition of opt presented in [9, 12, 5], F may +consist of real-valued functions on any domain, not just functions from [0, 1] to R. +The case where F contains functions f : [0, 1] → R whose derivatives have various bounded +norms was studied in [9, 12, 5]. For q ≥ 1, let Fq be the class of absolutely continuous functions +f : [0, 1] → R such that +� 1 +0 |f ′(x)|qdx ≤ 1, and let F∞ be the class of absolutely continuous functions +f : [0, 1] → R such that +sup +x∈(0,1) +|f ′(x)| ≤ 1. As noted in [12], F∞ contains exactly those f : [0, 1] → R +such that |f(x) − f(y)| ≤ |x − y| for all x, y ∈ [0, 1]. Also, by Jensen’s inequality, F∞ ⊆ Fq ⊆ Fr +for all q ≥ r ≥ 1. Hence optp(F∞) ≤ optp(Fq) ≤ optp(Fr) for all p ≥ 1 and q ≥ r ≥ 1. Previous +papers determined the exact values of optp(Fq) for p = 1, q = 1, and p, q ≥ 2, as well as bounds on +optp(Fq) for p ∈ (1, 2) and q ≥ 2. +The paper [9] proved that optp(F1) = ∞ for all p ≥ 1. They also showed that opt1(Fq) = +opt1(F∞) = ∞ for all q ≥ 1. In contrast, they found that optp(Fq) = optp(F∞) = 1 for all p ≥ 2 +and q ≥ 2. This was also proved in [4] using a different algorithm based on a generalization of +the Widrow-Hoff algorithm [8, 14], and a noisy version of this problem was studied in [3]. In this +paper, we extend the region of values of p, q for which it is known that optp(Fq) = 1. +Theorem 1.1. For any reals q > 1 and p ≥ 2 + +1 +q−1, we have optp(Fq) = 1. +For p = 1 + ε with ε ∈ (0, 1), the paper [9] proved that optp(Fq) = O(ε−1) for all q ≥ 2, +which implies that optp(F∞) = O(ε−1). However, these bounds are not sharp. In this paper, we +determine opt1+ε(Fq) up to a constant factor for all ε ∈ (0, 1) and q ≥ 2. +2 + +1 +2 +1 +2 +1 +Θ +� +1 +√p−1 +� +1 +O +� +1 +q−1 +� +p +q +Figure 2: Exact values and bounds on optp(Fq) for p, q > 1 including the results in this paper +Theorem 1.2. For all ε ∈ (0, 1), we have opt1+ε(F∞) = Θ(ε− 1 +2 ) and opt1+ε(Fq) = Θ(ε− 1 +2 ) for all +q ≥ 2, where the constants in the bound do not depend on q. +The proof of Theorem 1.2 splits into an upper bound and a lower bound. For the upper bound, +we use H¨older’s inequality combined with results from [9]. +For the lower bound, we modify a +construction used in [12], which obtained bounds on a finite variant of opt1(Fq) that depends on +the number of trials m. +The results of [9] and [12] left open the problem of determining optp(Fq) for q ∈ (1, 2). +It +was not even known up to a constant factor. We make progress on this problem by determining +opt2(F1+ε) up to a constant factor for ε ∈ (0, 1). Figure 1 shows the bounds and exact values +known for p, q > 1 prior to the results in our paper, while Figure 2 shows the bounds and exact +values known for p, q > 1 including the results in our paper. +Theorem 1.3. For ε ∈ (0, 1), we have opt2(F1+ε) = Θ(ε−1). +The paper [9] also discussed the problem of online learning for smooth functions of multiple +variables. Previous research on learning multi-variable functions [2, 6, 7] has focused on expected +loss rather than worst-case loss, using models where the inputs xi are determined by a probability +distribution. +We introduce a natural extension of the single-variable setup from [9] to multi-variable functions. +Specifically, for q ≥ 1 and d ∈ Z+, let Fq,d be the class of functions f : [0, 1]d → R such that for +any (d − 1)-tuple (x1, . . . , xd−1) ∈ [0, 1]d−1 and integer i with 1 ≤ i ≤ d, the function g : [0, 1] → R +given by g(x) = f(vi,x) is in Fq, where vi,x ∈ [0, 1]d is the vector formed when x is inserted at the +ith position of (x1, . . . , xd−1). +One of the most fundamental questions about optp(Fq,d) is to determine when it is finite and +when it is infinite. We answer this question almost completely when q = ∞. +Theorem 1.4. For any positive integer d, optp(F∞,d) is finite when p > d and infinite when +0 < p < d. +As a corollary, it immediately follows for 0 < p < d that optp(Fq,d) = ∞ for all q ≥ 1. Moreover, +it is easy to see that optp(F1,d) = ∞ for all positive integers d and p. +3 + +The papers [9] and [12] also investigated worst-case mistake bounds for online learning of smooth +functions when the number of trials is bounded. In particular, using the same notation as in the +first paragraph of this section, define +Lp(A, f, σ, m) = +m +� +t=1 +|ˆyt − f(st)|p. +Moreover, define +Lp(A, F, m) = +sup +f∈F,σ∈Sm+1 Lp(A, f, σ, m) +and optp(F, m) = inf +A Lp(A, F, m). +The paper [9] proved that opt1(Fq, m) = O(log(m)) for all q ≥ 2 and opt1(F2, m) = Ω( +� +log(m)). +The paper [12] sharpened these bounds by proving that opt1(Fq, m) = Θ( +� +log(m)) for all q ≥ 2 +and opt1(F2, m) = +√ +log2(m) +2 +± O(1). We obtain sharp bounds for online learning of smooth func- +tions with a bounded number of trials when 0 < p < d. In particular, these sharp bounds are also +new in the single-variable case. +Theorem 1.5. For any positive integer d and real number p with 0 < p < d, we have optp(F∞,d, m) = +Θ(m1− p +d ), where the constants in the bounds depend on p and d. +In Section 2, we focus on the single-variable setup. We prove Theorem 1.3 in Subsections 2.1 and +2.2. Subsection 2.1 establishes the lower bound, while Subsection 2.2 establishes the upper bound +along with several useful lemmas. Subsection 2.3 focuses on proving Theorem 1.1. In Subsection +2.4, we prove Theorem 1.2. In Section 3, we focus on the multi-variable setup, establishing various +bounds on optp(Fq,d). Finally, in Section 4, we discuss open problems. +2 +Results in the single-variable setup for q ∈ (1, 2) +First, we adopt some notation from [9]. For f : [0, 1] → R, define the q-action of f, denoted by +Jq[f], as +Jq[f] = +� 1 +0 +|f ′(x)|qdx, +so that Fq is exactly the set of absolutely continuous f : [0, 1] → R such that Jq[f] ≤ 1. +Also, for a nonempty set S = {(ui, vi) : 1 ≤ i ≤ m} of points in [0, 1]×R such that u1 < . . . < um, +define +fS(x) = + + + + + +v1 +x ≤ u1 +vi + (x−ui)(vi+1−vi) +ui+1−ui +x ∈ (ui, ui+1] +vm +x > um +and set f∅(x) ≡ 0. +Finally, define the learning algorithm LININT as follows: on trial 0, LININT guesses ˆy0 = 0, +and on trial i > 0, with the points in S = {(x0, f(x0)), . . . , (xi−1, f(xi−1))} having been revealed +and given xi, LININT guesses ˆyi = fS(xi). +4 + +2.1 +Lower bounds for optp(Fq) +First, for all p, q > 1 we have an obvious lower bound for optp(Fq). +Proposition 2.1. For p, q > 1, we have optp(Fq) ≥ 1. +The paper [9] proved that equality holds when p, q ≥ 2. As we will see, equality also holds when +q ∈ (1, 2) for sufficiently large values of p. +For q ∈ (1, 2) and p > 1, we also prove a lower bound for opt2(Fq), using an adversary strategy +similar to that in Theorem 8 of [9]. +Theorem 2.2. For q ∈ (1, 2), we have optp(Fq) ≥ +q +(p2pe ln 2)(q−1). +Proof. Fix q ∈ (1, 2) and an algorithm A for learning Fq. Consider the following family of adversary +strategies, depending on a parameter b ∈ (0, 1). The adversary picks x0 = 0 and reveals f(x0) = 0, +then picks x1 = 1 and reveals f(x1) = ±b such that |ˆy1 − f(x1)| ≥ b; without loss of generality, +suppose f(x1) = b. Then for the next k = +� +− q log2 b +q−1 +� +trials, the adversary recursively picks xi and +f(xi) as follows. On trial 2 ≤ i < k + 2, the adversary sets li to be the greatest real x ∈ [0, 1] such +that f(x) = 0 has been previously revealed, and similarly sets ri to be the least real x ∈ [0, 1] such +that f(x) = b has been previously revealed, then sets xi = li+ri +2 +. Upon receiving A’s guess ˆyi, the +adversary reveals f(xi) = 0 or f(xi) = b such that |f(xi) − ˆyi| ≥ b +2. +To see that this strategy is well-defined, note that all the xi are distinct, so it suffices to +show that there exists a function f ∈ Fq which is consistent with all (xi, f(xi)). +Indeed, take +f = f{(x0,f(x0)),...,(xk+1,f(xk+1))} which linearly interpolates between all points (xi, f(xi)); then f has +only one segment of nonzero slope, with +Jq[f] = 2−k +� b +2−k +�q += 2k(q−1)bq ≤ 2− q log2 b +q−1 ·(q−1)bq = 1. +Thus the adversary guarantees an error of at least +optp(Fq) ≥ +k+1 +� +i=1 +|f(xi) − ˆyi|p ≥ bp + k +�b +2 +�p +≥ − bpq log2 b +2p(q − 1). +Picking b = e− 1 +p yields optp(Fq) ≥ +q +(p2pe ln 2)(q−1). +■ +In particular, when p = 2 we get the following: +Corollary 2.3. For q ∈ (1, 2), we have +opt2(Fq) ≥ +q +(8e ln 2)(q − 1) > +1 +(8e ln 2)(q − 1). +2.2 +Bounds for opt2(Fq) +The main result of this section is that for ε ∈ (0, 1), opt2(F1+ε) = Θ(ε−1). For q ∈ (1, 2), Corollary +2.3 gives a lower bound for opt2(Fq); we now prove an upper bound for opt2(Fq) and use this to +derive the desired result. First, we show that a similar fact to Lemma 9 in [9] holds. +Lemma 2.4. Let u1 < . . . < um be reals in [0, 1] and v1, . . . , vm be reals, and define S = +{(u1, v1), . . . , (um, vm)}. Then for any q ∈ (1, 2) and absolutely continuous f : [0, 1] → R such +that f(ui) = vi for 1 ≤ i ≤ m, we have Jq[f] ≥ Jq[fS]. +5 + +Proof. If m = 1, then Jq[fS] = 0 and the result is clear. Otherwise, fix some absolutely continuous +f : [0, 1] → R which is consistent with the (ui, vi), and fix 1 ≤ i < m. Then by Jensen’s inequality, +� ui+1 +ui +|f ′(x)|qdx +ui+1 − ui +≥ +�� ui+1 +ui +|f ′(x)|dx +ui+1 − ui +�q +≥ + + +��� +� ui+1 +ui +f ′(x)dx +��� +ui+1 − ui + + +q += +���� +vi+1 − vi +ui+1 − ui +���� +q +. +Thus we obtain +Jq[f] ≥ +� um +u1 +|f ′(x)|qdx ≥ +m−1 +� +i=1 +(ui+1 − ui) +���� +vi+1 − vi +ui+1 − ui +���� +q += Jq[fS] +by summing over all 1 ≤ i < m. +■ +This leads to the following useful fact. +Lemma 2.5. For any q > 1, target function f ∈ Fq, integer m ≥ 1, and sequence of inputs +x0, . . . , xm ∈ [0, 1], LININT never produces an error |ˆyi − f(xi)| > 1 on any trial i ≥ 1. +Proof. Suppose otherwise, so that LININT produces an error |ˆyi − f(xi)| > 1 for i ≥ 1; then there +exists 0 ≤ j < i such that |f(xi)−f(xj)| > 1. Letting S = {(x0, f(x0)), . . . , (xi, f(xi))}, by Lemma +2.4 +Jq[f] ≥ Jq[fS] ≥ |xi − xj| +���� +f(xi) − f(xj) +xi − xj +���� +q +≥ |f(xi) − f(xj)| > 1, +upon which f ̸∈ Fq, contradiction. +■ +Corollary 2.6. For any q > 1 and p′ > p > 1, we have Lp′(LININT, Fq) ≤ Lp(LININT, Fq). +Proof. On every trial i ≥ 1, LININT produces an error |ˆyi−f(xi)| ≤ 1, so |ˆyi−f(xi)|p′ ≤ |ˆyi−f(xi)|p +for all i ≥ 1. +■ +With this, the proof proceeds similarly to the proof of Theorem 11 in [9]. Specifically, we will +compare changes in Jq[fS] as new points are added to S to the squared errors (ˆy−fS(x))2 produced +by LININT to bound L2(LININT, Fq). This requires the following inequalities. +Lemma 2.7. For reals a > 0, b > 0, q ∈ (1, 2), and x ∈ (−a, b), we have +a +� +1 + x +a +�q ++ b +� +1 − x +b +�q +− (a + b) ≥ 2q(q − 1) +a + b +· x2. +Proof. Fix a, b, q, and define the function +f(x) = a +� +1 + x +a +�q ++ b +� +1 − x +b +�q +− (a + b) − 2q(q − 1) +a + b +· x2 +for x ∈ (−a, b), so that we wish to show f(x) ≥ 0 for all x ∈ (−a, b). Compute +f ′(x) = q +�� +1 + x +a +�q−1 +− +� +1 − x +b +�q−1 +− 4(q − 1) +a + b x +� +f ′′(x) = q(q − 1) +�1 +a +� +1 + x +a +�q−2 ++ 1 +b +� +1 − x +b +�q−2 +− +4 +a + b +� +f (3)(x) = q(q − 1)(q − 2) +� 1 +a2 +� +1 + x +a +�q−3 +− 1 +b2 +� +1 − x +b +�q−3� +f (4)(x) = q(q − 1)(q − 2)(q − 3) +� 1 +a3 +� +1 + x +a +�q−4 ++ 1 +b3 +� +1 − x +b +�q−4� +. +6 + +First, we show f ′′(x) ≥ 0 for all x ∈ (−a, b). Note that f (4)(x) > 0 for all x and +lim +x→−a+ f ′′(x) = lim +x→b− f ′′(x) = ∞, +so f (3)(x) is increasing on (−a, b) and it suffices to check that f ′′(x) ≥ 0 at the point where +f (3)(x) = 0. Solving for this x yields +f (3)(x) = 0 ⇐⇒ +1 +a2 +� +1 + x +a +�q−3 += 1 +b2 +� +1 − x +b +�q−3 +⇐⇒ +�1 + x +a +1 − x +b +�3−q += b2 +a2 ⇐⇒ 1 + x +a +1 − x +b += b +2 +3−q +a +2 +3−q +⇐⇒ x = −a +2 +3−q + b +2 +3−q +a +q−1 +3−q + b +q−1 +3−q +. +At this x, +f ′′(x) = q(q − 1) + +1 +a +� +1 + −a +2 +3−q + b +2 +3−q +a +2 +3−q + ab +q−1 +3−q +�q−2 ++ 1 +b +� +1 − −a +2 +3−q + b +2 +3−q +a +q−1 +3−q b + b +2 +3−q +�q−2 +− +4 +a + b + + += q(q − 1) + +1 +a + + +(a + b)b +q−1 +3−q +a +� +a +q−1 +3−q + b +q−1 +3−q +� + + +q−2 ++ 1 +b + + +(a + b)a +q−1 +3−q +b +� +a +q−1 +3−q + b +q−1 +3−q +� + + +q−2 +− +4 +a + b + + += q(q − 1) + + +� +a +q−1 +3−q + b +q−1 +3−q +�2−q � +a1−qb +(q−1)(q−2) +3−q ++ a +(q−1)(q−2) +3−q +b1−q +� +(a + b)2−q +− +4 +a + b + + +≥ q(q − 1) + +22−q(ab) +(q−1)(2−q) +2(3−q) +· 2(ab) +(q−1)(2q−5) +2(3−q) +(a + b)2−q +− +4 +a + b + + += 4q(q − 1) +� +1 +(a + b)2−q(4ab) +q−1 +2 +− +1 +a + b +� +≥ 4q(q − 1) +� +1 +(a + b)2−q(a + b)q−1 − +1 +a + b +� += 0, +where all inequalities follow from the inequality (u + v)2 ≥ 4uv ⇐⇒ (u − v)2 ≥ 0 for all reals u, v. +Since f ′′(x) is minimized here, it follows that f ′′(x) ≥ 0 for all x ∈ (−a, b). +Since f ′(0) = 0 and f ′′(x) ≥ 0 for all x ∈ (−a, b), it follows that f ′(x) ≤ 0 for x < 0 and +f ′(x) ≥ 0 for x > 0, so f(x) ≥ f(0) = 0 for all x ∈ (−a, b). +■ +Lemma 2.8. For reals a, b ∈ (0, 1), q ∈ (1, 2), and x ̸∈ (−a, b), we have +a +���x +a + 1 +��� +q ++ b +���x +b − 1 +��� +q +− (a + b) ≥ (q − 1)|x|q +(a + b)q−1 . +Proof. Fix a, b, q; by symmetry, it suffices to consider x ≥ b. Define the function +f(x) = a +�x +a + 1 +�q ++ b +�x +b − 1 +�q +− (a + b) − (q − 1)xq +(a + b)q−1 +7 + +for x ≥ b, so that we wish to show f(x) ≥ 0 for all x ≥ b. Since +f ′(x) = q +�x +a + 1 +�q−1 ++ q +�x +b − 1 +�q−1 +− q(q − 1)xq−1 +(a + b)q−1 +≥ q +� +x +a + b +�q−1 ++ q +�x +b − 1 +�q−1 +− +qxq−1 +(a + b)q−1 > 0 +for all x > b, f is increasing, so it suffices to show +f(b) = a +�a + b +a +�q +− (a + b) − (q − 1)bq +(a + b)q−1 ≥ 0. +Dividing by a + b and substituting r = +a +a+b, we see that this is equivalent to +g(r) = +1 +rq−1 − 1 − (q − 1)(1 − r)q ≥ 0 +for r ∈ (0, 1). As g(1) > 0, it suffices for +g′(r) = r−q(q − 1) +� +qrq(1 − r)q−1 − 1 +� +< 0 +⇐⇒ qrq−1(1 − r)q−1 < 1 +r. +For r ∈ (0, 1), the quantity rq−1(1 − r)q−1 is maximized when r = 1 +2, so it suffices to prove that +q +� 1 +2 +�2q−2 < 1 for q ∈ (1, 2). Define h(q) = q +� 1 +2 +�2q−2, so h′(q) = 41−q(1 − q ln 4) < 0 for q ∈ (1, 2). +Since h(1) = 1, we have h(q) < 1 for q ∈ (1, 2). +■ +Corollary 2.9. For reals a, b ∈ (0, 1) such that a + b ≤ 1, q ∈ (1, 2), and x ̸∈ (−a, b), we have +a +���x +a + 1 +��� +q ++ b +���x +b − 1 +��� +q +− (a + b) ≥ (q − 1)|x|q. +Combining the above yields the following key result. +Lemma 2.10. Fix q ∈ (1, 2), a nonempty set S = {(u1, v1), . . . , (uk, vk)} of points in [0, 1] × R, +and (x, y) ∈ [0, 1] × R such that u1 < . . . < uk, x ̸= ui for any 1 ≤ i ≤ k, and Jq +� +fS∪{(x,y)} +� +≤ 1. +Then +Jq +� +fS∪{(x,y)} +� +− Jq[fS] ≥ (q − 1)(y − fS(x))2. +Proof. First, suppose x < u1. Then compared to fS, the function fS∪{(x,y)} contains a new line +segment of (possibly) nonzero slope between (x, y) and (u1, v1), so as |y − fS(x)| = |v1 − y| ≤ 1 (by +Lemma 2.5) and |u1 − x| ≤ 1, +Jq +� +fS∪{(x,y)} +� +− Jq[fS] = (u1 − x) +���� +v1 − y +u1 − x +���� +q +≥ (v1 − y)2 = (y − fS(x))2 ≥ (q − 1)(y − fS(x))2. +The case x > uk is similar. +Now suppose there exists an integer 1 ≤ i < k such that ui < x < ui+1. In this case, +Jq +� +fS∪{(x,y)} +� +− Jq[fS] = (x − ui) +���� +y − vi +x − ui +���� +q ++ (ui+1 − x) +���� +vi+1 − y +ui+1 − x +���� +q +− (ui+1 − ui) +���� +vi+1 − vi +ui+1 − ui +���� +q +. +8 + +Substituting a = x − ui, b = ui+1 − x, d = y − fS(x), and m = vi+1−vi +ui+1−ui = fS(x)−vi +a += vi+1−fS(x) +b +, we +can rewrite the above as +Jq +� +fS∪{(x,y)} +� +− Jq[fS] = a +����m + d +a +���� +q ++ b +����m − d +b +���� +q +− (a + b)|m|q += |m|q +� +a +����1 + d +ma +���� +q ++ b +����1 − d +mb +���� +q +− (a + b) +� +. +Then applying either Lemma 2.7 or Corollary 2.9 (depending on whether +d +m ∈ (−a, b)) yields +Jq +� +fS∪{(x,y)} +� +− Jq[fS] ≥ |m|q min +� +2q(q − 1) +a + b +· +� d +m +�2 +, (q − 1) +���� +d +m +���� +q� += min +� +2q(q − 1) +|m|2−q(a + b) · d2, (q − 1)|d|q +� +. +If 0 < |m| ≤ 1, then a + b = ui+1 − ui ≤ 1 =⇒ |m|2−q(a + b) ≤ 1, while if |m| ≥ 1, then +|m|2−q(a + b) ≤ |m|q(a + b) ≤ Jq [fS] ≤ Jq[fS∪{(x,y)}] ≤ 1 +by Lemma 2.4, so in either case +2q(q − 1) +|m|2−q(a + b) · d2 ≥ 2q(q − 1)d2 ≥ (q − 1)d2. +Moreover, since Jq +� +fS∪{(x,y)} +� +≤ 1, by Lemma 2.5 |d| ≤ 1. Hence (q −1)|d|q ≥ (q −1)d2 as well. +■ +This directly yields the desired upper bound. +Theorem 2.11. For q ∈ (1, 2), we have L2(LININT, Fq) ≤ +1 +q−1. +Proof. Fix a target function f ∈ Fq, an integer m ≥ 1, and a sequence of inputs σ = (x0, . . . , xm) ∈ +[0, 1]m+1. +Assume without loss of generality that all xi are distinct. +For 0 ≤ i ≤ m, define +Si = {(x0, f(x0)), . . . , (xi, f(xi))}, and suppose LININT produces guesses ˆy0, . . . , ˆym ∈ R. +By +Lemma 2.4 and Lemma 2.10, +1 ≥ Jq[f] ≥ Jq [fSm] = +m +� +i=1 +� +Jq [fSi] − Jq +� +fSi−1 +�� +≥ (q − 1) +m +� +i=1 +(ˆyi − f(xi))2, +so +L2(LININT, f, σ) = +m +� +i=1 +(ˆyi − f(xi))2 ≤ +1 +q − 1 +for any f ∈ Fq, integer m ≥ 1, and σ ∈ [0, 1]m+1. Thus L2(LININT, Fq) ≤ +1 +q−1. +■ +Finally, combining the above with the lower bound in Corollary 2.3, we get the following result. +Theorem 2.12. For ε ∈ (0, 1), we have opt2(F1+ε) = Θ(ε−1). +Proof. Combining Corollary 2.3 and Theorem 2.11, +ε−1 +8e ln 2 < +1 + ε +(8e ln 2)ε ≤ opt2(F1+ε) ≤ ε−1. +Hence, opt2(F1+ε) = Θ(ε−1). +■ +9 + +It is simple to generalize the upper bound in Theorem 2.11 to all p ≥ 2. +Corollary 2.13. For ε ∈ (0, 1) and p ≥ 2, we have optp(F1+ε) = O(ε−1). +Proof. By Lemma 2.6, optp(F1+ε) ≤ Lp(LININT, F1+ε) ≤ L2(LININT, F1+ε) = O(ε−1). +■ +2.3 +An exact result for large p +In this section, we prove that for q ∈ (1, 2) and p ≥ 2 + +1 +q−1, optp(Fq) = 1. This first requires the +following lemma. +Lemma 2.14. For reals q ∈ (1, 2), a ∈ (0, 1), and u, v satisfying |u − v| ≥ (q−1)q−1 +a(1−a) , we have +a|u|q + (1 − a)|v|q > 1. +Proof. Without loss of generality, suppose u > v, so that u ≥ v + (q−1)q−1 +a(1−a) . +First, suppose v < 0 < u; then |u| + |v| ≥ (q−1)q−1 +a(1−a) . By the weighted power mean inequality, +|u|(a|u|q−1) + |v|((1 − a)|v|q−1) +|u| + |v| +≥ +� +a− +1 +q−1 + (1 − a)− +1 +q−1 +|u| + |v| +�−(q−1) +=⇒ a|u|q + (1 − a)|v|q ≥ +(|u| + |v|)q +� +a− +1 +q−1 + (1 − a)− +1 +q−1 +�q−1 +≥ +(q − 1)q(q−1) +aq(1 − a)q +� +a− +1 +q−1 + (1 − a)− +1 +q−1 +�q−1 +≥ +(q − 1)q(q−1) +2q−1aq(1 − a)q max {a−1, (1 − a)−1} += +(q − 1)q(q−1) +2q−1 max {aq−1(1 − a)q, aq(1 − a)q−1}. +By the weighted arithmetic mean - geometric mean inequality, for r ∈ (0, 1) we have +rq(1 − r)q−1 = +qq +(q − 1)q +�(q − 1)r +q +�q +(1 − r)q−1 +≤ +qq +(q − 1)q + +q · (q−1)r +q ++ (q − 1)(1 − r) +(q − 1) + q + + +(q−1)+q += +qq +(q − 1)q +� q − 1 +2q − 1 +�2q−1 +. +Thus +max +� +aq−1(1 − a)q, aq(1 − a)q−1� +≤ +qq +(q − 1)q +� q − 1 +2q − 1 +�2q−1 += qq(q − 1)q−1 +(2q − 1)2q−1 , +so +a|u|q + (1 − a)|v|q ≥ (q − 1)(q−1)2(2q − 1)2q−1 +2q−1qq +. +10 + +Consider +f(q) = (q − 1)2 ln(q − 1) + (2q − 1) ln(2q − 1) − (q − 1) ln 2 − q ln q +over q ∈ (1, 2). Note that +f ′(q) = (2(q − 1) ln(q − 1) + (q − 1)) + (2 ln(2q − 1) + 2) − ln 2 − (ln q + 1) += 1 − ln 2 + 2(q − 1) ln(q − 1) + (q − 1 − ln q) + 2 ln(2q − 1) +≥ 1 − ln 2 + 2(q − 1) ln(q − 1) + 2 ln(2q − 1), +as ex ≥ 1 + x implies that x ≥ ln(1 + x) for x > −1. Since x ln x is decreasing on +� +0, 1 +e +� +and +increasing on +� 1 +e, ∞ +� +(so in particular x ln x ≥ − 1 +e for x > 0), +q ∈ (1, 1.004] =⇒ f ′(q) ≥ 1 − ln 2 + 0.008 ln 0.004 > 0 +q ∈ [1.004, 1.055] =⇒ f ′(q) ≥ 1 − ln 2 + 0.11 ln 0.055 + 2 ln 1.008 > 0 +q ∈ [1.055, 1.12] =⇒ f ′(q) ≥ 1 − ln 2 + 0.24 ln 0.12 + 2 ln 1.11 > 0 +q ∈ [1.12, 2) =⇒ f ′(q) ≥ 1 − ln 2 − 2 +e + 2 ln 1.24 > 0, +so for all q ∈ (1, 2), f ′(q) > 0. +As +lim +q→1+ f(q) = 0, it follows that f(q) > 0 for q ∈ (1, 2), so +a|u|q + (1 − a)|v|q ≥ ef(q) > 1 whenever v < 0 < u. +Now suppose v ≥ 0. As |x|q is increasing for x ≥ 0, +a|u|q + (1 − a)|v|q ≥ a +�(q − 1)q−1 +a(1 − a) +�q += (q − 1)q(q−1) +aq−1(1 − a)q . +Using the work above, +(q − 1)q(q−1) +aq−1(1 − a)q ≥ +(q − 1)q(q−1) +max {aq−1(1 − a)q, aq(1 − a)q−1} > 2q−1 > 1, +so the inequality holds whenever v ≥ 0. The case u ≤ 0 is identical, which completes the proof. +■ +With this, we have the following key result. +Lemma 2.15. Fix q ∈ (1, 2), a nonempty set S = {(u1, v1), . . . , (uk, vk)} of points in [0, 1] × R, +and (x, y) ∈ [0, 1] × R such that u1 < . . . < uk, x ̸= ui for any 1 ≤ i ≤ k, and Jq +� +fS∪{(x,y)} +� +≤ 1. +Let p = 2 + +1 +q−1. Then +Jq +� +fS∪{(x,y)} +� +− Jq[fS] ≥ |y − fS(x)|p. +Proof. We first show that |y − fS(x)| > (q − 1)q−1 and x ∈ (u1, uk) cannot both hold. Suppose +otherwise, so that there exists an integer 1 ≤ i < k such that ui < x < ui+1. We will derive a +contradiction by showing Jq +� +fS∪{(x,y)} +� +> 1. Clearly +Jq +� +fS∪{(x,y)} +� +≥ (x − ui) +���� +y − vi +x − ui +���� +q ++ (ui+1 − x) +���� +vi+1 − y +ui+1 − x +���� +q +. +Substituting a = x − ui, b = ui+1 − x, d = y − fS(x), and m = +vi+1−vi +ui+1−ui as in Lemma 2.10, this +rewrites as +Jq +� +fS∪{(x,y)} +� +≥ a +����m + d +a +���� +q ++ b +����m − d +b +���� +q +. +11 + +As a + b = ui+1 − ui ≤ 1 and q > 1, +Jq +� +fS∪{(x,y)} +� +≥ +a +a + b +����(a + b)m + d(a + b) +a +���� +q ++ +b +a + b +����(a + b)m − d(a + b) +b +���� +q +, +and because +|d| > (q − 1)q−1 =⇒ +����d(a + b) +�1 +a + 1 +b +����� = |d|(a + b)2 +ab +≥ (q − 1)q−1 +a +a+b · +b +a+b +, +applying Lemma 2.14 yields Jq +� +fS∪{(x,y)} +� +> 1, contradiction. +Thus at least one of |y − fS(x)| ≤ (q − 1)q−1 and x ̸∈ (u1, uk) holds. If +|y − fS(x)| ≤ (q − 1)q−1 =⇒ (q − 1)(y − fS(x))2 ≥ |y − fS(x)|p, +the result follows from Lemma 2.10. Otherwise, assume without loss of generality that x < u1 (the +case x > uk is similar); then +Jq +� +fS∪{(x,y)} +� +− Jq[fS] = (u1 − x) +���� +v1 − y +u1 − x +���� +q +≥ |v1 − y|q ≥ |v1 − y|p = |y − fS(x)|p +by Lemma 2.5 (as q < 2 < p) and the result holds in this case as well. +■ +This immediately yields the following. +Theorem 2.16. For any reals q > 1 and p ≥ 2 + +1 +q−1, we have optp(Fq) = 1. +Proof. By Proposition 2.1 and Corollary 2.6, it suffices to prove that for q ∈ (1, 2) and p = 2+ +1 +q−1, +Lp(LININT, Fq) ≤ 1. Fix p = 2 + +1 +q−1, a target function f ∈ Fq, an integer m ≥ 1, and a sequence +of inputs σ = (x0, . . . , xm) ∈ [0, 1]m+1. Assume without loss of generality that all xi are distinct. +For 0 ≤ i ≤ m, define Si = {(x0, f(x0)), . . . , (xi, f(xi))}, and suppose LININT produces guesses +ˆy0, . . . , ˆym ∈ R. By Lemma 2.4 and Lemma 2.15, +1 ≥ Jq[f] ≥ Jq [fSm] = +m +� +i=1 +� +Jq [fSi] − Jq +� +fSi−1 +�� +≥ +m +� +i=1 +|ˆyi − f(xi)|p, +so Lp(LININT, f, σ) ≤ 1 for any f ∈ Fq and σ. Thus Lp(LININT, Fq) ≤ 1. +■ +2.4 +Sharp bounds for p ∈ (1, 2) +The paper [9] showed that opt1+ε(Fq) = O(ε−1) for all ε ∈ (0, 1) and q ≥ 2. In this section, we +first improve their upper bound by proving that opt1+ε(Fq) = O(ε− 1 +2) for all ε ∈ (0, 1) and q ≥ 2. +Then we show that this bound is sharp by proving that opt1+ε(Fq) = Ω(ε− 1 +2) for all q ≥ 1. In +order to prove the upper bound, we use two lemmas from [9]. To state the lemmas and prove our +upper bound, we use the following notation. Let x0, . . . , xm be any sequence of distinct elements +of [0, 1], and let f ∈ F2. Let ˆy1, . . . , ˆym be LININT’s predictions on trials 1, . . . , m. For each i > 1, +let di = minj 1, we have �m +i=1 dx +i ≤ 1+ +1 +2x−2. +By combining Lemmas 2.17 and 2.18 with H¨older’s inequality, we obtain the following sharp +upper bound. +Theorem 2.19. If p = 1 + ε ∈ (1, 2), then optp(F2) = O(ε− 1 +2). +Proof. First, note that +m +� +i=1 +ep +i = +m +� +i=1 +ep +i +d +p +2 +i +· d +p +2 +i . +By H¨older’s inequality, we have +m +� +i=1 +ep +i +d +p +2 +i +· d +p +2 +i ≤ +� m +� +i=1 +e2 +i +di +� p +2 � m +� +i=1 +d +p +2−p +i +�1− p +2 +. +Note that �m +i=1 +e2 +i +di ≤ 1 by Lemma 2.17 and +m +� +i=1 +d +p +2−p +i +≤ 1 + +1 +2 +p +2−p − 2 +by Lemma 2.18, since p > 1 implies that +p +2−p > 1. Thus +� m +� +i=1 +d +p +2−p +i +�1− p +2 +≤ +� +1 + +1 +2 +p +2−p − 2 +�1− p +2 +. +Let δ = +p +2−p − 1, and note that 1 +δ = +2−p +2p−2. Thus +� +1 + +1 +2 +p +2−p − 2 +�1− p +2 += +� +1 + +1 +21+δ − 2 +�1− p +2 += O +�� +1 + 1 +δ +� 2−p +2 +� += O +�� +p +2p − 2 +� 2−p +2 +� +, +where the upper bound follows from the fact that eδ ln 2 ≥ 1 + δ ln 2. Thus we have proved that +m +� +i=1 +ep +i = O +�� +p +2p − 2 +� 2−p +2 +� +, +so optp(F2) = O +� +(2p − 2)− 2−p +2 +� +, where we use the fact that p +2−p +2 += Θ(1) for p ∈ (1, 2) to obtain +the last bound. Since p = 1 + ε, we have +optp(F2) = O +� +(2p − 2)− 2−p +2 +� += O +� +ε− 1−ε +2 +� += O(ε− 1 +2), +where we use the fact that εε = Θ(1) for ε ∈ (0, 1) to obtain the last bound. +■ +We obtain the next corollary since optp(F∞) ≤ optp(Fr) ≤ optp(Fq) whenever 1 ≤ q ≤ r. +Corollary 2.20. If ε ∈ (0, 1), then opt1+ε(F∞) = O(ε− 1 +2 ) and opt1+ε(Fq) = O(ε− 1 +2 ) for all q ≥ 2, +where the constant does not depend on q. +13 + +In order to show that the last corollary is sharp up to a constant factor, we construct a family +of functions in F∞. Our proof uses the following lemma from [9] which was also used in [12]. +Lemma 2.21 ([9]). Let S ⊆ [0, 1] × R with S = {(ui, vi) : 1 ≤ i ≤ m} and u1 < u2 < · · · < um. If +(x, y) ∈ [0, 1] × R and there exists 1 ≤ j ≤ m such that |x − uj| = |x − uj+1| = mini |x − ui|, then +J2[fS∪{(x,y)}] = J2[fS] + 2(y−fS(x))2 +mini |x−ui| . +The method in the following proof is similar to one used in [12] to obtain bounds for a finite +variant of opt1(Fq) for q ≥ 2 that depends on the number of trials m. +Theorem 2.22. If ε ∈ (0, 1), then opt1+ε(F∞) = Ω(ε− 1 +2 ). +Proof. Since opt1+ε(F∞) ≥ 1 for all ε ∈ (0, 1), it suffices to prove the theorem for ε ∈ +� +0, 1 +2 +� +. Define +x0 = 1 and y0 = 0. For natural numbers i, j with 0 ≤ j < 2i−1, define x2i−1+j = +1 +2i + +j +2i−1 . For +each i = 1, 2, . . ., we consider the trials for x2i−1, . . . , x2i−1 to be part of stage i, so that x1 = 1 +2 is +in stage 1, x2 = 1 +4 and x3 = 3 +4 are in stage 2, and so on. +Let A be any algorithm for learning F∞. Using A, we construct an infinite sequence of piecewise +linear functions f0, f1, . . . ∈ F∞ and an infinite sequence of numbers y0, y1, . . . ∈ R for which ft is +consistent with the xk and yk values for k ≤ t and A has total (1 + ε)-error at least +i +� +k=1 +2k−2 +�√ε(1 − ε) +k +2 +2k+1 +�1+ε +after i stages. This implies that +opt1+ε(F∞) ≥ +∞ +� +k=1 +2k−2 +�√ε(1 − ε) +k +2 +2k+1 +�1+ε +. +In order to analyze the functions fi, we will also define and analyze another infinite sequence +of piecewise linear functions gi,j with 0 ≤ j ≤ 2i−1 and another infinite sequence of numbers +v1, v2, . . . ∈ R. We start by letting f0 be the 0-function. Next, we inductively define both sequences +of piecewise linear functions. +Fix a stage i, and let gi,0 = f2i−1−1. Let t be a trial in stage i, and let vt be whichever of +ft−1(xt) ± +√ε(1−ε) +i +2 +2i+1 +is furthest from ˆyt. Let gi,t−2i−1+1 be the function which linearly interpolates +{(0, 0), (1, 0)} ∪ +� +(xs, ys) : s < 2i−1� +∪ +� +(xs, vs) : 2i−1 ≤ s ≤ t +� +. +For any t ≥ 1, let Lt and Rt be the elements of {0, 1} ∪ {xs : s < t} that are closest to xt on the +left and right respectively. If both |vt − ft−1(Lt)| ≤ 2−i and |vt − ft−1(Rt)| ≤ 2−i, then let yt = vt. +Otherwise we let yt = ft−1(xt). Finally, we define ft to be the function which linearly interpolates +{(0, 0), (1, 0)} ∪ {(xs, ys) : s ≤ t}. +By definition, we have ft ∈ F∞ for each t ≥ 0. We will prove next that for all i, j we have +J2[gi,j] ≤ 1 +4, and then we will use this to prove that yt = ft−1(xt) for at most half of the trials t +in stage i. The proof will use double induction, first on i and then on j, and we will prove the +stronger statement that +J2[gi,j] ≤ ε +4 +i−1 +� +k=0 +(1 − ε)k + jε(1 − ε)i +2i+1 +. +(1) +14 + +In order to prove this statement, we will also prove that +J2[f2i−1−1] ≤ ε +4 +i−1 +� +k=0 +(1 − ε)k +(2) +for all i ≥ 1. Note that this is equivalent to proving that +J2[gi,0] ≤ ε +4 +i−1 +� +k=0 +(1 − ε)k +for all i ≥ 1. Clearly this is true for i = 1, which is the base case of the induction on i. Fix some +stage i ≥ 1. We will assume that Inequality 2 is true for this fixed i, and use this to prove that +J2[f2i−1] ≤ ε +4 +i +� +k=0 +(1 − ε)k. +(3) +In order to prove Inequality 3, we will prove Inequality 1 for all 0 ≤ j ≤ 2i−1. This follows from +the inductive hypothesis for i and the definition of gi,0 when j = 0, which is the base case of the +induction on j. Fix some integer j with 0 ≤ j ≤ 2i−1 − 1 and assume that Inequality 1 is true for +this fixed j. By Lemma 2.21, we have +J2[gi,j+1] = J2[gi,j] + +2 +� √ε(1−ε) +i +2 +2i+1 +�2 +2−i += J2[gi,j] + ε(1 − ε)i +2i+1 +. +By the inductive hypothesis for j, we obtain +J2[gi,j+1] ≤ ε +4 +i−1 +� +k=0 +(1 − ε)k + jε(1 − ε)i +2i+1 ++ ε(1 − ε)i +2i+1 +, +which completes the inductive step for j. Substituting j = 2i−1, we obtain +J2[gi,2i−1] ≤ ε +4 +i +� +k=0 +(1 − ε)k. +Note that Lemma 2.21 implies that +J2[f2i−1−1+j] ≤ J2[gi,j] +for all j = 0, . . . , 2i−1, so we obtain Inequality 3, which completes the inductive step for i. By +Inequality 1, we obtain +J2[gi,j] ≤ ε +4 +i−1 +� +k=0 +(1 − ε)k + ε(1 − ε)i +4 += ε +4 +i +� +k=0 +(1 − ε)k +for all j with 0 ≤ j ≤ 2i−1. Note that +ε +4 +i +� +k=0 +(1 − ε)k < ε +4 +∞ +� +k=0 +(1 − ε)k = 1 +4. +15 + +Now that we have shown that J2[gi,j] ≤ 1 +4, we are ready to prove for each i ≥ 1 that yt = ft−1(xt) +for at most half of the trials t in stage i. For each trial t with yt = ft−1(xt), note that the absolute +value of the slope of gi,t−2i−1+1 must exceed 1 in at least one of the intervals of length 2−i on either +side of xt. If yt = ft−1(xt) for at least b of the trials in stage i, then restricting to intervals of slope +at least 1 implies that J2[gi,2i−1] ≥ b2−i. Since J2[gi,2i−1] ≤ 1 +4, we must have b ≤ 2i−2. Thus during +stage i, there are at most 2i−2 trials t with yt = ft−1(xt), which implies that there are at least 2i−2 +trials with yt = vt. In each of those trials, A was off by at least +√ε(1−ε) +i +2 +2i+1 +, so the total (1 + ε)-error +of A after i stages is at least �i +k=1 2k−2 +� √ε(1−ε) +k +2 +2k+1 +�1+ε +. Thus +opt1+ε(F∞) ≥ +∞ +� +k=1 +2k−2 +�√ε(1 − ε) +k +2 +2k+1 +�1+ε += +1 +2 +�√ +ε(1−ε) +4 +�1+ε +1 − 2 +� √1−ε +2 +�1+ε = Ω + + + +(ε(1 − ε)) +1+ε +2 +1 − 2 +� √1−ε +2 +�1+ε + + + . +Since εε = Θ(1) and (1 − ε)1+ε = Θ(1) for ε ∈ +� +0, 1 +2 +� +, we have opt1+ε(F∞) = Ω +� +√ε +1−2 +� √1−ε +2 +�1+ε +� +. +Since 2ε = Θ(1) for ε ∈ +� +0, 1 +2 +� +, we have opt1+ε(F∞) = Ω +� +√ε +2ε−√1−ε1+ε +� +. Note that (1 − ε) +1+ε +2 +≥ +1 − ε(1 + ε) for ε ∈ +� +0, 1 +2 +� +. To check this, note that it is true when ε = 0, and the derivative of +(1 − ε) +1+ε +2 − (1 − ε(1 + ε)) is +2ε + 1 + (1 − ε) +1+ε +2 +�1 +2 ln(1 − ε) − 1 +2 − +ε +1 − ε +� +> 0 +for ε ∈ +� +0, 1 +2 +� +. Thus, opt1+ε(F∞) = Ω +� +√ε +2ε−1+ε(1+ε) +� +. Also note that 2ε ≤ 1 + ε for ε ∈ (0, 1). +Equality holds at ε = 0 and ε = 1, and the derivative of 1 + ε − 2ε is 1 − 2ε ln 2, which is +positive for ε ∈ +� +0, − ln ln 2 +ln 2 +� +and negative for ε ∈ +� − ln ln 2 +ln 2 +, 1 +� +. Thus 2ε − 1 + ε(1 + ε) < 3ε, so +opt1+ε(F∞) = Ω(ε− 1 +2). +■ +The next corollary follows from Theorem 2.22, again using the fact that optp(F∞) ≤ optp(Fr) ≤ +optp(Fq) whenever 1 ≤ q ≤ r. +Corollary 2.23. If ε ∈ (0, 1), then opt1+ε(Fq) = Ω(ε− 1 +2) for all q ≥ 1, where the constant does +not depend on q. +Combining Corollaries 2.20 and 2.23, we have the following theorem. +Theorem 2.24. If ε ∈ (0, 1), then opt1+ε(F∞) = Θ(ε− 1 +2) and opt1+ε(Fq) = Θ(ε− 1 +2) for all q ≥ 2, +where the constant in the bound does not depend on q. +3 +A multi-variable generalization +In this section, we prove several results on optp(Fq,d). First, we prove a simple lower bound for +optp(Fq,d) in terms of optp(Fq). +Proposition 3.1. For any positive integer d, real number p > 0, and q ∈ [1, ∞) ∪ {∞}, we have +optp(Fq,d) ≥ dp · optp(Fq). +16 + +Proof. If optp(Fq,d) = ∞, there is nothing to prove, and if optp(Fq) = ∞, it is clear, by restricting +the inputs xi to the set {ce1 : c ∈ [0, 1]} ⊂ [0, 1]d (where e1 ∈ [0, 1]d has a 1 in the first component +and a 0 in the rest), that optp(Fq,d) = ∞ as well. +Now suppose that both optp(Fq,d) and optp(Fq) are finite. Fix any algorithm A for learning +Fq,d. Let 1 be the all-ones d-tuple and let a(xi : (x0, z0), . . . , (xi−1, zi−1)) denote the output of A +given the input xi1 after learning the pairs (xj1, zj) for j < i, given that there is a function in Fq,d +which passes through the points (xj1, zj) for j < i. Then let A′ be the algorithm for learning Fq +which, given the input xi after learning the pairs (xj, wj) for j < i, returns the output +a′(xi : (x0, w0), . . . , (xi−1, wi−1)) = a(xi : (x0, dw0), . . . , (xi−1, dwi−1)) +d +, +given that there is a function in Fq which passes through the points (xj, wj) for j < i. +Fix ε > 0. Then there exist f ∈ Fq and a sequence of inputs x0, x1, . . . , xM such that +M +� +i=1 +|a′(xi : (x0, f(x0)), . . . , (xi−1, f(xi−1))) − f(xi)|p ≥ optp(Fq) − ε. +Against A, the adversary uses the function +γ(a1, . . . , ad) = +d +� +i=1 +f(ai) +with the inputs x01, x11, . . . , xM1. First, suppose that q ∈ [1, ∞). Observe that for any 1 ≤ k ≤ d +and d − 1 reals xi ∈ [0, 1], where 1 ≤ i ≤ d but i ̸= k, +� 1 +0 +���� +dγ +dxk +���� +q +dxk = +� 1 +0 +|f ′(x)|qdx ≤ 1 +since f ∈ Fq; hence, γ ∈ Fq,d. Next, suppose that q = ∞. Observe that +��� dγ +dxk +��� = |f ′(xk)| ≤ 1 for all +xk ∈ [0, 1] since f ∈ Fq; hence, in this case we also have γ ∈ Fq,d. To finish the proof, let +ei = |a′(xi : (x0, f(x0)), . . . , (xi−1, f(xi−1))) − f(xi)| +and +ki = |a(xi : (x0, γ(x01)), . . . , (xi−1, γ(xi−11))) − γ(xi1)| +for each i. Thus, +ki = |da′(xi : (x0, f(x0)), . . . , (xi−1, f(xi−1))) − df(xi)| = dei +and +M +� +i=1 +ep +i ≥ optp(Fq) − ε. +Hence, +M +� +i=1 +kp +i ≥ dp(optp(Fq) − ε). +Taking ε → 0 finishes the proof. +■ +17 + +The next corollary follows from Proposition 3.1 since optp(F1) = ∞ [9]. +Corollary 3.2. For any positive integer d and real number p > 0, we have optp(F1,d) = ∞. +Now we directly prove some results about optp(F∞,d), depending on whether p < d or p > d. +The main negative result is the following. +Theorem 3.3. Let d > 0 be an integer and p be a real number with 0 < p < d. Then optp(F∞,d) = +∞. +Proof. Fix any algorithm A for learning F∞,d. Then choose any integer n ≥ 1, and let S be the +set of reals 0 < r < 1 such that 2nr is an odd integer (so |S| = n). The adversary first reveals +f(0, . . . , 0) = 0, then chooses xi ranging over all elements of Sd in lexicographic order, receives +input ˆyi from A, and reveals f(xi) = ± 1 +2n, whichever is farther from ˆyi. +Let {x} = x−⌊x⌋ denote the fractional part of x. At the end of the nd+1 trials, the algorithm’s +revealed values of f are consistent with a function f : [0, 1]d → R given by +f(x1, . . . , xd) = ± 1 +n min +1≤i≤d (min ({nxi}, {−nxi})) , +where the signs ± are chosen such that f(x) agrees with the adversary’s outputs for any x = +(x1, . . . , xd) ∈ Sd and f has constant sign in any region +�n1 +n , n1 + 1 +n +� +× . . . × +�nd +n , nd + 1 +n +� +⊂ [0, 1]d +for integers 0 ≤ ni < n. The consistency follows since {nxi} = {−nxi} = 1 +2 for all xi ∈ S. +First, we show f ∈ F∞,d. Fix any 1 ≤ i ≤ d and x = (x1, . . . , xd−1) ∈ [0, 1]d−1, and consider +the function g : [0, 1] → R given by g(x) = f(x′), where x′ ∈ [0, 1]d is formed by inserting x into +the ith position of x. Then g is given by +g(x) = ± 1 +n min (min ({nx}, {−nx}) , M) , +where +M = +min +1≤i≤d−1 (min ({nxi}, {−nxi})) . +Evidently g is piecewise linear, with finitely many points where g′ is not defined and |g′(x)| = 1 +or g′(x) = 0 everywhere else by definition of g; moreover, since the function min({nx}, {−nx}) +is continuous, it follows that g is continuous. Hence g ∈ F∞. Since this holds for any choice of +1 ≤ i ≤ d and x ∈ [0, 1]d, it follows that f ∈ F∞,d. +Now we find a lower bound for the error the adversary can guarantee. There are nd trials past +the first, each of which has |ˆyi − f(xi)| ≥ +1 +2n; hence the adversary guarantees +� +i>0 +|ˆyi − f(xi)|p ≥ +nd +(2n)p = 1 +2p · nd−p. +Because p < d, this grows arbitrarily large as n increases; hence optp(F∞,d) = ∞. +■ +As F∞ ⊆ Fq implies that F∞,d ⊆ Fq,d for any q ≥ 1, this bound extends to q ̸= ∞. +18 + +Corollary 3.4. Let d > 0 be an integer and p be a real number with 0 < p < d. For any q ≥ 1, we +have optp(Fq,d) = ∞. +In order to establish upper bounds on optp(F∞,d), we prove the following lemma. +Lemma 3.5. For f ∈ F∞,d and x1 = (x1,1, . . . , xd,1), x2 = (x1,2, . . . , xd,2) ∈ [0, 1]d, we have +|f(x1) − f(x2)| ≤ +d +� +i=1 +|xi,1 − xi,2|. +Proof. Fix such f, x1, x2. Define a sequence of x′ +i ∈ [0, 1]d, for 0 ≤ i ≤ d, such that x′ +i has its first +i components equal to the first i components of x2 and its last d − i components equal to the last +d − i components of x1 (so x′ +0 = x1 and x′ +d = x2). By the triangle inequality, +|f(x1) − f(x2)| = +����� +d +� +i=1 +� +f(x′ +i−1) − f(x′ +i) +� +����� ≤ +d +� +i=1 +��f(x′ +i−1) − f(x′ +i) +�� . +Now consider any 1 ≤ i ≤ d. Note that xi−1 and x′ +i only differ in their ith components, with one +being xi,1 and the other being xi,2. Then by definition of F∞,d and using the fact that for g ∈ F∞ +and x1, x2 ∈ [0, 1], |g(x1) − g(x2)| ≤ |x1 − x2|, it follows that +��f(x′ +i−1) − f(x′ +i) +�� ≤ |xi,1 − xi,2|. +Summing over 1 ≤ i ≤ d yields the result. +■ +Lemma 3.5 makes the class F∞,d particularly nice to work with. +Using a nearest neighbor +algorithm, we establish the following upper bound. +Theorem 3.6. Suppose p > d. Then optp(F∞,d) ≤ (2d−1)dp +1− 2d +2p +. +Proof. Consider the algorithm A which guesses 0 on the first input and, on trial i (after receiving +inputs x0, . . . , xi−1), picks the least index 0 ≤ j < i which minimizes the L1 distance between xj +and xi and guesses ˆyi = f(xj). We will show Lp(A, F∞,d) ≤ (2d−1)dp +1− 2d +2p +. +Fix f ∈ F∞,d and a sequence x0, . . . , xm of xi ∈ [0, 1]d. Assume all the xi are distinct. Then +for each 1 ≤ i ≤ m, there exists a least integer ki such that, if [0, 1]d is divided into 2kid regions +given by +{(x1, . . . , xd) ∈ [0, 1]d : ni ≤ 2kixi ≤ ni + 1} +over all d-tuples (n1, . . . , nd) of integers 0 ≤ ni < 2ki, then xi is not in the same region as any of +x0, . . . , xi−1. Note that because x0 and xi are both in [0, 1]d for any 1 ≤ i ≤ m, all ki are at least +1. For each integer k ≥ 1, let ck be the number of integers 1 ≤ i ≤ m such that ki = k. By the +Pigeonhole Principle, for any fixed integer k ≥ 0, there exist at most 2kd − 1 indices 1 ≤ i ≤ m +such that ki ≤ k; otherwise, at least 2kd + 1 of the xi (including x0) would be the first within their +containing length-2−k hypercube region. Thus +K +� +k=1 +ck ≤ 2Kd − 1 +(4) +for any integer K ≥ 1. Moreover, for any 1 ≤ i ≤ m, xi lies in the same length-2−(ki−1) hypercube +as one of x0, . . . , xi−1, and this hypercube has L1 distance +d +2ki−1 between two of its opposite vertices, +so by Lemma 3.5, +|ˆyi − f(xi)| ≤ +d +2ki−1 . +(5) +19 + +Combining these, +m +� +i=1 +|ˆyi − f(xi)|p ≤ +� +k≥1 +ck +� +d +2k−1 +�p += +� +K≥1 +�� K +� +k=1 +ck +� �� +d +2K−1 +�p +− +� d +2K +�p�� +≤ dp � +K≥1 +(2Kd − 1)(2−p(K−1) − 2−pK) = dp(2p − 1) +� +K≥1 +2−pK(2Kd − 1) += dp(2p − 1) +� +2d−p +1 − 2d−p − +2−p +1 − 2−p +� += (2d − 1)dp +1 − 2d +2p +. +This holds for all f ∈ F∞,d and sequences of xi; hence Lp(A, F∞,d) ≤ (2d−1)dp +1− 2d +2p +. +■ +The next corollary follows from Proposition 3.1 and Theorem 3.6 since optp(F∞) = 1 for all +p ≥ 2. +Corollary 3.7. For any fixed positive integer d and real number p ≥ d + 1, we have optp(F∞,d) = +Θ(dp), where the constant in the upper bound depends only on d. +We can also use Theorems 3.3 and 3.6 to obtain sharp bounds on the worst-case errors for +learning F∞,d when the number of trials is bounded. +Corollary 3.8. Let d > 0 be an integer and p be a real number with 0 < p < d. Then optp(F∞,d, m) = +Θ(m1− p +d ), where the constants in the bounds depend on p and d. +Proof. By Theorem 3.3, we have optp(F∞,d, m) ≥ +1 +2p nd−p for n = ⌊m1/d⌋, so we obtain the lower +bound optp(F∞,d, m) ≥ +1 +2p m +d−p +d (1−o(1)). For the upper bound, we use the algorithm and notation +of Theorem 3.6 with K = +� +log2(m+1) +d +� +to obtain +m +� +i=1 +|ˆyi − f(xi)|p ≤ +m +� +i=1 +� +d +2ki−1 +�p +≤ +K +� +k=1 +(2kd − 2(k−1)d) +� +d +2k−1 +�p += dp(2d − 1) +K +� +k=1 +2(k−1)(d−p) = dp(2d − 1)2K(d−p) − 1 +2d−p − 1 +< dp(2d − 1)2d−p +2d−p − 1 +m +d−p +d (1 + o(1)), +where the first inequality follows from Inequality 5 and the second inequality follows from Inequality +4 since +� +d +2k−1 +�p is decreasing in k. Thus +optp(F∞,d, m) ≤ dp(2d − 1)2d−p +2d−p − 1 +m +d−p +d (1 + o(1)). +■ +20 + +4 +Discussion and open problems +With the results in this paper, the value of optp(Fq) is now bounded up to a constant factor for all +p, q ≥ 1 except when q ∈ (1, 2) and p ∈ (1, 2) ∪ (2, 2 + +1 +q−1). In particular, by combining the results +in this paper with the results in [9], we now know that optp(Fq) = 1 for all (p, q) that lie in the +following regions. +• p, q ≥ 2 +• q ∈ (1, 2) and p ≥ 2 + +1 +q−1 +In addition to investigating the regions in which optp(Fq) is not bounded up to a constant factor, +it remains to narrow the constant gap between the upper and lower bounds for opt1+ε(Fq) = Θ(ε− 1 +2 ) +when ε ∈ (0, 1) and q ∈ [2, ∞) ∪ {∞}. Another similar problem is to narrow the constant gap +between the upper and lower bounds for opt2(Fq) = Θ(ε−1) when q ∈ (1, 2). +The results in this paper also help characterize the values of (p, q) for which optp(Fq) is finite. +Before this paper, it was only known that opt2(Fq) is finite for p > 1 and q ≥ 2, and optp(Fq) = ∞ +when p = 1 or q = 1. With our new results, we now know that optp(Fq) is also finite when p ≥ 2 +and q > 1. We make the following conjecture about this problem. +Conjecture 4.1. For all p > 1 and q > 1, optp(Fq) is finite. +Besides the new results about smooth functions of a single variable, we also introduced a +generalization of the model to multi-variable functions and found some bounds for this multi- +variable online learning scenario. We showed that optp(F∞,d) is infinite when 0 < p < d and finite +when p > d, but it remains to determine whether optd(F∞,d) is finite for d > 1. For finite q ≥ 1 and +0 < p < d, we also know that optp(Fq,d) is infinite, but it remains to determine whether optp(Fq,d) +is finite for p ≥ d and q ∈ (1, ∞). In addition, we proved for any fixed positive integer d that +optp(F∞,d) = Θ(dp) for p ≥ d + 1, where the constant in the upper bound depends on d. The +multiplicative gap between the upper and lower bounds is 2d+1 − 2. We conjecture that the lower +bound is sharp for p sufficiently large with respect to d and q. +Conjecture 4.2. For all q ∈ [1, ∞) ∪ {∞}, for all positive integers d, and for all real numbers p +sufficiently large with respect to q and d, we have optp(Fq,d) = dp. +The papers [9] and [12] investigated opt1(Fq, m) for q ≥ 2, where m is the number of trials. It +would be natural to study optp(Fq, m) for p = 1+ε with 0 < ε < 1 and q ≥ 1, since opt1+ε(Fq) can +grow arbitrarily large as ε → 0. We bounded optp(F∞,d, m) up to a constant factor for any fixed +positive integer d and fixed real number p with 0 < p < d, but the constants in the bounds depend +on p and d. It remains to narrow the gap between the upper bound of dp(2d−1)2d−p +2d−p−1 +m +d−p +d (1 + o(1)) +and the lower bound of +1 +2p m +d−p +d (1 − o(1)). It would also be interesting to investigate optp(Fq,d, m) +for finite values of q. +Another possible direction would be to investigate families of smooth functions with additional +restrictions. For example, let Eq ⊆ Fq be the family of exponential functions f(x) = eax+b with +f ∈ Fq. +Proposition 4.3. For all p > 0 and q ≥ 1, we have optp(Eq) = 1. +21 + +Proof. The upper bound optp(Eq) ≤ 1 follows by Lemma 2.5, since the learner knows the function +after two rounds with different inputs and the first round does not count for the total error. For +the lower bound, consider an adversary that chooses some ε ∈ (0, 1), defines ̺ = 1 − +1−ε√1 − ε, and +reveals f(0) = − +1−ε +ln(1−ε). On the second turn, they either reveal f(1) = − +1 +ln(1−ε) or f(1) = − +1−̺ +ln(1−̺), +whichever maximizes the error for the learner’s guess. +If f(1) = − +1 +ln(1−ε), then f(x) = eax+b with a = − ln(1 − ε) and b = ln(1 − ε) − ln(− ln(1 − ε)). +Note that f ′(x) = aeax+b ∈ [1−ε, 1] for all x ∈ [0, 1], so f ∈ Eq for all q ≥ 1. If f(1) = − +1−̺ +ln(1−̺), then +f(x) = eax+b with a = ln(1 − ̺) and b = − ln(− ln(1 − ̺)). Note that f ′(x) = aeax+b ∈ [−1, −1 + ̺] +for all x ∈ [0, 1], so f ∈ Eq for all q ≥ 1. Moreover, note that +lim +ε→0 +� +− +1 +ln(1 − ε) + +1 − ̺ +ln(1 − ̺) +� += lim +ε→0 +�−1 + (1 − ε) 1−ε√1 − ε +ln(1 − ε) +� += 2, +by L’Hˆopital’s rule. Thus optp(Eq) ≥ 1. +■ +Let Pq,m ⊆ Fq be the family of polynomial functions f ∈ Fq of degree at most m. It is easy to +see that optp(Pq,1) = 1 for all p > 0 and q ≥ 1, but it would be interesting to investigate optp(Pq,m) +for m > 1. Note that we have optp(Pq,m) ≤ optp(Fq, m) for all p > 0, q ≥ 1, and m ≥ 1, since the +learner will know f ∈ Pq,m with certainty after being tested on m+1 different inputs. Let Pq ⊆ Fq +be the family of all polynomial functions f ∈ Fq. We make the following conjecture about this +family. +Conjecture 4.4. For all p > 0 and q ≥ 1, we have optp(Pq) = optp(Fq). +Some other possible subsets of Fq that could be investigated are piecewise functions with at +most k pieces where the pieces are polynomials of degree at most m, sums of exponential functions, +and sums of trigonometric functions. +Finally, we return to the problem from the introduction of predicting the next day’s temperature +range at a given location. In particular, consider the single-variable problem where we predict the +next day’s temperature range based only on the time of year. An issue with using the model from +[9] for this temperature prediction problem is that the same input for time of year could have +different outputs for the temperature range in different years. A more realistic way to model this +problem would be to choose the output from a probability distribution which depends on the input. +In order for the learner to be able to guarantee a finite bound on the worst-case error, the number +of trials would be bounded and restrictions would be placed on the probability distribution. For +example, the density function for the probability distribution could be required to have smoothness +properties like the functions from [9], and the support of the density function could be required to +be a subset of [0, r] for some r > 0. Investigating such a model would be an interesting direction +for future research. Note that this model reduces to the model from [9] when the support consists +of a single point. +5 +Acknowledgments +Most of this research was performed in PRIMES 2022. We thank the organizers for this research +opportunity. Our paper subsumes [5], which proved Theorem 1.2. We also thank the anonymous +reviewers for helpful comments which improved the clarity and presentation of the results in this +paper. +22 + +References +[1] D. Angluin, Queries and concept learning. Machine Learning 2 (1988) 319–342. +[2] A. Barron, Approximation and estimation bounds for artificial neural networks. Workshop on +Computational Learning Theory (1991) +[3] N. Cesa-Bianchi, P.M. Long, and M.K. Warmuth, Worst-case quadratic loss bounds for pre- +diction using linear functions and gradient descent. IEEE Transactions on Neural Networks 7 +(1996) 604–619. +[4] V. Faber and J. Mycielski, Applications of learning theorems. Fundamenta Informaticae 15 +(1991) 145–167. +[5] J. Geneson, Sharper bounds for online learning of smooth functions of a single variable. CoRR +abs/2105.14648 (2021) +[6] W. Hardle, Smoothing techniques. Springer Verlag (1991) +[7] D. Haussler, Generalizing the PAC model: sample size bounds from metric dimension-based +uniform convergence results. Proceedings of the 30th Annual Symposium on the Foundations of +Computer Science (1989) +[8] S. Kaczmarz, Angenaherte Aufl¨osung von systemen linearer gleichungen. Bull. Acad. Polon. +Sci. Lett. A 35 (1937) 355–357. +[9] D. Kimber and P. M. Long, On-line learning of smooth functions of a single variable. Theoretical +Computer Science 148 (1995) 141–156. +[10] N. Littlestone, Learning quickly when irrelevant attributes abound: A new linear-threshold +algorithm. Machine Learning 2 (1988) 285–318. +[11] N. Littlestone and M.K. Warmuth, The weighted majority algorithm. Proceedings of the 30th +Annual Symposium on the Foundations of Computer Science (1989) +[12] P. M. Long, Improved bounds about on-line learning of smooth functions of a single variable. +Theoretical Computer Science 241 (2000) 25–35. +[13] J. Mycielski, A learning algorithm for linear operators. Proceedings of the American Mathe- +matical Society 103 (1988) 547–550. +[14] B. Widrow and M.E. Hoff, Adaptive switching circuits. 1960 IRE WESCON Conv. Record +(1960) 96–104. +23 + diff --git a/zNAzT4oBgHgl3EQfePy_/content/tmp_files/load_file.txt b/zNAzT4oBgHgl3EQfePy_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dbc1fd92bb91e6eae9ec46175be23bf29dc35d06 --- /dev/null +++ b/zNAzT4oBgHgl3EQfePy_/content/tmp_files/load_file.txt @@ -0,0 +1,953 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf,len=952 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='01434v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='LG] 4 Jan 2023 Online Learning of Smooth Functions Jesse Geneson and Ethan Zhou January 5, 2023 Abstract In this paper, we study the online learning of real-valued functions where the hidden function is known to have certain smoothness properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Specifically, for q ≥ 1, let Fq be the class of absolutely continuous functions f : [0, 1] → R such that ∥f ′∥q ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For q ≥ 1 and d ∈ Z+, let Fq,d be the class of functions f : [0, 1]d → R such that any function g : [0, 1] → R formed by fixing all but one parameter of f is in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For any class of real-valued functions F and p > 0, let optp(F) be the best upper bound on the sum of pth powers of absolute prediction errors that a learner can guarantee in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In the single-variable setup, we find new bounds for optp(Fq) that are sharp up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We show for all ε ∈ (0, 1) that opt1+ε(F∞) = Θ(ε− 1 2 ) and opt1+ε(Fq) = Θ(ε− 1 2 ) for all q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We also show for ε ∈ (0, 1) that opt2(F1+ε) = Θ(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In addition, we obtain new exact results by proving that optp(Fq) = 1 for q ∈ (1, 2) and p ≥ 2 + 1 q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In the multi-variable setup, we establish inequalities relating optp(Fq,d) to optp(Fq) and show that optp(F∞,d) is infinite when p < d and finite when p > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We also obtain sharp bounds on learning F∞,d for p < d when the number of trials is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 1 Introduction Consider a learner that wants to predict the next day’s temperature range at a given location based on inputs such as the current day’s temperature range, humidity, atmospheric pressure, precipitation, wind speed, solar radiation, location, and time of year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In our model, this learner is tested daily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' On a given day, the learner gets inputs for that day, which it uses to output a prediction for the next day’s temperature range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' when the next day arrives, it sees the correct temperature range, then uses this feedback to update future predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' As this is repeated, the learner accumulates information to help it make better predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' A natural question arises: can the learner guarantee that its predictions become better over time, and if so, how quickly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We investigate a model of online learning of real-valued functions previously studied in [9, 12, 13, 1, 10, 11] where an algorithm A learns a real-valued function f from some class F in trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Past research on this model focused on functions of one input, for example, predicting the temperature range solely based on the time of year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The research showed that, as long as the function is sufficiently smooth, the learner can become a good predictor fairly rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Suppose that F consists of functions f : S → R for some set S, and fix some f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In each trial t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , m, A receives an input st ∈ S, guesses ˆyt for the value of f(st), and receives the actual value of f(st).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Following [9], we focus on an error function which measures how difficult it is for a learner to predict functions accurately in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The error function depends on two parameters, p and q, which determine how harshly the learner is punished for errors and the types of functions that the learner might encounter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Small values of p and q are more difficult for the 1 1 2 1 2 1 O � 1 p−1 � p q Figure 1: Exact values and bounds on optp(Fq) for p, q > 1 prior to the results in this paper learner, leading to higher values of the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For each algorithm A, p > 0, f ∈ F, and σ = (s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , sm) ∈ Sm+1, define Lp(A, f, σ) = m � t=1 |ˆyt − f(st)|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' When f and σ are clear from the context, we refer to Lp(A, f, σ) as the total p-error of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Define Lp(A, F) = sup f∈F,σ∈∪m∈Z+Sm Lp(A, f, σ) and optp(F) = inf A Lp(A, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Note that unlike the definition of opt presented in [9, 12, 5], F may consist of real-valued functions on any domain, not just functions from [0, 1] to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The case where F contains functions f : [0, 1] → R whose derivatives have various bounded norms was studied in [9, 12, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For q ≥ 1, let Fq be the class of absolutely continuous functions f : [0, 1] → R such that � 1 0 |f ′(x)|qdx ≤ 1, and let F∞ be the class of absolutely continuous functions f : [0, 1] → R such that sup x∈(0,1) |f ′(x)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' As noted in [12], F∞ contains exactly those f : [0, 1] → R such that |f(x) − f(y)| ≤ |x − y| for all x, y ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Also, by Jensen’s inequality, F∞ ⊆ Fq ⊆ Fr for all q ≥ r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Hence optp(F∞) ≤ optp(Fq) ≤ optp(Fr) for all p ≥ 1 and q ≥ r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Previous papers determined the exact values of optp(Fq) for p = 1, q = 1, and p, q ≥ 2, as well as bounds on optp(Fq) for p ∈ (1, 2) and q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The paper [9] proved that optp(F1) = ∞ for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' They also showed that opt1(Fq) = opt1(F∞) = ∞ for all q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In contrast, they found that optp(Fq) = optp(F∞) = 1 for all p ≥ 2 and q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' This was also proved in [4] using a different algorithm based on a generalization of the Widrow-Hoff algorithm [8, 14], and a noisy version of this problem was studied in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In this paper, we extend the region of values of p, q for which it is known that optp(Fq) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For any reals q > 1 and p ≥ 2 + 1 q−1, we have optp(Fq) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For p = 1 + ε with ε ∈ (0, 1), the paper [9] proved that optp(Fq) = O(ε−1) for all q ≥ 2, which implies that optp(F∞) = O(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' However, these bounds are not sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In this paper, we determine opt1+ε(Fq) up to a constant factor for all ε ∈ (0, 1) and q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 2 1 2 1 2 1 Θ � 1 √p−1 � 1 O � 1 q−1 � p q Figure 2: Exact values and bounds on optp(Fq) for p, q > 1 including the results in this paper Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For all ε ∈ (0, 1), we have opt1+ε(F∞) = Θ(ε− 1 2 ) and opt1+ε(Fq) = Θ(ε− 1 2 ) for all q ≥ 2, where the constants in the bound do not depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2 splits into an upper bound and a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For the upper bound, we use H¨older’s inequality combined with results from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For the lower bound, we modify a construction used in [12], which obtained bounds on a finite variant of opt1(Fq) that depends on the number of trials m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The results of [9] and [12] left open the problem of determining optp(Fq) for q ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' It was not even known up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We make progress on this problem by determining opt2(F1+ε) up to a constant factor for ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Figure 1 shows the bounds and exact values known for p, q > 1 prior to the results in our paper, while Figure 2 shows the bounds and exact values known for p, q > 1 including the results in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For ε ∈ (0, 1), we have opt2(F1+ε) = Θ(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The paper [9] also discussed the problem of online learning for smooth functions of multiple variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Previous research on learning multi-variable functions [2, 6, 7] has focused on expected loss rather than worst-case loss, using models where the inputs xi are determined by a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We introduce a natural extension of the single-variable setup from [9] to multi-variable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Specifically, for q ≥ 1 and d ∈ Z+, let Fq,d be the class of functions f : [0, 1]d → R such that for any (d − 1)-tuple (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , xd−1) ∈ [0, 1]d−1 and integer i with 1 ≤ i ≤ d, the function g : [0, 1] → R given by g(x) = f(vi,x) is in Fq, where vi,x ∈ [0, 1]d is the vector formed when x is inserted at the ith position of (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , xd−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' One of the most fundamental questions about optp(Fq,d) is to determine when it is finite and when it is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We answer this question almost completely when q = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For any positive integer d, optp(F∞,d) is finite when p > d and infinite when 0 < p < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' As a corollary, it immediately follows for 0 < p < d that optp(Fq,d) = ∞ for all q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Moreover, it is easy to see that optp(F1,d) = ∞ for all positive integers d and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 3 The papers [9] and [12] also investigated worst-case mistake bounds for online learning of smooth functions when the number of trials is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In particular, using the same notation as in the first paragraph of this section, define Lp(A, f, σ, m) = m � t=1 |ˆyt − f(st)|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Moreover, define Lp(A, F, m) = sup f∈F,σ∈Sm+1 Lp(A, f, σ, m) and optp(F, m) = inf A Lp(A, F, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The paper [9] proved that opt1(Fq, m) = O(log(m)) for all q ≥ 2 and opt1(F2, m) = Ω( � log(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The paper [12] sharpened these bounds by proving that opt1(Fq, m) = Θ( � log(m)) for all q ≥ 2 and opt1(F2, m) = √ log2(m) 2 ± O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We obtain sharp bounds for online learning of smooth func- tions with a bounded number of trials when 0 < p < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In particular, these sharp bounds are also new in the single-variable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For any positive integer d and real number p with 0 < p < d, we have optp(F∞,d, m) = Θ(m1− p d ), where the constants in the bounds depend on p and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In Section 2, we focus on the single-variable setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3 in Subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 establishes the lower bound, while Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2 establishes the upper bound along with several useful lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3 focuses on proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In Section 3, we focus on the multi-variable setup, establishing various bounds on optp(Fq,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Finally, in Section 4, we discuss open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 2 Results in the single-variable setup for q ∈ (1, 2) First, we adopt some notation from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For f : [0, 1] → R, define the q-action of f, denoted by Jq[f], as Jq[f] = � 1 0 |f ′(x)|qdx, so that Fq is exactly the set of absolutely continuous f : [0, 1] → R such that Jq[f] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Also, for a nonempty set S = {(ui, vi) : 1 ≤ i ≤ m} of points in [0, 1]×R such that u1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' < um, define fS(x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 v1 x ≤ u1 vi + (x−ui)(vi+1−vi) ui+1−ui x ∈ (ui, ui+1] vm x > um and set f∅(x) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Finally, define the learning algorithm LININT as follows: on trial 0, LININT guesses ˆy0 = 0, and on trial i > 0, with the points in S = {(x0, f(x0)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , (xi−1, f(xi−1))} having been revealed and given xi, LININT guesses ˆyi = fS(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 Lower bounds for optp(Fq) First, for all p, q > 1 we have an obvious lower bound for optp(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For p, q > 1, we have optp(Fq) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The paper [9] proved that equality holds when p, q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' As we will see, equality also holds when q ∈ (1, 2) for sufficiently large values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For q ∈ (1, 2) and p > 1, we also prove a lower bound for opt2(Fq), using an adversary strategy similar to that in Theorem 8 of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For q ∈ (1, 2), we have optp(Fq) ≥ q (p2pe ln 2)(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Fix q ∈ (1, 2) and an algorithm A for learning Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Consider the following family of adversary strategies, depending on a parameter b ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The adversary picks x0 = 0 and reveals f(x0) = 0, then picks x1 = 1 and reveals f(x1) = ±b such that |ˆy1 − f(x1)| ≥ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' without loss of generality, suppose f(x1) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then for the next k = � − q log2 b q−1 � trials, the adversary recursively picks xi and f(xi) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' On trial 2 ≤ i < k + 2, the adversary sets li to be the greatest real x ∈ [0, 1] such that f(x) = 0 has been previously revealed, and similarly sets ri to be the least real x ∈ [0, 1] such that f(x) = b has been previously revealed, then sets xi = li+ri 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Upon receiving A’s guess ˆyi, the adversary reveals f(xi) = 0 or f(xi) = b such that |f(xi) − ˆyi| ≥ b 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' To see that this strategy is well-defined, note that all the xi are distinct, so it suffices to show that there exists a function f ∈ Fq which is consistent with all (xi, f(xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Indeed, take f = f{(x0,f(x0)),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=',(xk+1,f(xk+1))} which linearly interpolates between all points (xi, f(xi));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' then f has only one segment of nonzero slope, with Jq[f] = 2−k � b 2−k �q = 2k(q−1)bq ≤ 2− q log2 b q−1 ·(q−1)bq = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Thus the adversary guarantees an error of at least optp(Fq) ≥ k+1 � i=1 |f(xi) − ˆyi|p ≥ bp + k �b 2 �p ≥ − bpq log2 b 2p(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Picking b = e− 1 p yields optp(Fq) ≥ q (p2pe ln 2)(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ In particular, when p = 2 we get the following: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For q ∈ (1, 2), we have opt2(Fq) ≥ q (8e ln 2)(q − 1) > 1 (8e ln 2)(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2 Bounds for opt2(Fq) The main result of this section is that for ε ∈ (0, 1), opt2(F1+ε) = Θ(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For q ∈ (1, 2), Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3 gives a lower bound for opt2(Fq);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' we now prove an upper bound for opt2(Fq) and use this to derive the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' First, we show that a similar fact to Lemma 9 in [9] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Let u1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' < um be reals in [0, 1] and v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , vm be reals, and define S = {(u1, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , (um, vm)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then for any q ∈ (1, 2) and absolutely continuous f : [0, 1] → R such that f(ui) = vi for 1 ≤ i ≤ m, we have Jq[f] ≥ Jq[fS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' If m = 1, then Jq[fS] = 0 and the result is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Otherwise, fix some absolutely continuous f : [0, 1] → R which is consistent with the (ui, vi), and fix 1 ≤ i < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then by Jensen’s inequality, � ui+1 ui |f ′(x)|qdx ui+1 − ui ≥ �� ui+1 ui |f ′(x)|dx ui+1 − ui �q ≥ \uf8eb \uf8ed ��� � ui+1 ui f ′(x)dx ��� ui+1 − ui \uf8f6 \uf8f8 q = ���� vi+1 − vi ui+1 − ui ���� q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Thus we obtain Jq[f] ≥ � um u1 |f ′(x)|qdx ≥ m−1 � i=1 (ui+1 − ui) ���� vi+1 − vi ui+1 − ui ���� q = Jq[fS] by summing over all 1 ≤ i < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ This leads to the following useful fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For any q > 1, target function f ∈ Fq, integer m ≥ 1, and sequence of inputs x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , xm ∈ [0, 1], LININT never produces an error |ˆyi − f(xi)| > 1 on any trial i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Suppose otherwise, so that LININT produces an error |ˆyi − f(xi)| > 1 for i ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' then there exists 0 ≤ j < i such that |f(xi)−f(xj)| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Letting S = {(x0, f(x0)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , (xi, f(xi))}, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4 Jq[f] ≥ Jq[fS] ≥ |xi − xj| ���� f(xi) − f(xj) xi − xj ���� q ≥ |f(xi) − f(xj)| > 1, upon which f ̸∈ Fq, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For any q > 1 and p′ > p > 1, we have Lp′(LININT, Fq) ≤ Lp(LININT, Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' On every trial i ≥ 1, LININT produces an error |ˆyi−f(xi)| ≤ 1, so |ˆyi−f(xi)|p′ ≤ |ˆyi−f(xi)|p for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ With this, the proof proceeds similarly to the proof of Theorem 11 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Specifically, we will compare changes in Jq[fS] as new points are added to S to the squared errors (ˆy−fS(x))2 produced by LININT to bound L2(LININT, Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' This requires the following inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For reals a > 0, b > 0, q ∈ (1, 2), and x ∈ (−a, b), we have a � 1 + x a �q + b � 1 − x b �q − (a + b) ≥ 2q(q − 1) a + b x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Fix a, b, q, and define the function f(x) = a � 1 + x a �q + b � 1 − x b �q − (a + b) − 2q(q − 1) a + b x2 for x ∈ (−a, b), so that we wish to show f(x) ≥ 0 for all x ∈ (−a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Compute f ′(x) = q �� 1 + x a �q−1 − � 1 − x b �q−1 − 4(q − 1) a + b x � f ′′(x) = q(q − 1) �1 a � 1 + x a �q−2 + 1 b � 1 − x b �q−2 − 4 a + b � f (3)(x) = q(q − 1)(q − 2) � 1 a2 � 1 + x a �q−3 − 1 b2 � 1 − x b �q−3� f (4)(x) = q(q − 1)(q − 2)(q − 3) � 1 a3 � 1 + x a �q−4 + 1 b3 � 1 − x b �q−4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 6 First, we show f ′′(x) ≥ 0 for all x ∈ (−a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Note that f (4)(x) > 0 for all x and lim x→−a+ f ′′(x) = lim x→b− f ′′(x) = ∞, so f (3)(x) is increasing on (−a, b) and it suffices to check that f ′′(x) ≥ 0 at the point where f (3)(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Solving for this x yields f (3)(x) = 0 ⇐⇒ 1 a2 � 1 + x a �q−3 = 1 b2 � 1 − x b �q−3 ⇐⇒ �1 + x a 1 − x b �3−q = b2 a2 ⇐⇒ 1 + x a 1 − x b = b 2 3−q a 2 3−q ⇐⇒ x = −a 2 3−q + b 2 3−q a q−1 3−q + b q−1 3−q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' At this x,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='= q(q − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='q−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3−q + b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='q−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3−q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='�2−q � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='a1−qb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(q−1)(q−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3−q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='+ a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(q−1)(q−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3−q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='b1−q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(a + b)2−q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='a + b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='≥ q(q − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8f022−q(ab) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(q−1)(2−q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2(3−q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2(ab) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(q−1)(2q−5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2(3−q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(a + b)2−q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='a + b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='= 4q(q − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(a + b)2−q(4ab) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='q−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='a + b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='≥ 4q(q − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='(a + b)2−q(a + b)q−1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='a + b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' where all inequalities follow from the inequality (u + v)2 ≥ 4uv ⇐⇒ (u − v)2 ≥ 0 for all reals u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Since f ′′(x) is minimized here, it follows that f ′′(x) ≥ 0 for all x ∈ (−a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Since f ′(0) = 0 and f ′′(x) ≥ 0 for all x ∈ (−a, b), it follows that f ′(x) ≤ 0 for x < 0 and f ′(x) ≥ 0 for x > 0, so f(x) ≥ f(0) = 0 for all x ∈ (−a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For reals a, b ∈ (0, 1), q ∈ (1, 2), and x ̸∈ (−a, b), we have a ���x a + 1 ��� q + b ���x b − 1 ��� q − (a + b) ≥ (q − 1)|x|q (a + b)q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Fix a, b, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' by symmetry, it suffices to consider x ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Define the function f(x) = a �x a + 1 �q + b �x b − 1 �q − (a + b) − (q − 1)xq (a + b)q−1 7 for x ≥ b, so that we wish to show f(x) ≥ 0 for all x ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Since f ′(x) = q �x a + 1 �q−1 + q �x b − 1 �q−1 − q(q − 1)xq−1 (a + b)q−1 ≥ q � x a + b �q−1 + q �x b − 1 �q−1 − qxq−1 (a + b)q−1 > 0 for all x > b, f is increasing, so it suffices to show f(b) = a �a + b a �q − (a + b) − (q − 1)bq (a + b)q−1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Dividing by a + b and substituting r = a a+b, we see that this is equivalent to g(r) = 1 rq−1 − 1 − (q − 1)(1 − r)q ≥ 0 for r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' As g(1) > 0, it suffices for g′(r) = r−q(q − 1) � qrq(1 − r)q−1 − 1 � < 0 ⇐⇒ qrq−1(1 − r)q−1 < 1 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For r ∈ (0, 1), the quantity rq−1(1 − r)q−1 is maximized when r = 1 2, so it suffices to prove that q � 1 2 �2q−2 < 1 for q ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Define h(q) = q � 1 2 �2q−2, so h′(q) = 41−q(1 − q ln 4) < 0 for q ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Since h(1) = 1, we have h(q) < 1 for q ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For reals a, b ∈ (0, 1) such that a + b ≤ 1, q ∈ (1, 2), and x ̸∈ (−a, b), we have a ���x a + 1 ��� q + b ���x b − 1 ��� q − (a + b) ≥ (q − 1)|x|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Combining the above yields the following key result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Fix q ∈ (1, 2), a nonempty set S = {(u1, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , (uk, vk)} of points in [0, 1] × R, and (x, y) ∈ [0, 1] × R such that u1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' < uk, x ̸= ui for any 1 ≤ i ≤ k, and Jq � fS∪{(x,y)} � ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then Jq � fS∪{(x,y)} � − Jq[fS] ≥ (q − 1)(y − fS(x))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' First, suppose x < u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then compared to fS, the function fS∪{(x,y)} contains a new line segment of (possibly) nonzero slope between (x, y) and (u1, v1), so as |y − fS(x)| = |v1 − y| ≤ 1 (by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='5) and |u1 − x| ≤ 1, Jq � fS∪{(x,y)} � − Jq[fS] = (u1 − x) ���� v1 − y u1 − x ���� q ≥ (v1 − y)2 = (y − fS(x))2 ≥ (q − 1)(y − fS(x))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The case x > uk is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Now suppose there exists an integer 1 ≤ i < k such that ui < x < ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In this case, Jq � fS∪{(x,y)} � − Jq[fS] = (x − ui) ���� y − vi x − ui ���� q + (ui+1 − x) ���� vi+1 − y ui+1 − x ���� q − (ui+1 − ui) ���� vi+1 − vi ui+1 − ui ���� q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 8 Substituting a = x − ui, b = ui+1 − x, d = y − fS(x), and m = vi+1−vi ui+1−ui = fS(x)−vi a = vi+1−fS(x) b , we can rewrite the above as Jq � fS∪{(x,y)} � − Jq[fS] = a ����m + d a ���� q + b ����m − d b ���� q − (a + b)|m|q = |m|q � a ����1 + d ma ���� q + b ����1 − d mb ���� q − (a + b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then applying either Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='7 or Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='9 (depending on whether d m ∈ (−a, b)) yields Jq � fS∪{(x,y)} � − Jq[fS] ≥ |m|q min � 2q(q − 1) a + b � d m �2 , (q − 1) ���� d m ���� q� = min � 2q(q − 1) |m|2−q(a + b) · d2, (q − 1)|d|q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' If 0 < |m| ≤ 1, then a + b = ui+1 − ui ≤ 1 =⇒ |m|2−q(a + b) ≤ 1, while if |m| ≥ 1, then |m|2−q(a + b) ≤ |m|q(a + b) ≤ Jq [fS] ≤ Jq[fS∪{(x,y)}] ≤ 1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4, so in either case 2q(q − 1) |m|2−q(a + b) · d2 ≥ 2q(q − 1)d2 ≥ (q − 1)d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Moreover, since Jq � fS∪{(x,y)} � ≤ 1, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='5 |d| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Hence (q −1)|d|q ≥ (q −1)d2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ This directly yields the desired upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For q ∈ (1, 2), we have L2(LININT, Fq) ≤ 1 q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Fix a target function f ∈ Fq, an integer m ≥ 1, and a sequence of inputs σ = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , xm) ∈ [0, 1]m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Assume without loss of generality that all xi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For 0 ≤ i ≤ m, define Si = {(x0, f(x0)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , (xi, f(xi))}, and suppose LININT produces guesses ˆy0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , ˆym ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='10, 1 ≥ Jq[f] ≥ Jq [fSm] = m � i=1 � Jq [fSi] − Jq � fSi−1 �� ≥ (q − 1) m � i=1 (ˆyi − f(xi))2, so L2(LININT, f, σ) = m � i=1 (ˆyi − f(xi))2 ≤ 1 q − 1 for any f ∈ Fq, integer m ≥ 1, and σ ∈ [0, 1]m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Thus L2(LININT, Fq) ≤ 1 q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ Finally, combining the above with the lower bound in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3, we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For ε ∈ (0, 1), we have opt2(F1+ε) = Θ(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Combining Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='11, ε−1 8e ln 2 < 1 + ε (8e ln 2)ε ≤ opt2(F1+ε) ≤ ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Hence, opt2(F1+ε) = Θ(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ 9 It is simple to generalize the upper bound in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='11 to all p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For ε ∈ (0, 1) and p ≥ 2, we have optp(F1+ε) = O(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='6, optp(F1+ε) ≤ Lp(LININT, F1+ε) ≤ L2(LININT, F1+ε) = O(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='3 An exact result for large p In this section, we prove that for q ∈ (1, 2) and p ≥ 2 + 1 q−1, optp(Fq) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' This first requires the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For reals q ∈ (1, 2), a ∈ (0, 1), and u, v satisfying |u − v| ≥ (q−1)q−1 a(1−a) , we have a|u|q + (1 − a)|v|q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Without loss of generality, suppose u > v, so that u ≥ v + (q−1)q−1 a(1−a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' First, suppose v < 0 < u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' then |u| + |v| ≥ (q−1)q−1 a(1−a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' By the weighted power mean inequality, |u|(a|u|q−1) + |v|((1 − a)|v|q−1) |u| + |v| ≥ � a− 1 q−1 + (1 − a)− 1 q−1 |u| + |v| �−(q−1) =⇒ a|u|q + (1 − a)|v|q ≥ (|u| + |v|)q � a− 1 q−1 + (1 − a)− 1 q−1 �q−1 ≥ (q − 1)q(q−1) aq(1 − a)q � a− 1 q−1 + (1 − a)− 1 q−1 �q−1 ≥ (q − 1)q(q−1) 2q−1aq(1 − a)q max {a−1, (1 − a)−1} = (q − 1)q(q−1) 2q−1 max {aq−1(1 − a)q, aq(1 − a)q−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' By the weighted arithmetic mean - geometric mean inequality, for r ∈ (0, 1) we have rq(1 − r)q−1 = qq (q − 1)q �(q − 1)r q �q (1 − r)q−1 ≤ qq (q − 1)q \uf8eb \uf8edq · (q−1)r q + (q − 1)(1 − r) (q − 1) + q \uf8f6 \uf8f8 (q−1)+q = qq (q − 1)q � q − 1 2q − 1 �2q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Thus max � aq−1(1 − a)q, aq(1 − a)q−1� ≤ qq (q − 1)q � q − 1 2q − 1 �2q−1 = qq(q − 1)q−1 (2q − 1)2q−1 , so a|u|q + (1 − a)|v|q ≥ (q − 1)(q−1)2(2q − 1)2q−1 2q−1qq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 10 Consider f(q) = (q − 1)2 ln(q − 1) + (2q − 1) ln(2q − 1) − (q − 1) ln 2 − q ln q over q ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Note that f ′(q) = (2(q − 1) ln(q − 1) + (q − 1)) + (2 ln(2q − 1) + 2) − ln 2 − (ln q + 1) = 1 − ln 2 + 2(q − 1) ln(q − 1) + (q − 1 − ln q) + 2 ln(2q − 1) ≥ 1 − ln 2 + 2(q − 1) ln(q − 1) + 2 ln(2q − 1), as ex ≥ 1 + x implies that x ≥ ln(1 + x) for x > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Since x ln x is decreasing on � 0, 1 e � and increasing on � 1 e, ∞ � (so in particular x ln x ≥ − 1 e for x > 0), q ∈ (1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='004] =⇒ f ′(q) ≥ 1 − ln 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='008 ln 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='004 > 0 q ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='004, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='055] =⇒ f ′(q) ≥ 1 − ln 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='11 ln 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='055 + 2 ln 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='008 > 0 q ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='055, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='12] =⇒ f ′(q) ≥ 1 − ln 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='24 ln 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='12 + 2 ln 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='11 > 0 q ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='12, 2) =⇒ f ′(q) ≥ 1 − ln 2 − 2 e + 2 ln 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='24 > 0, so for all q ∈ (1, 2), f ′(q) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' As lim q→1+ f(q) = 0, it follows that f(q) > 0 for q ∈ (1, 2), so a|u|q + (1 − a)|v|q ≥ ef(q) > 1 whenever v < 0 < u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Now suppose v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' As |x|q is increasing for x ≥ 0, a|u|q + (1 − a)|v|q ≥ a �(q − 1)q−1 a(1 − a) �q = (q − 1)q(q−1) aq−1(1 − a)q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Using the work above, (q − 1)q(q−1) aq−1(1 − a)q ≥ (q − 1)q(q−1) max {aq−1(1 − a)q, aq(1 − a)q−1} > 2q−1 > 1, so the inequality holds whenever v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' The case u ≤ 0 is identical, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ With this, we have the following key result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Fix q ∈ (1, 2), a nonempty set S = {(u1, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , (uk, vk)} of points in [0, 1] × R, and (x, y) ∈ [0, 1] × R such that u1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' < uk, x ̸= ui for any 1 ≤ i ≤ k, and Jq � fS∪{(x,y)} � ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Let p = 2 + 1 q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then Jq � fS∪{(x,y)} � − Jq[fS] ≥ |y − fS(x)|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We first show that |y − fS(x)| > (q − 1)q−1 and x ∈ (u1, uk) cannot both hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Suppose otherwise, so that there exists an integer 1 ≤ i < k such that ui < x < ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' We will derive a contradiction by showing Jq � fS∪{(x,y)} � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Clearly Jq � fS∪{(x,y)} � ≥ (x − ui) ���� y − vi x − ui ���� q + (ui+1 − x) ���� vi+1 − y ui+1 − x ���� q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Substituting a = x − ui, b = ui+1 − x, d = y − fS(x), and m = vi+1−vi ui+1−ui as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='10, this rewrites as Jq � fS∪{(x,y)} � ≥ a ����m + d a ���� q + b ����m − d b ���� q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' 11 As a + b = ui+1 − ui ≤ 1 and q > 1, Jq � fS∪{(x,y)} � ≥ a a + b ����(a + b)m + d(a + b) a ���� q + b a + b ����(a + b)m − d(a + b) b ���� q , and because |d| > (q − 1)q−1 =⇒ ����d(a + b) �1 a + 1 b ����� = |d|(a + b)2 ab ≥ (q − 1)q−1 a a+b · b a+b , applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='14 yields Jq � fS∪{(x,y)} � > 1, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Thus at least one of |y − fS(x)| ≤ (q − 1)q−1 and x ̸∈ (u1, uk) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' If |y − fS(x)| ≤ (q − 1)q−1 =⇒ (q − 1)(y − fS(x))2 ≥ |y − fS(x)|p, the result follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Otherwise, assume without loss of generality that x < u1 (the case x > uk is similar);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' then Jq � fS∪{(x,y)} � − Jq[fS] = (u1 − x) ���� v1 − y u1 − x ���� q ≥ |v1 − y|q ≥ |v1 − y|p = |y − fS(x)|p by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='5 (as q < 2 < p) and the result holds in this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ This immediately yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For any reals q > 1 and p ≥ 2 + 1 q−1, we have optp(Fq) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='6, it suffices to prove that for q ∈ (1, 2) and p = 2+ 1 q−1, Lp(LININT, Fq) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Fix p = 2 + 1 q−1, a target function f ∈ Fq, an integer m ≥ 1, and a sequence of inputs σ = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , xm) ∈ [0, 1]m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Assume without loss of generality that all xi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For 0 ≤ i ≤ m, define Si = {(x0, f(x0)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , (xi, f(xi))}, and suppose LININT produces guesses ˆy0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , ˆym ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='15, 1 ≥ Jq[f] ≥ Jq [fSm] = m � i=1 � Jq [fSi] − Jq � fSi−1 �� ≥ m � i=1 |ˆyi − f(xi)|p, so Lp(LININT, f, σ) ≤ 1 for any f ∈ Fq and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Thus Lp(LININT, Fq) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content='4 Sharp bounds for p ∈ (1, 2) The paper [9] showed that opt1+ε(Fq) = O(ε−1) for all ε ∈ (0, 1) and q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In this section, we first improve their upper bound by proving that opt1+ε(Fq) = O(ε− 1 2) for all ε ∈ (0, 1) and q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Then we show that this bound is sharp by proving that opt1+ε(Fq) = Ω(ε− 1 2) for all q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' In order to prove the upper bound, we use two lemmas from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' To state the lemmas and prove our upper bound, we use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Let x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , xm be any sequence of distinct elements of [0, 1], and let f ∈ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' Let ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , ˆym be LININT’s predictions on trials 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQfePy_/content/2301.01434v1.pdf'} +page_content=' For each i > 1, let di = minj 𝑡ℎ, +0, 𝑧 ≤ 𝑡ℎ, +(3) +where 𝑡ℎ is the pre-defined threshold. GM picks exemplar patches centered on the corner points of +the averaged edge map rather than traditional uniform selection. After exemplar patches selection, +we perform global patch matching using the convolution operation of deep learning frameworks. +As shown in Fig. 2 a, each exemplar edge patch (∇ℎ 𝒑 and ∇𝑣 𝒑) acts as a convolution kernel +to convolute the edge map. Note that the convolution operator has slightly different from the +convolution in mathematics. It specifically refers to the ‘Conv2d’ in deep convolutional networks. +Benefiting from the bloom of deep learning framework, we can perform parallel convolution +by concatenating the exemplar edge patches ∇ 𝒑 into a multi-channel convolutional kernel ∇𝑷. +Figure 2 b demonstrates that GM enables increasing matching precision and finding more +structurally similar patches than BM. In the case of multi-channel images, both the edge maps +and the exemplar edge patches are of three dimensions. We can conduct the batch convolution +with the multi-channel edge maps as batch inputs. The similarity heat map is obtained as +𝑯 = +∑︁ +𝑎=(ℎ,𝑣),𝑏=(+,−) +Conv2d +� +∇𝑏 +𝑎𝒙, ∇𝑏 +𝑎𝑷 +� +. +(4) +The sizes of ∇𝑏 +𝑎𝒙, ∇𝑏 +𝑎𝑷, and 𝐻 are H×W×C, P×P×C×N, and (H-P+1)×(W-P+1)×N respectively, +where H×W is the size of input images, P×P is the size of extracted patches, C represents the + +Block matching +Corner-based block matching +Global matching +a +Top 30 most similar patches to the exemplar patch via block matching +b +... +... +#1 +#2 +#3 +#4 +#27 +#28 +#29 +#30 +#1 +#2 +#3 +#4 +#27 +#28 +#29 +#30 +Top 30 most similar patches to the exemplar patch via global matching +Extracted similar patches using block and global matching +Comparison of various exemplar patch selection strategies +Searching +window +Searching +window +Center of +exemplar patch +Center of +exemplar patch +Edge +Edge +Block +matching +Global +matching +Similarity heat map +a +-10 +-5 +0 +5 +10 +15 +20 +-10 +-5 +0 +5 +10 +15 +20 +0 +50 +100 +150 +200 +250 +300 +350 +0 +50 +100 +150 +200 +250 +300 +350 +Comparison between block and global matching +Exemplar +patch +Exemplar +edge patch +b +Similarity heat map +0 +50 +100 +150 +200 +250 +300 +350 +0 +50 +100 +150 +200 +250 +300 +350 +-1 0 +1 +-2 0 +2 +-1 0 +1 +-1 0 +1 +-2 0 +2 +-1 0 +1 +1 +2 +1 +0 +0 +0 +-1 -2 -1 +1 +2 +1 +0 +0 +0 +-1 -2 -1 +x +p +h ++ + x +h ++ + p +v ++ + x +v ++ + p +H +h + +v + +3 +v +− + x +v +− + p +d +Global structural matching via neural-based convolution +c +Structural matching using edge maps +0×1=0 +1×1 =1 +0×0=0 +h +− + x +h +− + p +Fig. 2. Comparison between BM and GM. a, BM searches similar patches by measuring +the MSE of each patch against the exemplar patch within a local window. GM obtains +the similarity heat maps by correlating the edge maps with exemplar edge patches. The +similarity heat map for the exemplar patch is the summation heat maps corresponding +to horizontal and vertical edge maps which are obtained by neural-based Conv2d +operation. b, Comparison between extracted similar patches from BM and the reported +GM. The patches extracted by GM have higher structural similarity to the exemplar +patch in a. +input channels, and N denotes the number of exemplar patches. GM obtains all the heat maps of +N exemplar patches with 4 Conv2d operations while BM requires N times KNN-based matching. +So the neural-based convolution contributes to reduce computation load and boost the running +efficiency of GLR. Besides, modern deep learning frameworks provide several acceleration +approaches to speed up the Conv2d layer including Im2Col + matrix manipulation [25], fast +Fourier transform (FFT), Winograd [26], etc. A larger value in the heat map means that the image +patch centered at the corresponding position has a higher structural similarity to the exemplar +patch. +We use the operator � +𝑹𝑖 to represent the 𝑖-th global matching operation. The resulted grouped + +T1 +0口 +十工matrix is � +𝑹𝑖𝒙, where 𝒙 denotes the to-be-update image. The objective function of low-rank +approximation is +𝒙 = arg min +𝒙 +∑︁ +𝑖 +rank +� +� +𝑹𝑖𝒙 +� +, s.t. 𝒚 = 𝚽𝒙, +(5) +where 𝒚 is the measurement, and 𝚽 stands for the sensing matrix. We adopt WNNM [15] for +solving the above low-rank approximation in Eq. (5). The reconstructed image is derived by +the iterative optimization framework such as generalized alternative projection (GAP) [27] and +alternating direction multiplier (ADMM) [28]. +3. +Results +We applied GLR and the existing CS-based algorithms on both simulation and experiment data +of three computational imaging tasks including coded aperture compressive temporal imaging +(CACTI) [29], magnetic resonance imaging (MRI) [30], and multispectral filter array (MSFA) +demosaicing [31]. In the following evaluations, we employed the peak signal-to-noise ratio +(PSNR) and structural similarity index (SSIM) [32] to quantify reconstruction accuracy. In the +evaluations of the Results and the following Discussion sections, GLR and NLR shared the same +hyper-parameters. The deep learning platform MatConvNet [25] is utilized for convolution-based +GM. All the calculations are done on a desktop PC with an Intel i7-9700K CPU and 64G RAM +for a fair comparison between GLR and conventional nonlocal low-rank techniques. We believe +that GLR can be faster on convolution-accelerating hardware. +3.1. +Coded aperture compressive temporal imaging +CACTI [29] is a typical temporal compressive imaging modality. As shown in Fig. 2 a, +CACTI compresses temporally continuous 2D scenes into one snapshot via aperture coding. +Mathematically, it can be modeled as +𝒚 = +𝑁 +∑︁ +𝑖=1 +𝑨𝑖𝒙𝑖 +(6) +where 𝒙𝒊 is the scene at moment 𝒊, 𝑨 denotes the spatial variant mask generated by the digital +micro-mirror device (DMD), and 𝒚 represents the measurement. +We test the corner-based exemplar patch selection in the state-of-the-art NLR-based algorithm +DeSCI [20] for CACTI reconstruction. Denote DeSCI with corner-based BM as DeSCI-C. +Accept from the corner-based exemplar patches, a modest number of uniformly selected patches +are included. The interval of two neighbor exemplar patches is 3 times larger than vanilla BM. +The common-used benchmark datasets (Kobe, Runner, Traffic, Drop) [18,33] are +applied for simulation. One snapshot of the simulation data encodes 8 frames of the scenes at +256 × 256 resolution. The commonly used CS-based algorithms GAP-TV [27] and Plug-and-play +(PnP) [18] are included as the baselines. FFDNet [34] is employed for the learning-based +regularization of PnP. All the hyper-parameters of above-mentioned algorithms are set as the +default values provided by ref. [18]. Sub-figures 3 b & c show the visual and quantitative results +of the simulation data. It can be seen that DeSCI with corner-based BM achieved competitive +results compared to conventional DeSCI. The differences between the average PSNR and SSIM of +DeSCI and DeSCI-C are 0.27 dB and 0.002, respectively. This small numerical difference is hard +to be noticed in visual perception as shown in Fig. 3 b. Furthermore, we build a proof-of-concept +prototype shown in Fig. 3 e. The prototype consists of one camera lens, two relay lenses, one +DMD to generate the spatial variant masks, and one CCD sensor to capture the measurement. +With the real-captured measurement encoded with 10 masks, we retrieved 10 temporal successive +frames of the target scene. It can be seen from Fig. 3 f that DeSCI and DeSCI-C can recover + +a +Coded aperture compressive temporal imaging +Camera lens +Relay lens 1 +DMD +Relay lens 2 +Sensor +Kobe #2 +GAP-TV +DeSCI +DeSCI-C +Runner #5 +Traffic #3 +Drop #7 +GAP-TV +PnP +DeSCI +DeSCI-C +PSNR +27.37 +31.67 +35.28 +35.04 +SSIM +0.8985 +0.9301 +0.9752 +0.9739 +PSNR +20.96 +24.21 +28.11 +27.82 +SSIM +0.7351 +0.8389 +0.9321 +0.9261 +PSNR +28.4 +32.47 +38.77 +38.53 +SSIM +0.9044 +0.9248 +0.9697 +0.9685 +PSNR +34.43 +40.83 +43.16 +42.85 +SSIM +0.9694 +0.9841 +0.9914 +0.9906 +Drop +Image +Kobe +Traffic +Runner +b +c +d +Objective lens +DMD Relay lens CCD +Measurement +e +f +Reconstructed images from simulation data +Quantitative results on simulation data +Experiment setup +Reconstructed images from experiment data +#1 +#4 +#7 +#10 +DeSCI +DeSCI-C +PnP +PnP +Measurement +2728 +3599 +6327 +846 +2264 +3111 +0 +2000 +4000 +6000 + DeSCI + DeSCI-C +Matching +Low-rank +Total +Time (min) +77 +53 +133 +10 +23 +35 +Average running time +Fig. 3. Reconstruction results for CACTI simulations and experiments. a, The imaging +scheme of CACTI. b, Reconstruction results on benchmark simulation datasets (Kobe, +Runner, Traffic, Drop). c, The average reconstruction accuracy on the test +data of different algorithms. d, The average running time of nonlocal algorithms with +and without corner-based exemplar patch selection. e, The proof-of-concept prototype +of the CACTI system. f, The reconstruction results of the real-captured measurement +via different algorithms. +more sharp textures compared to the prevalent PnP method. DeSCI-C obtained similar results +as DeSCI, which proved the effectiveness of corner-based BM. However, DeSCI-C has higher +reconstruction efficiency due to fewer exemplar patches. The matching speed of corner-based +BM is 7.7 times faster compared to vanilla BM, and the total reconstruction speed of DeSCI-C is +3.8 times faster than that of DeSCI, according to the average running times listed in Fig. 3 d. All +the above simulations and experiments demonstrate the efficiency of structural feature attention + +国30 +8130 +81 +'0030 +81 +00QDQD网from corner-based exemplar patch selection. +3.2. +Magnetic resonance imaging +The Fourier spectrum of natural images has the nature of sparsity. One can reconstruct target +scenes with magnetic resonance data even below the Nyquist sampling ratio via CS algorithms [37]. +As shown in Fig. 4 a, the Fourier domain compressive imaging model is formulated as +𝒚 = 𝑴ℱ(𝒙) +(7) +where 𝒙 denotes the target scene, ℱ is the Fourier transform function, 𝑴 represents the sampling +mask, and 𝒚 is the corresponding measurement. We evaluated the reported reconstruction +technique on magnetic resonance imaging (MRI). In the experiments, the sampling strategy +is set as the prevalent radial mask. We applied Dong et al.’s NLR algorithm [16] for MRI +reconstruction. GLR is formed by replacing BM with GM in NLR. +Figure 4 shows the experiment results. +GLR holds a similar performance to NLR in +a +Reconstructed images from MRI data +c +Quantitative reconstruction results +d +Average running time +Time (s) +Matching +Low-rank +Total +b +Fourier spectrum +MRI principle +0.1 +0.2 +0.3 +0.4 +28 +32 +36 +40 +44 +Ratio + GAP-TV + DCT + NLR + GLR +PSNR +NLR +GLR +Ratio = 13% +Ratio = 20% +Ratio = 29% +Ground Truth +DCT +GLR +NLR +Close-up +Fig. 4. MRI reconstruction results. a, The mathematical principle of MRI. We focus +on the radial sampling strategy in the Fourier domain. b, The visual comparison of +reconstructed images (256×256 pixels) from Dong et al.’s [16] dataset and fastMRI +[35,36]. c, The quantitative results of different reconstruction techniques. d, Average +running time for each procedure of NLR and GLR. + +reconstruction accuracy (Fig. 4 b, c). For single-channel reconstructions, GM exhibits about 7 +times acceleration against BM. The low-rank approximation of GLR shows a 3 times speed-up +compared to the conventional approach. GLR has an average 4.5 times improvement in running +time compared to NLR. Specifically, the speed-up of low-rank approximation mainly comes from +the reduction of exemplar patches by corner-based exemplar patch selection. The acceleration +of the matching process results from the joint promotion of both corner-based exemplar patch +selection and convolution-based structural matching. +3.3. +Multi-spectral filter array demosaicing +Multi-spectral filter array (MSFA) is an extension of the conventional color camera that covers a +Bayer filter [39]. MSFA cameras can acquire multi-spectral information in one snapshot [40]. +Compared to other existing multi-spectral imaging systems [41], it provides an integrated, +low-cost, exact registration, and full frame rate solution [42,43]. The imaging model is denoted +as +𝒚 = +𝑁 +∑︁ +𝑖=1 +𝑨𝑖𝒙𝑖 +(8) +where 𝑨𝑖 stands for the filter mask of the 𝑖-th spectral channel. The masks are designed to be +orthogonal, +𝑨𝑖⊙𝑨 𝑗 = 0 +(9) +where ⊙ denotes the Hadamard product. However, there exhibits a fundamental compromise +between spatial and spectral resolution for MSFA camera. +There is a rising challenge to +reconstruct full-resolution images from multi-spectral mosaic data because it is more ill-posed +than demosaicing for RGB cameras (Fig. 5 a). Bian et al. [44] experimentally proved that +NLR algorithms with multi-channel block matching can achieve state-of-the-art performance +for multi-spectral demosaicing. Inheriting Bian et al.’s framework, GLR replaces the matching +strategy with multi-channel global matching (see Supplement 1). Here we conduct a series of +simulations and experiments to demonstrate the superiority of GLR over conventional NLR. +As shown in both simulations and experiments of Fig. 5 b, c, and e, the visual and numerical +results indicate GLR outperforms NLR with various channel numbers and types of MSFA. The +recovered closeups in Fig. 5 b show that global matching contributes to the recovery of more +delicate textures and details than conventional block matching. In multi-channel cases, GM +presents obvious efficiency superiority over vanilla BM. GM provides more speed boost as the +channel number increases (Fig. 5 d). It is mainly attributed to the high efficiency of batch +convolution. For large-scale reconstruction with more than 256×256 spatial resolution and 6 +channels, GM achieves up to above one magnitude of improvement in running time (Fig. 5 d, e). +Besides, with corner-based exemplar patch selection, GM extracts fewer groups of self-repetitive +patches than BM. Corner-based exemplar patch selection enables decreasing the running time +of both matching and low-rank approximation. Combining the above two improvements, GLR +achieves about 4 to 8 times efficiency gain while obtaining state-of-the-art performance. Taking +the single-channel cases of MRI into consideration, GLR exhibits an increasing priority over +NLR in terms of both running time and reconstruction accuracy as the channel number goes +larger. Experiments of real-captured data validated GLR’s advancement in both accuracy and +efficiency (Fig. 6). +4. +Conclusion and discussion +In this work, we engaged in a generalized global low-rank optimization for computational +reconstruction. It enables fast calculation and global self-similarity regularization, offering an +impetus to efficient nonlocal techniques. As validated by extensive simulations and experiments +on the above three computational imaging modalities (CACTI, MRI, and MSFA demosaicing), the + +Multi-spectral filter array demosaicing +b +c +Random MSFA +Regular MSFA +BT MSFA +GAP-TV +DeSCI +NLR +GLR +400nm +400nm +400nm +400nm +400nm +d +0 +0.5 +a +Demosaicing +Quantitative results of reconstruction +Visual comparison of reconstruction +Average running time of different channel numbers at 256×256 resolution +MSFA +Method +4 channels +6 channels +9 channels +BTES +13.45 +12.43 +11.66 +GAP-TV +29.81 +27.60 +25.89 +DeSCI +31.91 +28.86 +26.56 +NLR +38.30 +35.53 +30.53 +GLR +37.18 +35.98 +32.27 +BTES +20.35 +12.53 +11.72 +GAP-TV +31.36 +28.07 +26.97 +DeSCI +31.98 +28.33 +27.19 +NLR +36.30 +30.90 +28.66 +GLR +37.35 +31.39 +29.44 +BTES +29.47 +28.07 +26.56 +GAP-TV +31.36 +29.50 +27.81 +DeSCI +31.98 +30.37 +28.61 +NLR +36.28 +34.41 +30.85 +GLR +37.42 +35.95 +32.19 + +Random + +Regular + +BT +Time +(min) +Matching +Low rank +Total +e +Average running time of 9-channel BT MFSA demosaicing at different resolutions +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +160 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +160 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +2 +4 +6 +8 +10 +12 +14 +16 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +2 +4 +6 +8 +10 +12 +14 +16 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +GLR +NLR +channels +Time (min) +Reconstruction results for real-captured data +a +NLR +GLR +470nm +590nm +NLR +GLR +470nm +590nm +NLR +GLR +NLR +GLR +Measurement #2 +Measurement #1 +b +Average running time +Time (min) +Matching +Low-rank +Total +203 +24 +227 +16 +9 +26 +0 +50 +100 +150 +200 +250 + NLR + GLR +Fig. 5. The demosaicing simulations under different MSFAs. a, The demosaicing +process for a multispectral mosaic camera. b, The PSNR values of reconstructed +multi-channel images. c, The reconstruction images for various 6-channel MSFAs. +The closeups and corresponding error maps are shown on the right side of each image. +d, Average running time (200 iterations in total) of different channels at 256 × 256 +resolution for each procedure of NLR and GLR. e, Average running time of 9-channel +BT MSFA demosaicing at different resolutions. + +工151.9 +102.5 +68.8 +19.0 +14.8 +12.815.1 +12.2 +9.3 +7.5 +5.6 +4.6136.6 +90.1 +59.3 +11.3 +8.9 +8.0579.6 +400 +300 +149.2 +200 +84.9 +100 +19.7 +20.4 +512 +5.3 +solution59.2 +60 +50 +Ot +30 +19.7 +15.6 +20 +70 +8.0 +3.1 +512 +1.4520.2 +AOO +300 +133.3 +200 +65.0 +16.3 +12.1 +512 +NLR +3.7 +- +256 +GLR +iiMulti-spectral filter array demosaicing +b +c +Random MSFA +Regular MSFA +BT MSFA +GAP-TV +DeSCI +NLR +GLR +400nm +400nm +400nm +400nm +400nm +d +0 +0.5 +a +Demosaicing +Quantitative results of reconstruction +Visual comparison of reconstruction +Average running time of different channel numbers at 256×256 resolution +MSFA +Method +4 channels +6 channels +9 channels +BTES +13.45 +12.43 +11.66 +GAP-TV +29.81 +27.60 +25.89 +DeSCI +31.91 +28.86 +26.56 +NLR +38.30 +35.53 +30.53 +GLR +37.18 +35.98 +32.27 +BTES +20.35 +12.53 +11.72 +GAP-TV +31.36 +28.07 +26.97 +DeSCI +31.98 +28.33 +27.19 +NLR +36.30 +30.90 +28.66 +GLR +37.35 +31.39 +29.44 +BTES +29.47 +28.07 +26.56 +GAP-TV +31.36 +29.50 +27.81 +DeSCI +31.98 +30.37 +28.61 +NLR +36.28 +34.41 +30.85 +GLR +37.42 +35.95 +32.19 + +Random + +Regular + +BT +Time +(min) +Matching +Low rank +Total +e +Average running time of 9-channel BT MFSA demosaicing at different resolutions +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +160 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +160 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +2 +4 +6 +8 +10 +12 +14 +16 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +2 +4 +6 +8 +10 +12 +14 +16 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +GLR +NLR +channels +Time (min) +4 +6 +9 +0 +20 +40 +60 +80 +100 +120 +140 +GLR +NLR +channels +Time (min) +Reconstruction results for real-captured data +a +NLR +GLR +550nm +NLR +GLR +Meas #2 +b +Average running time +Time (min) +Matching Low-rank +Total +203 +24 +227 +16 +9 +26 +0 +50 +100 +150 +200 +250 + NLR + GLR +Meas #1 +Fig. 6. The demosaicing results for real-captured data. The measurement captured with +a 4×4 IMEC snapshot camera is provided by Feng et al. [38]. The measurements are of +256×256×16 pixels. a, Visual comparisons between reconstruction results of NLR and +GLR. b, Average running time of both algorithms. +BM-C +BM-C-U +BM +GM +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Ground truth +Error maps of reconstructed images +Reconstructed spectra +Comparison of running time +a +b +c +Toy +Character +Matching +Low-rank +Total + BM + BM-C-U + BM-C + GM +4899 +827 +5726 +2626 +631 +3257 +803 +385 +1187 +505 +342 +847 +Matching +Low-rank +Total +Time +(min) + BM + BM-C-U + BM-C + GM +4899 +827 +5726 +2626 +631 +3257 +803 +385 +1187 +505 +342 +847 + GT + BM + BM-C-U + BM-C + GM + BM + BM-C-U + BM-C + GM +93 +13 +106 +42 +10 +52 +11 +6 +17 +9 +4 +14 +Toy +Character +400 +500 +600 +700 + GT + BM + BM-C-U + BM-C + GM +Fig. 7. Comparison of various matching strategies. a, The spectra of the randomly +selected point (marked with a hexagon in the ground truth image of b) in the reconstructed +multispectral scene. b, Error maps of reconstructed images with different matching +strategies "BM-C" and "BM-C-U" stand for block matching with corner-based exemplar +patches and with both corner-based exemplar patches and the same number of uniform +ones. c, Comparison of running time of nonlocal techniques with different matching +strategies. +core contributions of GLR (structural feature attention and neural-based matching) can promote + +工520.2 +AOO +300 +133.3 +200 +65.0 +16.3 +12.1 +512 +NLR +3.7 +- +256 +GLR +ii151.9 +102.5 +68.8 +19.0 +14.8 +12.815.1 +12.2 +9.3 +7.5 +5.6 +4.6136.6 +90.1 +59.3 +11.3 +8.9 +8.0579.6 +400 +300 +149.2 +200 +84.9 +100 +19.7 +20.4 +512 +5.3 +solution59.2 +60 +50 +Ot +30 +19.7 +15.6 +20 +70 +8.0 +3.1 +512 +1.4工COJOL2the running efficiency of nonlocal techniques by nearly an order of magnitude. Benefiting +from inherent fusion of deep learning strategies and iterative optimization, GLR exhibits +unique advantages in efficient and high-fidelity reconstruction for high-dimensional large-scale +computational imaging tasks. +4.1. +From block matching to global matching +To reveal the core advantages of GLR, we evaluated GM, uniform, corner-based, and corner- +based+uniform BM for 6-channel BT MSFA demosaicing. For the corner-based+uniform BM, the +number of uniformly picked exemplar patches is set to be similar to that of corner-based patches. +As shown in Fig. 7, GLR/GM outperforms other approaches in both reconstruction accuracy +and efficiency. The reconstructed images with GM exhibit more sharp textures (Fig. 7 b) and +high-fidelity spectra (Fig. 7 a) especially in the regions of rich textures. NLR with corner-based +block matching has less running time than with uniform block matching, due to fewer exemplar +patches. However, too few self-repetitive patches for block matching leads to reduced accuracy +(Fig. 7 b). With similar numbers of corner-based exemplar patches, the comparisons between +GM and corner-based BM indicate that GM outperforms BM, demonstrating the significance of +global structural self-similarity. +4.2. +Combining deep learning and optimization +By directly incorporating the neural network as a regularization operator into optimization, PnP +[18,46] alternatives iterations between the model-based fidelity and learning-based regularization +(Fig. 8 a). The optimization framework improves the interpretability of neural networks. PnP +techniques have achieved tremendous success in computational reconstruction. However, the +enhancing process of the network in PnP iterations remains unexplainable due to the black-box +nature of neural networks. Different from PnP, GLR presents the innovative neural strategies that +combine feature detection and neural computation for explainable global low-rank optimization. +The use of the ‘Conv2d’ neural operator here has clear physical significance. The ‘Conv2d’ +between edge maps and exemplar edge patches measures the similarity between these inputs. +Besides, the commonly used enhancing networks (such as FFDNet [34]) for PnP have limited +effective receptive field, while GLR holds global perception. As shown in Fig. 8 b, GLR +outperforms PnP, end-to-end networks (U-Net [47] and TENet [48]), and conventional NLR in +reconstruction accuracy. Above all, GLR provides a generalized and effective perspective of +explainable applications of deep learning strategies. +4.3. +Outlook +GLR can be extended to various challenging reconstruction tasks, especially in high-dimensional +and high-precision applications. To further improve the calculating speed, we can run the +technique on convolution-accelerate hardware such as FPGA [49,50] and specific SOC [51,52]. +Next, the low-complexity low-rank approximation deserves further study under the framework. +The low-rank network may be a promising alternative for further promoting efficiency. What’s +more, the collision between deep learning and classical optimization will lead to a spark of +inspiration. The bloom of learning-based methods can drive the progress of optimization +techniques. +References +1. +Y. Altmann, S. McLaughlin, M. J. Padgett, V. K. Goyal, A. O. Hero, and D. Faccio, “Quantum-inspired computational +imaging,” Science 361, eaat2298 (2018). +2. +J. Wu, Y. Guo, C. Deng, A. Zhang, H. Qiao, Z. Lu, J. Xie, L. Fang, and Q. Dai, “An integrated imaging sensor for +aberration-corrected 3d photography,” Nature 612, 62–71 (2022). +3. +A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” +Science 343, 58–61 (2014). + +Model-based fidelity +a +b +Ground truth +U-Net +TENet +NLR +GLR +PnP +Ch.4 +Ch.6 +Ch.9 +Channels Method PSNR +SSIM +UNet +31.25 +0.968 +TENet +35.28 +0.984 +PnP +37.12 +0.989 +NLR +40.05 +0.995 +GLR +41.29 +0.996 +UNet +30.30 +0.848 +TENet +34.74 +0.983 +PnP +36.15 +0.989 +NLR +40.58 +0.996 +GLR +43.08 +0.997 +UNet +27.67 +0.922 +TENet +31.91 +0.933 +PnP +34.29 +0.946 +NLR +35.38 +0.961 +GLR +35.63 +0.966 +9 +4 +6 +Combing deep learning and optimization: +from direct neural network operators to innovative neural strategies +Error maps of reconstruction images +0 +0.02 +0.04 +0.06 +0.08 +0.1 +c +Quantitative results +Neural network: +Direct operator in alternating iterations +Neural strategies: +Feature attention & Neural computation +Model-based fidelity +Fig. 8. Comparison between end-to-end networks, PnP, and GLR. a, The diagrams of +PnP (direct neural network operator) and GLR (neural strategies) schemes that combine +deep learning and optimization. b, The error maps of BT MSFA demosaicing images +from Monno’s dataset [43]. c, The quantitative comparisons of different methods. We +compare the reported GLR with end-to-end networks (U-Net and TENet trained on +CAVE [45] and Monno’s dataset), PnP (FFDNet as the regularization network), and +conventional NLR. +4. +P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” +Nat. communications 6, 1–6 (2015). +5. +G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution fourier ptychographic microscopy,” Nat. +photonics 7, 739–745 (2013). +6. +C. Qiao, D. Li, Y. Guo, C. Liu, T. Jiang, Q. Dai, and D. Li, “Evaluation and development of deep neural networks for +image super-resolution in optical microscopy,” Nat. Methods 18, 194–202 (2021). +7. +M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” +Nature 555, 338–341 (2018). +8. +C. Saunders, J. Murray-Bruce, and V. K. Goyal, “Computational periscopy with an ordinary digital camera,” Nature +565, 472–475 (2019). +9. +J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10, 409–483 (2018). +10. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D: nonlinear +phenomena 60, 259–268 (1992). +11. A. N. Tikhonov, A. S. Leonov, and A. G. Yagola, “Nonlinear ill-posed problems,” (1997). +12. A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in 2005 IEEE computer society +conference on computer vision and pattern recognition (CVPR’05), vol. 2 (Ieee, 2005), pp. 60–65. +13. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising with block-matching and 3d filtering,” in Image + +D6COG0processing: algorithms and systems, neural networks, and machine learning, vol. 6064 (SPIE, 2006), pp. 354–365. +14. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative +filtering,” IEEE Trans. on image processing 16, 2080–2095 (2007). +15. S. Gu, L. Zhang, W. Zuo, and X. Feng, “Weighted nuclear norm minimization with application to image denoising,” +in Proceedings of the IEEE conference on computer vision and pattern recognition, (2014), pp. 2862–2869. +16. W. Dong, G. Shi, X. Li, Y. Ma, and F. Huang, “Compressive sensing via nonlocal low-rank regularization,” IEEE +transactions on image processing 23, 3618–3632 (2014). +17. C. Zhang, W. Hu, T. Jin, and Z. Mei, “Nonlocal image denoising via adaptive tensor nuclear norm minimization,” +Neural Comput. Appl. 29, 3–19 (2018). +18. X. Yuan, Y. Liu, J. Suo, and Q. Dai, “Plug-and-play algorithms for large-scale snapshot compressive imaging,” in +Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2020), pp. 1447–1457. +19. X. Yuan, D. J. Brady, and A. K. Katsaggelos, “Snapshot compressive imaging: Theory, algorithms, and applications,” +IEEE Signal Process. Mag. 38, 65–88 (2021). +20. Y. Liu, X. Yuan, J. Suo, D. J. Brady, and Q. Dai, “Rank minimization for snapshot compressive imaging,” IEEE +transactions on pattern analysis machine intelligence 41, 2990–3006 (2018). +21. G. Barbastathis, A. Ozcan, and G. Situ, “On the use of deep learning for computational imaging,” Optica 6, 921–943 +(2019). +22. C. Belthangady and L. A. Royer, “Applications, promises, and pitfalls of deep learning for fluorescence image +reconstruction,” Nat. methods 16, 1215–1225 (2019). +23. G. Wang, J. C. Ye, and B. De Man, “Deep learning for tomographic image reconstruction,” Nat. Mach. Intell. 2, +737–748 (2020). +24. C. Rudin, “Stop explaining black box machine learning models for high stakes decisions and use interpretable models +instead,” Nat. Mach. Intell. 1, 206–215 (2019). +25. A. Vedaldi and K. Lenc, “Matconvnet: Convolutional neural networks for matlab,” in Proceedings of the 23rd ACM +international conference on Multimedia, (2015), pp. 689–692. +26. A. Lavin and S. Gray, “Fast algorithms for convolutional neural networks,” in Proceedings of the IEEE conference on +computer vision and pattern recognition, (2016), pp. 4013–4021. +27. X. Yuan, “Generalized alternating projection based total variation minimization for compressive sensing,” in 2016 +IEEE International Conference on Image Processing (ICIP), (IEEE, 2016), pp. 2539–2543. +28. S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein et al., “Distributed optimization and statistical learning via the +alternating direction method of multipliers,” Found. Trends Mach. learning 3, 1–122 (2011). +29. P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive +temporal imaging,” Opt. express 21, 10526–10545 (2013). +30. N. Weiskopf, L. J. Edwards, G. Helms, S. Mohammadi, and E. Kirilina, “Quantitative magnetic resonance imaging +of brain anatomy and in vivo histology,” Nat. Rev. Phys. 3, 570–588 (2021). +31. P.-J. Lapray, X. Wang, J.-B. Thomas, and P. Gouton, “Multispectral filter arrays: Recent advances and practical +implementation,” Sensors 14, 21626–21659 (2014). +32. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to +structural similarity,” IEEE transactions on image processing 13, 600–612 (2004). +33. J. Ma, X.-Y. Liu, Z. Shou, and X. Yuan, “Deep tensor admm-net for snapshot compressive imaging,” in Proceedings +of the IEEE/CVF International Conference on Computer Vision, (2019), pp. 10223–10232. +34. K. Zhang, W. Zuo, and L. Zhang, “Ffdnet: Toward a fast and flexible solution for cnn-based image denoising,” IEEE +Trans. on Image Process. 27, 4608–4622 (2018). +35. J. Zbontar, F. Knoll, A. Sriram, T. Murrell, Z. Huang, M. J. Muckley, A. Defazio, R. Stern, P. Johnson, M. Bruno +et al., “fastmri: An open dataset and benchmarks for accelerated mri,” arXiv preprint arXiv:1811.08839 (2018). +36. F. Knoll, J. Zbontar, A. Sriram, M. J. Muckley, M. Bruno, A. Defazio, M. Parente, K. J. Geras, J. Katsnelson, +H. Chandarana et al., “fastmri: A publicly available raw k-space and dicom dataset of knee images for accelerated mr +image reconstruction using machine learning,” Radiol. Artif. intelligence 2 (2020). +37. M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing mri,” IEEE signal processing magazine +25, 72–82 (2008). +38. K. Feng, Y. Zhao, J. C.-W. Chan, S. G. Kong, X. Zhang, and B. Wang, “Mosaic convolution-attention network for +demosaicing multispectral filter array images,” IEEE Trans. on Comput. Imaging 7, 864–878 (2021). +39. G. Sharma and R. Bala, Digital color imaging handbook (CRC press, 2017). +40. X. Cao, T. Yue, X. Lin, S. Lin, X. Yuan, Q. Dai, L. Carin, and D. J. Brady, “Computational snapshot multispectral +cameras: Toward dynamic capture of the spectral world,” IEEE Signal Process. Mag. 33, 95–108 (2016). +41. L. Huang, R. Luo, X. Liu, and X. Hao, “Spectral imaging with deep learning,” Light. Sci. & Appl. 11, 1–19 (2022). +42. L. Miao, H. Qi, R. Ramanath, and W. E. Snyder, “Binary tree-based generic demosaicking algorithm for multispectral +filter arrays,” IEEE Trans. on Image Process. 15, 3550–3558 (2006). +43. Y. Monno, D. Kiku, S. Kikuchi, M. Tanaka, and M. Okutomi, “Multispectral demosaicking with novel guide image +generation and residual interpolation,” in 2014 IEEE International Conference on Image Processing (ICIP), (IEEE, +2014), pp. 645–649. +44. L. Bian, Y. Wang, and J. Zhang, “Generalized msfa engineering with structural and adaptive nonlocal demosaicing,” +IEEE Trans. on Image Process. 30, 7867–7877 (2021). + +45. F. Yasuma, T. Mitsunaga, D. Iso, and S. K. Nayar, “Generalized assorted pixel camera: postcapture control of +resolution, dynamic range, and spectrum,” IEEE transactions on image processing 19, 2241–2253 (2010). +46. X. Chang, L. Bian, Y. Gao, L. Cao, J. Suo, and J. Zhang, “Plug-and-play pixel super-resolution phase retrieval for +digital holography,” Opt. Lett. 47, 2658–2661 (2022). +47. O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in +International Conference on Medical image computing and computer-assisted intervention, (Springer, 2015), pp. +234–241. +48. G. Qian, J. Gu, J. S. Ren, C. Dong, F. Zhao, and J. Lin, “Trinity of pixel enhancement: a joint solution for +demosaicking, denoising and super-resolution,” arXiv preprint arXiv:1905.02538 1, 4 (2019). +49. A. Rahman, J. Lee, and K. Choi, “Efficient fpga acceleration of convolutional neural networks using logical-3d +compute array,” in 2016 Design, Automation & Test in Europe Conference & Exhibition (DATE), (IEEE, 2016), pp. +1393–1398. +50. C. Zhang, G. Sun, Z. Fang, P. Zhou, P. Pan, and J. Cong, “Caffeine: Toward uniformed representation and acceleration +for deep convolutional neural networks,” IEEE Trans. on Comput. Des. Integr. Circuits Syst. 38, 2072–2085 (2018). +51. G. Hegde and N. Kapre, “Caffepresso: Accelerating convolutional networks on embedded socs,” ACM Trans. on +Embed. Comput. Syst. (TECS) 17, 1–26 (2017). +52. P. Meloni, A. Capotondi, G. Deriu, M. Brian, F. Conti, D. Rossi, L. Raffo, and L. Benini, “Neuraghe: Exploiting +cpu-fpga synergies for efficient and flexible cnn inference acceleration on zynq socs,” ACM Trans. on Reconfigurable +Technol. Syst. (TRETS) 11, 1–24 (2018). + diff --git a/zNE1T4oBgHgl3EQfRAN4/content/tmp_files/load_file.txt b/zNE1T4oBgHgl3EQfRAN4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b30a82e0ed1084e991dfff80005e7cf75e255791 --- /dev/null +++ b/zNE1T4oBgHgl3EQfRAN4/content/tmp_files/load_file.txt @@ -0,0 +1,1599 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf,len=1598 +page_content='Large-scale Global Low-rank Optimization for Computational Compressed Imaging DAOYU LI,1,2 HANWEN XU,1,2 MIAO CAO,3 XIN YUAN,3 DAVID J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' BRADY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 AND LIHENG BIAN1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='* 1School of Information and Electronics & Advanced Research Institute of Multidisciplinary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Beijing Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 100081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' China 2MIIT Key Laboratory of Complex-field Intelligent Sensing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Beijing Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Beijing 100081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' China 3Research Center for Industries of the Future and School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Westlake University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' China 4Wyant College of Optical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' AZ 85721,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Tucson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' USA bian@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='cn Abstract: Computational reconstruction plays a vital role in computer vision and computational photography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Most of the conventional optimization and deep learning techniques explore local information for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Recently, nonlocal low-rank (NLR) reconstruction has achieved remarkable success in improving accuracy and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, the computational cost has inhibited NLR from seeking global structural similarity, which consequentially keeps it trapped in the tradeoff between accuracy and efficiency and prevents it from high-dimensional large-scale tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' To address this challenge, we report here the global low-rank (GLR) optimization technique, realizing highly-efficient large-scale reconstruction with global self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Inspired by the self-attention mechanism in deep learning, GLR extracts exemplar image patches by feature detection instead of conventional uniform selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' This directly produces key patches using structural features to avoid burdensome computational redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Further, it performs patch matching across the entire image via neural-based convolution, which produces the global similarity heat map in parallel, rather than conventional sequential block-wise matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As such, GLR improves patch grouping efficiency by more than one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We experimentally demonstrate GLR’s effectiveness on temporal, frequency, and spectral dimensions, including different computational imaging modalities of compressive temporal imaging, magnetic resonance imaging, and multispectral filter array demosaicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' This work presents the superiority of inherent fusion of deep learning strategies and iterative optimization, and breaks the persistent dilemma of the tradeoff between accuracy and efficiency for various large-scale reconstruction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' © 2023 Optica Publishing Group 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Introduction By jointly optimizing computation and optics, computational imaging [1] improves the efficiency and information capacity of optical systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' With computational imaging, one can achieve detection under extremely complex lighting conditions [3,4], capture invisible high-dimensional information [4–6], look through obstacles [7,8], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As a critical tool for computational imaging, computational reconstruction recovers high-dimensional large-scale data from compressed or aliased low-dimensional signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A typical reconstruction technique first formulates a physical model to describe how to obtain the measurement with a specific target scene based on geometrical or fluctuating optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' It derives target information of interest by solving the inverse problem of the forward imaging model [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Despite a few cases that can be solved directly by inverse transformation, prior constraints are vital to regulate the reconstruction and eliminate noise and disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='03047v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='IV] 8 Jan 2023 To date, there exist two kinds of most prevalent priors, including model-based and statistics- based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Conventional reconstruction techniques with model-based prior such as total variation (TV) [10] and Tikhonov regularization [11], mainly focus on local information of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Motivated by rich self-repetitive structures in natural images, exploiting the nonlocal self-similarity (NSS) model has led to the well-known nonlocal means (NLM) [12] and block matching and 3D filtering (BM3D) [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The bloom of compressive sensing theories inspired the low-rank regularization [15–17] of self-repetitive patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Nonlocal low-rank (NLR) methods have shown state-of-the-art accuracy and strong generalization for various computational photography tasks [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, further extending the conventional nonlocal techniques to global perception results in unacceptable computational complexity especially in high-dimensional large-scale cases [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' With the bloom of deep learning, the most commonly used statistics-based approaches for computational reconstruction are learning-based methods [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Training a neural network on a large-scale dataset has been proven to be an effective way to learn the inverse imaging model [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, there remains a challenge for learning-based approaches to construct a generalized network robust to different system settings and noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Besides, concerns have been raised regarding the use of black-box deep learning for high-stakes tasks in healthcare, criminal justice, etc [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bearing these concerns in mind, we present the global low-rank (GLR) optimization for computational reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GLR regulates the optimization with global self-similarity prior, which incorporates the structural information of the entire image into reconstruction for each local image patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GLR realizes efficient reconstruction with high accuracy and robustness, by attention from structural feature detection and similar patch matching with neural computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' These two operations promote running efficiency by up to above one magnitude of order for high-dimensional large-scale computational reconstruction with improved accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GLR first conducts structural feature attention by extracting exemplar edge patches according to corners of the edge maps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Conventional NLR first uniformly picks exemplar patches as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mean square error (MSE) is utilized to measure the similarity between the exemplar patch and other patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Corner-based exemplar patch selection is inspired by common sense that natural images have sparse edges and corners which locate in the area of significant textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Corner-based exemplar patches contribute to focus on the recovery of sharp edges which are tougher than that of smooth areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We experimentally demonstrate that nonlocal techniques with fewer corner-based exemplar patches can achieve competitive performance and reduced running time against more evenly arranged patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Next, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1 b, GLR groups the patches across the entire image following structural similarity order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The block matching (BM) performs a k-nearest-neighbor (KNN) search for each exemplar patch within a local window [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' [18,19] have experimentally proved that the nonlocal low-rank reconstruction based on vanilla BM for snapshot compressed imaging requires more than one hour to restore a video clip at 256×256×8 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Benefiting from the bloom of deep learning framework, we can realize parallel patch matching with the ‘Conv2d’ layer (a neural layer performing batch convolution for 2-dimensional input), enabling fast and accurate computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GLR improves the matching speed by an order of magnitude, with advanced accuracy to vanilla BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The simulation and experiment results demonstrate GLR exhibits boosted efficiency, high precision, and strong generalization for reconstruction, facilitating diverse computational pho- tography applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' This work presents the superiority of inherent fusion of deep learning strategies and iterative optimization, and breaks the persistent dilemma of the trade-off between accuracy and efficiency for various large-scale reconstruction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Block matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Corner-based block matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Global matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Comparison of various exemplar patch selection strategies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Center of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='exemplar patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Similarity heat 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='\uf0d1 x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='\uf0d1 p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Illustration of global matching (GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, The comparison of various exemplar patch selection strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Vanilla block matching (BM) uniformly picks exemplar patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Corner-based BM selects exemplar patches based on the corner points of the edge map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GM obtains the similarity heat maps by correlating the edge map of the image with exemplar edge patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The search range is extended to the whole image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, Both horizontal and vertical gradient edge maps (∇ℎ𝒙 and ∇𝑣𝒙) and exemplar edge patches (∇ℎ 𝒑 and ∇𝑣 𝒑) are extracted by the Sobel operators (∇ℎ and ∇𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The similarity heat map for the exemplar patch is the summation heat maps corresponding to horizontal and vertical edge maps which are obtained by Conv2d operation �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' c, The exemplar edge patch slides over the edge map, performing an elementwise multiplication and obtaining the numbers of points of overlapped edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' d, The similarity heatmap of the exemplar patch in a extracted by GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 0Y00000口 十工2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Method Recall that the conventional nonlocal technique performs a KNN search for each exemplar patch within a local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' It first uniformly picks exemplar patches as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The exemplar patch is the center patch of every local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mean square error (MSE) is utilized to measure the similarity between the exemplar patch and other patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, it has relatively low efficiency when using low-value patches which do not contain textures and details as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' For further improvement in the efficiency of matching, we introduce the exemplar patch selection based on corner points of the image’s edge map (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Corner-based exemplar patch selection is inspired by common sense that natural images have sparse edges and corners which locate in the area of significant textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Corner-based exemplar patches enable structural feature attention by focusing on the recovery of sharp edges which are tougher than that of smooth areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We experimentally demonstrate that nonlocal techniques with fewer corner-based exemplar patches can achieve competitive performance and reduced running time against more evenly arranged patches in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The second limitation of nonlocal algorithms lies in the local receptive field of vanilla BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Denote the 𝑖-th exemplar patch as 𝑥𝑖, and 𝑥 𝑗 stands for the neighbor patches within a local window of 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 𝐺𝑖 = � 𝑖 𝑗|∥𝑥𝑖 − 𝑥 𝑗 ∥2 2 < 𝑇𝑖 � , (1) where 𝑇𝑖 is a pre-defined threshold, and 𝐺𝑖 is the collection of positions corresponding to those similar patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Due to BM’s high computational complexity (𝑂(𝑛2)), common practice sets the size of the local window to a relatively small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' To address this issue, GM efficiently search similar patches across the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Given the to-be-updated image 𝒙, we first extract the binary edge maps of the image (∇ℎ𝒙 and ∇𝑣𝒙) by the Sobel operators (∇ℎ and ∇𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' ∇+ ℎ𝒙 = 𝑇 (𝑎𝑏𝑠 (∇ℎ𝒙)) , ∇− ℎ𝒙 = 𝑇 (𝑎𝑏𝑠 (−∇ℎ𝒙)) , ∇+ 𝑣𝒙 = 𝑇 (𝑎𝑏𝑠 (∇𝑣𝒙)) , ∇− 𝑣𝒙 = 𝑇 (𝑎𝑏𝑠 (−∇𝑣𝒙)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' (2) 𝑇 is the threshold function 𝑇 (𝑧) = � 1, 𝑧 > 𝑡ℎ, 0, 𝑧 ≤ 𝑡ℎ, (3) where 𝑡ℎ is the pre-defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GM picks exemplar patches centered on the corner points of the averaged edge map rather than traditional uniform selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' After exemplar patches selection, we perform global patch matching using the convolution operation of deep learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2 a, each exemplar edge patch (∇ℎ 𝒑 and ∇𝑣 𝒑) acts as a convolution kernel to convolute the edge map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Note that the convolution operator has slightly different from the convolution in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' It specifically refers to the ‘Conv2d’ in deep convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Benefiting from the bloom of deep learning framework, we can perform parallel convolution by concatenating the exemplar edge patches ∇ 𝒑 into a multi-channel convolutional kernel ∇𝑷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Figure 2 b demonstrates that GM enables increasing matching precision and finding more structurally similar patches than BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' In the case of multi-channel images, both the edge maps and the exemplar edge patches are of three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We can conduct the batch convolution with the multi-channel edge maps as batch inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The similarity heat map is obtained as 𝑯 = ∑︁ 𝑎=(ℎ,𝑣),𝑏=(+,−) Conv2d � ∇𝑏 𝑎𝒙, ∇𝑏 𝑎𝑷 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' (4) The sizes of ∇𝑏 𝑎𝒙, ∇𝑏 𝑎𝑷, and 𝐻 are H×W×C, P×P×C×N, and (H-P+1)×(W-P+1)×N respectively, where H×W is the size of input images, P×P is the size of extracted patches, C represents the Block matching Corner-based block matching Global matching a Top 30 most similar patches to the exemplar patch via block matching b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='#30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Top 30 most similar patches to the exemplar patch via global matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Extracted similar patches using block and global matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Comparison of various exemplar patch selection strategies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Center of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='exemplar patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Center of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='exemplar patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Global ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1×1 =1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0×0=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='\uf0d1 x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='\uf0d1 p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Comparison between BM and GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, BM searches similar patches by measuring the MSE of each patch against the exemplar patch within a local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GM obtains the similarity heat maps by correlating the edge maps with exemplar edge patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The similarity heat map for the exemplar patch is the summation heat maps corresponding to horizontal and vertical edge maps which are obtained by neural-based Conv2d operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, Comparison between extracted similar patches from BM and the reported GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The patches extracted by GM have higher structural similarity to the exemplar patch in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' input channels, and N denotes the number of exemplar patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GM obtains all the heat maps of N exemplar patches with 4 Conv2d operations while BM requires N times KNN-based matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' So the neural-based convolution contributes to reduce computation load and boost the running efficiency of GLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Besides, modern deep learning frameworks provide several acceleration approaches to speed up the Conv2d layer including Im2Col + matrix manipulation [25], fast Fourier transform (FFT), Winograd [26], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A larger value in the heat map means that the image patch centered at the corresponding position has a higher structural similarity to the exemplar patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We use the operator � 𝑹𝑖 to represent the 𝑖-th global matching operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The resulted grouped T1 0口 十工matrix is � 𝑹𝑖𝒙, where 𝒙 denotes the to-be-update image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The objective function of low-rank approximation is 𝒙 = arg min 𝒙 ∑︁ 𝑖 rank � � 𝑹𝑖𝒙 � , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 𝒚 = 𝚽𝒙, (5) where 𝒚 is the measurement, and 𝚽 stands for the sensing matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We adopt WNNM [15] for solving the above low-rank approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The reconstructed image is derived by the iterative optimization framework such as generalized alternative projection (GAP) [27] and alternating direction multiplier (ADMM) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Results We applied GLR and the existing CS-based algorithms on both simulation and experiment data of three computational imaging tasks including coded aperture compressive temporal imaging (CACTI) [29], magnetic resonance imaging (MRI) [30], and multispectral filter array (MSFA) demosaicing [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' In the following evaluations, we employed the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) [32] to quantify reconstruction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' In the evaluations of the Results and the following Discussion sections, GLR and NLR shared the same hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The deep learning platform MatConvNet [25] is utilized for convolution-based GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' All the calculations are done on a desktop PC with an Intel i7-9700K CPU and 64G RAM for a fair comparison between GLR and conventional nonlocal low-rank techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We believe that GLR can be faster on convolution-accelerating hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Coded aperture compressive temporal imaging CACTI [29] is a typical temporal compressive imaging modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2 a, CACTI compresses temporally continuous 2D scenes into one snapshot via aperture coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mathematically, it can be modeled as 𝒚 = 𝑁 ∑︁ 𝑖=1 𝑨𝑖𝒙𝑖 (6) where 𝒙𝒊 is the scene at moment 𝒊, 𝑨 denotes the spatial variant mask generated by the digital micro-mirror device (DMD), and 𝒚 represents the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We test the corner-based exemplar patch selection in the state-of-the-art NLR-based algorithm DeSCI [20] for CACTI reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Denote DeSCI with corner-based BM as DeSCI-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Accept from the corner-based exemplar patches, a modest number of uniformly selected patches are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The interval of two neighbor exemplar patches is 3 times larger than vanilla BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The common-used benchmark datasets (Kobe, Runner, Traffic, Drop) [18,33] are applied for simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' One snapshot of the simulation data encodes 8 frames of the scenes at 256 × 256 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The commonly used CS-based algorithms GAP-TV [27] and Plug-and-play (PnP) [18] are included as the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' FFDNet [34] is employed for the learning-based regularization of PnP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' All the hyper-parameters of above-mentioned algorithms are set as the default values provided by ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sub-figures 3 b & c show the visual and quantitative results of the simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' It can be seen that DeSCI with corner-based BM achieved competitive results compared to conventional DeSCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The differences between the average PSNR and SSIM of DeSCI and DeSCI-C are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='27 dB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='002, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' This small numerical difference is hard to be noticed in visual perception as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Furthermore, we build a proof-of-concept prototype shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The prototype consists of one camera lens, two relay lenses, one DMD to generate the spatial variant masks, and one CCD sensor to capture the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' With the real-captured measurement encoded with 10 masks, we retrieved 10 temporal successive frames of the target scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3 f that DeSCI and DeSCI-C can recover a Coded aperture compressive temporal imaging Camera lens Relay lens 1 DMD Relay lens 2 Sensor Kobe #2 GAP-TV DeSCI DeSCI-C Runner #5 Traffic #3 Drop #7 GAP-TV PnP DeSCI DeSCI-C PSNR 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='37 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='67 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='28 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='04 SSIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='8985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9739 PSNR 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='96 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='21 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='11 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='82 SSIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='7351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='8389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9261 PSNR 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='47 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='77 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='53 SSIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9685 PSNR 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='43 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='83 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='16 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='85 SSIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9906 Drop Image Kobe Traffic Runner b c d Objective lens DMD Relay lens CCD Measurement e f Reconstructed images from simulation data Quantitative results on simulation data Experiment setup Reconstructed images from experiment data #1 #4 #7 #10 DeSCI DeSCI-C PnP PnP Measurement 2728 3599 6327 846 2264 3111 0 2000 4000 6000 DeSCI DeSCI-C Matching Low-rank Total Time (min) 77 53 133 10 23 35 Average running time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Reconstruction results for CACTI simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, The imaging scheme of CACTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, Reconstruction results on benchmark simulation datasets (Kobe, Runner, Traffic, Drop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' c, The average reconstruction accuracy on the test data of different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' d, The average running time of nonlocal algorithms with and without corner-based exemplar patch selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' e, The proof-of-concept prototype of the CACTI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' f, The reconstruction results of the real-captured measurement via different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' more sharp textures compared to the prevalent PnP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' DeSCI-C obtained similar results as DeSCI, which proved the effectiveness of corner-based BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, DeSCI-C has higher reconstruction efficiency due to fewer exemplar patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The matching speed of corner-based BM is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='7 times faster compared to vanilla BM, and the total reconstruction speed of DeSCI-C is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='8 times faster than that of DeSCI, according to the average running times listed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=" All the above simulations and experiments demonstrate the efficiency of structural feature attention 国30 8130 81 '0030 81 00QDQD网from corner-based exemplar patch selection." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Magnetic resonance imaging The Fourier spectrum of natural images has the nature of sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' One can reconstruct target scenes with magnetic resonance data even below the Nyquist sampling ratio via CS algorithms [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4 a, the Fourier domain compressive imaging model is formulated as 𝒚 = 𝑴ℱ(𝒙) (7) where 𝒙 denotes the target scene, ℱ is the Fourier transform function, 𝑴 represents the sampling mask, and 𝒚 is the corresponding measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We evaluated the reported reconstruction technique on magnetic resonance imaging (MRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' In the experiments, the sampling strategy is set as the prevalent radial mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We applied Dong et al.’s NLR algorithm [16] for MRI reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GLR is formed by replacing BM with GM in NLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Figure 4 shows the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GLR holds a similar performance to NLR in a Reconstructed images from MRI data c Quantitative reconstruction results d Average running time Time (s) Matching Low-rank Total b Fourier spectrum MRI principle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 28 32 36 40 44 Ratio GAP-TV DCT NLR GLR PSNR NLR GLR Ratio = 13% Ratio = 20% Ratio = 29% Ground Truth DCT GLR NLR Close-up Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' MRI reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, The mathematical principle of MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We focus on the radial sampling strategy in the Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, The visual comparison of reconstructed images (256×256 pixels) from Dong et al.’s [16] dataset and fastMRI [35,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' c, The quantitative results of different reconstruction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' d, Average running time for each procedure of NLR and GLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' reconstruction accuracy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4 b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' For single-channel reconstructions, GM exhibits about 7 times acceleration against BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The low-rank approximation of GLR shows a 3 times speed-up compared to the conventional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GLR has an average 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 times improvement in running time compared to NLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Specifically, the speed-up of low-rank approximation mainly comes from the reduction of exemplar patches by corner-based exemplar patch selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The acceleration of the matching process results from the joint promotion of both corner-based exemplar patch selection and convolution-based structural matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Multi-spectral filter array demosaicing Multi-spectral filter array (MSFA) is an extension of the conventional color camera that covers a Bayer filter [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' MSFA cameras can acquire multi-spectral information in one snapshot [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Compared to other existing multi-spectral imaging systems [41], it provides an integrated, low-cost, exact registration, and full frame rate solution [42,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The imaging model is denoted as 𝒚 = 𝑁 ∑︁ 𝑖=1 𝑨𝑖𝒙𝑖 (8) where 𝑨𝑖 stands for the filter mask of the 𝑖-th spectral channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The masks are designed to be orthogonal, 𝑨𝑖⊙𝑨 𝑗 = 0 (9) where ⊙ denotes the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, there exhibits a fundamental compromise between spatial and spectral resolution for MSFA camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' There is a rising challenge to reconstruct full-resolution images from multi-spectral mosaic data because it is more ill-posed than demosaicing for RGB cameras (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 5 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' [44] experimentally proved that NLR algorithms with multi-channel block matching can achieve state-of-the-art performance for multi-spectral demosaicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Inheriting Bian et al.’s framework, GLR replaces the matching strategy with multi-channel global matching (see Supplement 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Here we conduct a series of simulations and experiments to demonstrate the superiority of GLR over conventional NLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As shown in both simulations and experiments of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 5 b, c, and e, the visual and numerical results indicate GLR outperforms NLR with various channel numbers and types of MSFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The recovered closeups in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 5 b show that global matching contributes to the recovery of more delicate textures and details than conventional block matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' In multi-channel cases, GM presents obvious efficiency superiority over vanilla BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' GM provides more speed boost as the channel number increases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 5 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' It is mainly attributed to the high efficiency of batch convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' For large-scale reconstruction with more than 256×256 spatial resolution and 6 channels, GM achieves up to above one magnitude of improvement in running time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 5 d, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Besides, with corner-based exemplar patch selection, GM extracts fewer groups of self-repetitive patches than BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Corner-based exemplar patch selection enables decreasing the running time of both matching and low-rank approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Combining the above two improvements, GLR achieves about 4 to 8 times efficiency gain while obtaining state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Taking the single-channel cases of MRI into consideration, GLR exhibits an increasing priority over NLR in terms of both running time and reconstruction accuracy as the channel number goes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Experiments of real-captured data validated GLR’s advancement in both accuracy and efficiency (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Conclusion and discussion In this work, we engaged in a generalized global low-rank optimization for computational reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' It enables fast calculation and global self-similarity regularization, offering an impetus to efficient nonlocal techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As validated by extensive simulations and experiments on the above three computational imaging modalities (CACTI, MRI, and MSFA demosaicing), the Multi-spectral filter array demosaicing b c Random MSFA Regular MSFA BT MSFA GAP-TV DeSCI NLR GLR 400nm 400nm 400nm 400nm 400nm d 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 a Demosaicing Quantitative results of reconstruction Visual comparison of reconstruction Average running time of different channel numbers at 256×256 resolution MSFA Method 4 channels 6 channels 9 channels BTES 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='45 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='66 GAP-TV 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='81 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='60 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='89 DeSCI 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='91 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='86 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='56 NLR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='30 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='53 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='53 GLR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='18 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='98 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='27 BTES 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='35 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='53 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='72 GAP-TV 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='36 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='07 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='97 DeSCI 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='98 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='33 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='19 NLR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='90 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='66 GLR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='35 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='39 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='44 BTES 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='47 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='07 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='56 GAP-TV 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='36 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='50 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='81 DeSCI 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='98 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='37 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='61 NLR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='28 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='41 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='85 GLR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='42 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='95 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Random ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Regular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='BT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='(min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Low rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Average running time of 9-channel BT MFSA demosaicing at different resolutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='channels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='channels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Reconstruction results for real-captured data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='470nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='590nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='470nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='590nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Measurement #2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Measurement #1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Average running time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Low-rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='203 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The demosaicing simulations under different MSFAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, The demosaicing process for a multispectral mosaic camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, The PSNR values of reconstructed multi-channel images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' c, The reconstruction images for various 6-channel MSFAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The closeups and corresponding error maps are shown on the right side of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' d, Average running time (200 iterations in total) of different channels at 256 × 256 resolution for each procedure of NLR and GLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' e, Average running time of 9-channel BT MSFA demosaicing at different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 工151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 400 300 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 200 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 100 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 512 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 solution59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 60 50 Ot 30 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 20 70 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 512 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 AOO 300 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 200 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 512 NLR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='7 256 GLR iiMulti-spectral filter array demosaicing b c Random MSFA Regular MSFA BT MSFA GAP-TV DeSCI NLR GLR 400nm 400nm 400nm 400nm 400nm d 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 a Demosaicing Quantitative results of reconstruction Visual comparison of reconstruction Average running time of different channel numbers at 256×256 resolution MSFA Method 4 channels 6 channels 9 channels BTES 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='45 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='66 GAP-TV 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='81 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='60 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='89 DeSCI 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='91 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='86 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='56 NLR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='30 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='53 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='53 GLR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='18 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='98 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='27 BTES 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='35 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='53 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='72 GAP-TV 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='36 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='07 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='97 DeSCI 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='98 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='33 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='19 NLR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='90 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='66 GLR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='35 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='39 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='44 BTES 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='47 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='07 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='56 GAP-TV 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='36 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='50 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='81 DeSCI 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='98 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='37 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='61 NLR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='28 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='41 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='85 GLR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='42 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='95 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Random ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='channels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='channels ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='channels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='channels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='channels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Reconstruction results for real-captured data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='550nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Meas #2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Average running time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Matching Low-rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='203 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='NLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='GLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Meas #1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The demosaicing results for real-captured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The measurement captured with a 4×4 IMEC snapshot camera is provided by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The measurements are of 256×256×16 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, Visual comparisons between reconstruction results of NLR and GLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, Average running time of both algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' BM-C BM-C-U BM GM 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Ground truth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Error maps of reconstructed images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Reconstructed spectra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='Comparison of running time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='b ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Comparison of various matching strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, The spectra of the randomly selected point (marked with a hexagon in the ground truth image of b) in the reconstructed multispectral scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, Error maps of reconstructed images with different matching strategies "BM-C" and "BM-C-U" stand for block matching with corner-based exemplar patches and with both corner-based exemplar patches and the same number of uniform ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' c, Comparison of running time of nonlocal techniques with different matching strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' core contributions of GLR (structural feature attention and neural-based matching) can promote 工520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 AOO 300 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 200 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 512 NLR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='7 256 GLR ii151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 90.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 512 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4工COJOL2the running efficiency of nonlocal techniques by nearly an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Benefiting from inherent fusion of deep learning strategies and iterative optimization, GLR exhibits unique advantages in efficient and high-fidelity reconstruction for high-dimensional large-scale computational imaging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' From block matching to global matching To reveal the core advantages of GLR, we evaluated GM, uniform, corner-based, and corner- based+uniform BM for 6-channel BT MSFA demosaicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' For the corner-based+uniform BM, the number of uniformly picked exemplar patches is set to be similar to that of corner-based patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 7, GLR/GM outperforms other approaches in both reconstruction accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The reconstructed images with GM exhibit more sharp textures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 7 b) and high-fidelity spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 7 a) especially in the regions of rich textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' NLR with corner-based block matching has less running time than with uniform block matching, due to fewer exemplar patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, too few self-repetitive patches for block matching leads to reduced accuracy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 7 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' With similar numbers of corner-based exemplar patches, the comparisons between GM and corner-based BM indicate that GM outperforms BM, demonstrating the significance of global structural self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Combining deep learning and optimization By directly incorporating the neural network as a regularization operator into optimization, PnP [18,46] alternatives iterations between the model-based fidelity and learning-based regularization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 8 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The optimization framework improves the interpretability of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' PnP techniques have achieved tremendous success in computational reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' However, the enhancing process of the network in PnP iterations remains unexplainable due to the black-box nature of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Different from PnP, GLR presents the innovative neural strategies that combine feature detection and neural computation for explainable global low-rank optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The use of the ‘Conv2d’ neural operator here has clear physical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The ‘Conv2d’ between edge maps and exemplar edge patches measures the similarity between these inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Besides, the commonly used enhancing networks (such as FFDNet [34]) for PnP have limited effective receptive field, while GLR holds global perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 8 b, GLR outperforms PnP, end-to-end networks (U-Net [47] and TENet [48]), and conventional NLR in reconstruction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Above all, GLR provides a generalized and effective perspective of explainable applications of deep learning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Outlook GLR can be extended to various challenging reconstruction tasks, especially in high-dimensional and high-precision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' To further improve the calculating speed, we can run the technique on convolution-accelerate hardware such as FPGA [49,50] and specific SOC [51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Next, the low-complexity low-rank approximation deserves further study under the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The low-rank network may be a promising alternative for further promoting efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' What’s more, the collision between deep learning and classical optimization will lead to a spark of inspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' The bloom of learning-based methods can drive the progress of optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Altmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' McLaughlin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Padgett, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Goyal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Hero, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Faccio, “Quantum-inspired computational imaging,” Science 361, eaat2298 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Deng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Qiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Xie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Fang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dai, “An integrated imaging sensor for aberration-corrected 3d photography,” Nature 612, 62–71 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Kirmani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Venkatraman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Shin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Colaço, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Shapiro, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Goyal, “First-photon imaging,” Science 343, 58–61 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Model-based fidelity a b Ground truth U-Net TENet NLR GLR PnP Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='4 Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='6 Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='9 Channels Method PSNR SSIM UNet 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='968 TENet 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='984 PnP 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='989 NLR 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='995 GLR 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='996 UNet 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='848 TENet 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='983 PnP 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='989 NLR 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='996 GLR 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='997 UNet 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='922 TENet 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='933 PnP 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='946 NLR 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='961 GLR 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='966 9 4 6 Combing deep learning and optimization: from direct neural network operators to innovative neural strategies Error maps of reconstruction images 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='1 c Quantitative results Neural network: Direct operator in alternating iterations Neural strategies: Feature attention & Neural computation Model-based fidelity Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Comparison between end-to-end networks, PnP, and GLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' a, The diagrams of PnP (direct neural network operator) and GLR (neural strategies) schemes that combine deep learning and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' b, The error maps of BT MSFA demosaicing images from Monno’s dataset [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' c, The quantitative comparisons of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' We compare the reported GLR with end-to-end networks (U-Net and TENet trained on CAVE [45] and Monno’s dataset), PnP (FFDNet as the regularization network), and conventional NLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Morris, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Aspden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Boyd, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Padgett, “Imaging with a small number of photons,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' communications 6, 1–6 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Horstmeyer, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yang, “Wide-field, high-resolution fourier ptychographic microscopy,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' photonics 7, 739–745 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Qiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Jiang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dai, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Li, “Evaluation and development of deep neural networks for image super-resolution in optical microscopy,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Methods 18, 194–202 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' O’Toole, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lindell, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555, 338–341 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Saunders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Murray-Bruce, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Goyal, “Computational periscopy with an ordinary digital camera,” Nature 565, 472–475 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mait, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Euliss, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Athale, “Computational imaging,” Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Photonics 10, 409–483 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Rudin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Osher, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' D: nonlinear phenomena 60, 259–268 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Tikhonov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Leonov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yagola, “Nonlinear ill-posed problems,” (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Buades, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Coll, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Morel, “A non-local algorithm for image denoising,” in 2005 IEEE computer society conference on computer vision and pattern recognition (CVPR’05), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2 (Ieee, 2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 60–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dabov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Foi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Katkovnik, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Egiazarian, “Image denoising with block-matching and 3d filtering,” in Image D6COG0processing: algorithms and systems, neural networks, and machine learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 6064 (SPIE, 2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 354–365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dabov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Foi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Katkovnik, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on image processing 16, 2080–2095 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Gu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zuo, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Feng, “Weighted nuclear norm minimization with application to image denoising,” in Proceedings of the IEEE conference on computer vision and pattern recognition, (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2862–2869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Ma, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Huang, “Compressive sensing via nonlocal low-rank regularization,” IEEE transactions on image processing 23, 3618–3632 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Hu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Jin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mei, “Nonlocal image denoising via adaptive tensor nuclear norm minimization,” Neural Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 29, 3–19 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Suo, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dai, “Plug-and-play algorithms for large-scale snapshot compressive imaging,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1447–1457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Brady, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Katsaggelos, “Snapshot compressive imaging: Theory, algorithms, and applications,” IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 38, 65–88 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Suo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Brady, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dai, “Rank minimization for snapshot compressive imaging,” IEEE transactions on pattern analysis machine intelligence 41, 2990–3006 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Barbastathis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Ozcan, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Situ, “On the use of deep learning for computational imaging,” Optica 6, 921–943 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Belthangady and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Royer, “Applications, promises, and pitfalls of deep learning for fluorescence image reconstruction,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' methods 16, 1215–1225 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Ye, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' De Man, “Deep learning for tomographic image reconstruction,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2, 737–748 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Rudin, “Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1, 206–215 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Vedaldi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lenc, “Matconvnet: Convolutional neural networks for matlab,” in Proceedings of the 23rd ACM international conference on Multimedia, (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 689–692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lavin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Gray, “Fast algorithms for convolutional neural networks,” in Proceedings of the IEEE conference on computer vision and pattern recognition, (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 4013–4021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan, “Generalized alternating projection based total variation minimization for compressive sensing,” in 2016 IEEE International Conference on Image Processing (ICIP), (IEEE, 2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 2539–2543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Boyd, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Parikh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Chu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Peleato, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Eckstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=', “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Trends Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' learning 3, 1–122 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Llull, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Liao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Kittle, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Carin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sapiro, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Brady, “Coded aperture compressive temporal imaging,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' express 21, 10526–10545 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Weiskopf, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Edwards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Helms, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mohammadi, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Kirilina, “Quantitative magnetic resonance imaging of brain anatomy and in vivo histology,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 3, 570–588 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lapray, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Thomas, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Gouton, “Multispectral filter arrays: Recent advances and practical implementation,” Sensors 14, 21626–21659 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bovik, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sheikh, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE transactions on image processing 13, 600–612 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Shou, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan, “Deep tensor admm-net for snapshot compressive imaging,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 10223–10232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zuo, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, “Ffdnet: Toward a fast and flexible solution for cnn-based image denoising,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 27, 4608–4622 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zbontar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Knoll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sriram, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Murrell, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Muckley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Defazio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Stern, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Johnson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bruno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=', “fastmri: An open dataset and benchmarks for accelerated mri,” arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='08839 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Knoll, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zbontar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sriram, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Muckley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bruno, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Defazio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Parente, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Geras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Katsnelson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Chandarana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=', “fastmri: A publicly available raw k-space and dicom dataset of knee images for accelerated mr image reconstruction using machine learning,” Radiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Artif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' intelligence 2 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lustig, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Donoho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Santos, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Pauly, “Compressed sensing mri,” IEEE signal processing magazine 25, 72–82 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Chan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Kong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wang, “Mosaic convolution-attention network for demosaicing multispectral filter array images,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Imaging 7, 864–878 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sharma and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bala, Digital color imaging handbook (CRC press, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Carin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Brady, “Computational snapshot multispectral cameras: Toward dynamic capture of the spectral world,” IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 33, 95–108 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Huang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Luo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Liu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Hao, “Spectral imaging with deep learning,” Light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' & Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 11, 1–19 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Miao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Qi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Ramanath, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Snyder, “Binary tree-based generic demosaicking algorithm for multispectral filter arrays,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 15, 3550–3558 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Monno, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Kiku, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Kikuchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Tanaka, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Okutomi, “Multispectral demosaicking with novel guide image generation and residual interpolation,” in 2014 IEEE International Conference on Image Processing (ICIP), (IEEE, 2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 645–649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, “Generalized msfa engineering with structural and adaptive nonlocal demosaicing,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 30, 7867–7877 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Yasuma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Mitsunaga, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Iso, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Nayar, “Generalized assorted pixel camera: postcapture control of resolution, dynamic range, and spectrum,” IEEE transactions on image processing 19, 2241–2253 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Bian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Gao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Suo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, “Plug-and-play pixel super-resolution phase retrieval for digital holography,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 47, 2658–2661 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Ronneberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Fischer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Brox, “U-net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical image computing and computer-assisted intervention, (Springer, 2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Qian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Gu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Ren, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Dong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lin, “Trinity of pixel enhancement: a joint solution for demosaicking, denoising and super-resolution,” arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content='02538 1, 4 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Rahman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Lee, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Choi, “Efficient fpga acceleration of convolutional neural networks using logical-3d compute array,” in 2016 Design, Automation & Test in Europe Conference & Exhibition (DATE), (IEEE, 2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 1393–1398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Fang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Zhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Pan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Cong, “Caffeine: Toward uniformed representation and acceleration for deep convolutional neural networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Circuits Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 38, 2072–2085 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Hegde and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Kapre, “Caffepresso: Accelerating convolutional networks on embedded socs,” ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on Embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' (TECS) 17, 1–26 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Meloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Capotondi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Deriu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Brian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Conti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Rossi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Raffo, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Benini, “Neuraghe: Exploiting cpu-fpga synergies for efficient and flexible cnn inference acceleration on zynq socs,” ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' on Reconfigurable Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'} +page_content=' (TRETS) 11, 1–24 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfRAN4/content/2301.03047v1.pdf'}